TRAINING MATERIALS
Partial Equilibrium Analysis of Policy
Impacts (part I)
Federico Perali
With the support of
Partial Equilibrium Analysis of Policy Impacts (part I)
Foreword
The present volume is part of the series “Training Materials”, published by the National
Agriculture Policy Center (NAPC) with the support of the FAO Project GCP/SYR/OO6/ITA. The
series includes notes and handouts produced as part of the training activities carried out at the
NAPC by the international experts recruited by the Project. Even though they cannot be
considered as comprehensive textbooks, the NAPC decided to make these materials available for
a wider public, considering them as a useful reference for the study and the practice of
agricultural economics and policy analysis.
The FAO Project, which is generously funded by the Italian Government and executed in close
coordination with the Syrian Ministry of Agriculture and Agrarian Reform (MAAR) has been
supporting the establishment of a cadre of professional agricultural policy analysts for the NAPC
and other institutions involved in the Syrian agricultural policy making process. This
undertaking encompassed an intensive training activity articulated over two programs
involving, in a five year period, a total of about 130 officials of the MAAR. Each training program
comprised a set of intensive courses to provide theoretical background and familiarize with
issues, concepts, methods and tools needed to carry out policy analyses. The set of courses was
completed by on-the-job research experiences on issues of relevance for Syrian agricultural
development, whose results have been published by the NAPC’s Working Papers series. The
formal training programs were also accompanied by seminars, shorter intensive courses and
participation in research activities, which are still on-going as part of NAPC’s staff capacity
building process.
Training was part of a wider undertaking in institutions’ building for agricultural policy analysis.
Indeed, the Project has been providing support to the institutional development of the NAPC, its
technical capacity to analyze, formulate and monitor agricultural policies, and its capacity to
maintain and develop a comprehensive set of statistical information for the economic analysis of
policies (the Syrian Agriculture Database).
The program of study on “Partial Equilibrium Analysis” has been delivered in two modules. In
module I, Prof Perali illustrated the analysis of supply and demand within a partial equilibrium
setting, while in module II Prof Conforti focuses on the theory of market equilibrium, with
reference to the analysis of agricultural policies, within the most common quantitative
frameworks.
This volume presents part of the training material of the module I of the program of study on
“Partial Equilibrium Analysis”. In it, Prof Perali provides the theoretical foundations of demand,
supply and market behavior, both at the individual and aggregate levels. This is a necessary step
to build the readers’ capacity to apply demand, supply and market analysis to real cases, within
the economic, social and institutional context of Syria. The analytical tools applied are those
used by analysts to interpret economic results,
The reader should note that exercises pertaining the issues presented in the volume are available
at NAPC in electronic format. Furthermore, at NAPC, are also available the slides used in class
during the lectures and the slides of the seminar Prof Perali delivered on Drug Consumption
and Intra-household Resource Allocation: the case of Djibouti.
Damascus, December 2003
i
Partial Equilibrium Analysis of Policy Impacts (part I)
Table of Contents
Chapter 1 - Introduction .............................................................................. 1
1.1.
1.2.
Motivation and Objectives of the Course .......................................................................... 1
An example of Applied Partial Equilibrium Analysis...................................................... 2
1.2.2.
1.2.3.
1.3.
The Model .......................................................................................................................................... 3
Results ................................................................................................................................................ 4
A Methodological Note ..................................................................................................... 6
1.3.1.
1.3.2.
1.4.
Data Collection.................................................................................................................................. 6
Familiarization with the Data........................................................................................................ 7
Description of the Aggregate Data Base (the Colombian Rice Economy) ...................... 8
Chapter 2 - Demand Analysis..................................................................... 19
2.1.
Introductory Demand Analysis .......................................................................................19
2.1.1.
2.1.2.
2.2.
Some Basic Notions ........................................................................................................................ 20
Introductory Applications............................................................................................................. 23
Advanced Demand Analysis ............................................................................................35
2.2.1.
2.2.2.
2.2.3.
2.2.4.
Duality Theory ................................................................................................................................ 36
Empirical Implementation: The Almost Ideal Demand System (AIDS)................................ 37
Cost of Living Indexes and Compensating Variations ............................................................. 39
A GAUSSX program for advanced demand analysis............................................................... 40
Chapter 3 - Supply Response .....................................................................45
3.1.
Introductory Supply Analysis......................................................................................... 45
3.1.1.
3.2.
Approaches to the estimation of supply response ..................................................................... 45
An Introductory Exercise ............................................................................................... 49
3.2.1.
3.2.2.
The Colombian Rice Economy: Supply side............................................................................... 49
Estimation of a Nerlovian Supply Response Model for groundnuts in Senegal .................. 53
Chapter 4 - The Market Model ................................................................... 61
4.1.
4.1.1.
4.2.
4.2.1.
4.3.
4.4.
Causality in Economic Analysis ......................................................................................61
Exogenous and Endogenous Variables in a Model ................................................................... 61
A Structural Market Model with Exogenous Variables ................................................. 62
Structural and Reduced Form Models ........................................................................................ 63
The Colombian Rice Economy: the market model ........................................................ 65
Welfare Analysis of Technical Change ........................................................................... 66
References ............................................................................................... 69
iii
Partial Equilibrium Analysis of Policy Impacts (part I)
Chapter 1 - Introduction
1.1.
Motivation and Objectives of the Course
The specific objectives of the module in “Partial Equilibrium Analysis” aim at (a) providing the
theoretical foundations of demand, supply and market behavior both at the individual and
aggregate levels, (b) building the capacity to apply demand, supply and market analysis to real
cases within the economic, social and institutional context of Syria, and (c) enabling the analyst
to interpret the economic results, to verify the robustness of the applied methodology and to
provide sound policy recommendations. To pursue these objectives the course provides basic
theory notions required to specify, estimate and interpret demand, supply and market models
illustrated by means of computer based applications. This learning by doing approach is a
necessary step if the objective to bridge the gap between basic theory and applied analysis to real
economic problems is to be attained.
The learning process is intended to be incremental. The material is proposed both at the basic
and advanced level. The advanced material is not required for the exam. It is intended to be a
complete reference for both the future applied work of the trainees and the future activities of
the Center of Policy Analysis. Special emphasis is placed on a) concepts, b) methodology, and c)
economic intuition and interpretation. The presentation of the teaching material follows
rigorously this sequence. Active participation to the discussion is expected.
Let us start our journey through the analysis of partial market equilibrium with a definition.
Definition.
Partial Equilibrium Analysis:
It is the analysis of a market in equilibrium considered in isolation from other product or input
markets.
In general, the existence of an equilibrium implies competitive (Walrasian) conditions. Our
study will be confined to the situation of a perfectly functioning market satisfying both
Fundamental Theorems of Welfare Economics. The working assumption of Pareto efficiency is
far from the reality of imperfect markets. It is adopted because it is instructive. It is useful to
sharpen the understanding of the market paradigm. Also, it is of practical interest as a
benchmark reference model that can be used to gauge how far second best solutions are from
their first best.
The market, being the sum of individual demand and supply functions, describes the aggregate
behavior of both consumers and producers as summed up by the behavior of a representative
consumer. Consumers and producers are not distinct by their characteristics. Not all consumers
consume a little bit of all goods and producers do not produce in some positive quantities all the
products. As a matter of fact, the effects of a consumer or producer subsidy on a good reach only
those who consume or produce the good. This limitation of the aggregate analysis is a major
shortcoming for the implementation of a policy interesting welfare and distributive analysis.
1
Training Materials
Every market is both vertically and horizontally integrated with both the product and factor
markets. Consider the corn market. It is vertically connected from below to the labor, capital,
land and other inputs markets such as fertilizers and pesticides. It can be directly used for
human consumption and be itself an input of the feed industry for livestock production or of the
industry producing combustible from biomass. It is horizontally integrated with those markets
that are close substitutes or complements in production or consumption such as soybean. The
partial equilibrium analysis can be static or dynamic as we will see in the second part of the
course.
The advantage of the approach lies on its empirical simplicity. When the vertical and horizontal
links are weak, then the partial equilibrium effects are a reasonable approximation of the
general effects. If the linkages are expected to be strong, then the markets are better analyzed
jointly through a multi-market approach which is a partial equilibrium approach extended
vertically or horizontally to include other markets. However, the analysis is partial also in the
sense that only the price effect is considered. Income and cost changes that shift the demand
and supply functions, exchange rate effects, savings-investment, public investments,
government transfers are out of the classical graphical market analysis. To capture these indirect
effects the partial analysis must be extended to the general equilibrium analysis taking under
considerations all the markets in the economy.
To implement the partial equilibrium approach we need first to learn how the two sides of the
markets behave and how to estimate them. Finally, we will combine demand and supply to
understand the basic mechanics of market equilibrium and the fundament of welfare analysis
applying the concepts of consumer and producer surplus to the measurement of the welfare
effects of technological change. This is the organization of the course.
To appreciate the power of the partial equilibrium approach as an analytical tool, let us consider
an example that we will implement together and will accompany us throughout the course.
1.2.
An example of Applied Partial Equilibrium Analysis
This example is taken from a seminal article in the analysis of partial equilibrium written by
Scobie and
Posada (1977) as the product of a research conducted at CIAT (International Center of Tropical
Agriculture based in Cali - Colombia) on the impact of high-yielding rice varieties in Latin
America with special emphasis to Colombia. This study is a precursor of the modern debate on
the effects induced by genetically modified material and the value of indigenous genetic
patrimony.
The generation of technical change through public and private agricultural research is an
economic activity raising both efficiency, expressed in terms of rate of returns from the
investment, and equity issues. The study is concerned with the measurement of the distribution
of social benefits derived from public investments of the international community on
agricultural research. The authors elect Colombian rice production as the industry of interest
and examine the distributive impact on both producers and households considering both costs
and benefits of the research program. The rice example can be adapted to other economic
realities. Syrian trainees, for example, may find many similarities between the Colombian rice
industry and the local wheat industry.
1.2.1. Background
In the late 1950s Colombian rice production was plagued by a virus disease causing large losses.
Imports of rice rose and the real retail price of rice increased dramatically. This critical situation
primed the formation and funding of a national rice research program with the objective of
selecting varieties resistant to the virus capable of increasing productivity thus reducing the
2
Partial Equilibrium Analysis of Policy Impacts (part I)
dependence on imports and curbing the upward pressure on prices. In 1967 the newly founded
Centro Internacional de Agricultura Tropical (CIAT) joined in a collaborative effort with the
Colombian program contributing the dwarf lines developed at the International Rice Research
Institute (IRRI) in the Philippines. This effort led to the development of disease resistant dwarf
rices. These modern varieties were widely and rapidly adopted mainly by the irrigated rice
sector. Because in rainfed rice areas located in the uplands the modern varieties could not
express their potential. Rationally, they were not adopted in non irrigated rice fields. As a
consequence, while the irrigated sector increased both yields and production, the disadvantaged
upland sector did not enjoy the advantages pf the technological evolution and declined in
importance from 50 percent of the national output in 1966 to 10 percent in 1974. Rice is the
most important foodstuff in Colombia. Rice is the major source of calories and the second major
source of protein (beef is the first) in the Colombian diet.
1.2.2. The Model
The model can be represented as a set of demand and supply equations with an exponential
form (linear in the logarithms):
Inverse Demand:
P = A QD 1/η
Supply:
QS = B P β
Equilibrium
QS = QD
where P is the price of the good, Q is the quantity demanded or supplied, A and B are exogenous
shifters. In the demand equations, A includes exogenous variables such as income and
demographic effects. The B shifters, on the other hand, includes technical change. Note that in
general the demand function is specified as:
Q = (A* P) η
where A*=1/A The demand function is here inverted to be represented in a two dimensional
space while keeping all other exogenous factors such as other prices, income, tastes, and income
distribution data constant. A partial equilibrium model always assumes the coeteris paribus
(everything else equal) condition.
Figure 1.1 The market model of the rice economy with technical change
P
D
S
S’
P0
P1
Q
3
Training Materials
The model showing the impact of high yield varieties on equilibrium prices and quantities can
be represented as in Figure 1.1 The curves are exponential. The model can be represented in
linear form after taking a logarithmic transformation.
The graph also shows the welfare impact of technical change. It visualizes both the change of
consumer surplus and the and producer surplus. These concepts along with the definition and
neutral or factor biased nature of technical change have been developed already in other courses
and will be taken up again in the part of the course devoted to the analysis and welfare
interpretation of the market. The graph is aggregate because it refers to a representative
consumer and producer. Interestingly, we may distinguish the supply of upland rice producers
from the one of irrigated rice producers in order to separate the different impacts. Similarly, we
can proceed for the consumers trying to do as best as we can give the aggregate information we
have. Our goal is to build this graph using the same data set of the authors by estimating both
the demand and the supply curve and the shift in technical change.
1.2.3. Results
Table 1 shows the changes in consumer and producer surpluses resulting from the introduction
of virus-resistant and dwarf varieties estimated for the period 1964-1974. Consumers benefit
from the introduction of the modern varieties. If the modern varieties were not developed, the
quantity of rice produced would have been markedly lower and domestic prices would have been
higher. Producers suffered a substantial loss. However, it is likely that “early birds,” that is
farmers of the irrigated sector, which first adopted the modern varieties, enjoyed some short run
benefits. It must be emphasized that the partial equilibrium framework does not allow us to
judge whether higher imports would have curbed the upward pressure on prices due to a lower
availability of domestic rice. Higher imports would have increased the demand for foreign
exchange and would have pushed up the exchange rate. These side effects can be properly
accounted for only within a general equilibrium framework.
Table 1. Gross Benefits of modern rice varieties in Colombia ($ Col. Million)
Gross
benefits
Consumers
Producers
Irrigated
Upland
Total
1964-69
1970-74
1404
17542
-368
-517
519
-6468
-3878
7196
Gross benefits include also the research costs that are borne by the Colombian taxpayer (both
consumers and producers) funding the national rice research institutions and the international
community funding the international research centers. Table 2 reports the distribution of gross
social benefits, research costs and net benefits for producers and consumers for the period 19571974 without accounting for the international research costs. Producers borne about two times
the cost of the research financed by consumers taxes.
Table 2. Size and Distribution of Benefits and Costs of modern rice varieties in Colombia 1957-1974
($ Col. Million)
Item
Gross Benefits
Costs of
research
Net Benefits
4
Colombia
Intl
Cooper
-8835
40
Consumer
s
14939
22
6104
63
19
-8875
14917
6042
Upland
Producers
Irrigated
Total
-3542
9
-5293
32
-3551
-5235
Partial Equilibrium Analysis of Policy Impacts (part I)
It is of paramount importance that the estimates of producer and consumer surpluses critically
depend on the quality of the estimated demand and supply slopes. This point is clearly shown in
table 3. The net benefits are more than halved when the demand elasticity moves from -0.3 to 0.449. The internal rate of return from the investment on rice research, shown in italic in Table
3, remains high independently from the choice of the demand and supply elasticity. Such high
returns are not infrequent in agriculture.
Table 3 Net Benefits in 1974 and Internal Rates of return for differing elasticities
Elasticity of
supply ε
0.235
1.5
-0.3
9052
89%
8627
96%
Elasticity of demand η
-0.449
3981
94%
3556
87%
-0.754
2174
89%
1749
79%
Note: Internal rate of returns are in italic; net benefits are in bold; results in table 1 and 2 use ε =-0.449 and ε =0.235
as in the shadow area.
Net benefits and costs have been distributed across income deciles of consumers and rainfed
and irrigated producers as illustrated in table 4. A decile corresponds to 1/10 of the income
range. Each income decile is equally spaced. For example, the first decile counts 19 percent of
the household accounting for only 2 percent of total household income. The benefits accruing to
each income group were assumed to be proportional to the quantity of rice consumed. Of
course, the benefits received depend on whether all consumers consume at least some rice and
on the frequency of consumption. This information is in general not available from aggregate
Time series data. A higher disaggregation can be achieved only by combining the market data
with household level consumption information.
Income
deciles
I
II
III
IV
V
VI
VII
VIII
IX
X
Table 4 Distribution of Net Benefits across income deciles
Cumulative % of
Net benefits as
Annual avg
Net Benefits Households
% of income
net benefits
385
12.8
18
19
642
7.1
50
39
530
3.5
67
52
333
1.6
77
64
348
1.3
83
71
353
1.2
88
76
342
0.8
93
82
200
0.4
95
86
128
0.2
96
89
138
0.2
100
100
Total Hh
Income
2
8
15
23
29
35
43
51
57
100
Rice in Colombia is consumed in higher proportion by the poor. Considering that the poor make
a smaller tax contribution, the net benefits were reaching mostly the poor. Poor Colombian
households may consume also 70-80 percent of their total budget on food and about 50 percent
of food expenses on rice. This reasoning provides an explanation for the fact that net benefits
accruing to very poor consumers range around 10 percent of the household income.
Rice is a staple food If we assume as an acceptable poverty line PL the level of income
corresponding to PL= 0.5 mean(income), then the poverty line is close to the upper bound of the
third decile. If so, about 50 percent of the Colombian population would be in poverty. Note also
that the income distribution in Colombia is highly concentrated. The affluent Colombians living
in the upper two deciles own 50 percent of household income.
The lowest 50 percent of Colombian households belonging to the three lowest deciles of the
income distribution account for 15 percent of household income but receive almost 70 percent of
the net benefits generated by the research program. This is in line with our expectations because
5
Training Materials
a decline in the price of a food staple due to technological change targets those most in need and
tends to reduce the income polarization.
In contrast, if we consider the distribution of benefits accruing to producers disaggregated by
farm size (and farm incomes) shown in Table, the group most severely hit was the small rainfed
producer. Small upland producers participating to the market were “takers” of lower prices
without receiving the benefits stemming from the technological advances. The irrigated
producers, on the other hand also had a substantial reduction of producer surplus that were
partially offset by the cost reducing and output increasing effects of technical change.
Table 5 Annual average Distributional Impact of rice research program on producers
Farm Size (Ha)
Avg Income
0-1
1-2
......
1000-2000
2000+
1500
3647
......
532389
1480199
Change in Prod Surplus+Research costs (%
income)
Upland Sector
Irrigated Sector
-58%
-56
-53
-39
......
......
-19
-49
-11
-36
In conclusion, small producers were the most severely affected while poor households benefitted
the most. In terms of social welfare, given that the losses were distributed across about 12000
small producers with less than 5 hectares while the gains reached more than one-million
households belonging to the lowest deciles, the economic situation with technical change is
socially preferable.
Let us now try to estimate the partial equilibrium model used by Scobie and Posada to carry out
their study. We first need to delineate some of the main traits of a general methodology that
should be followed when carrying out applied partial equilibrium analysis.
1.3.
A Methodological Note
The first step to be undertaken is data collection and preparation for the econometric analysis.
Before starting the econometric execution we have to be familiar with the data and be sure that
the data reflect rational behavior of both producers and consumers.
In market economies characterized by a strong government intervention or in economies in
transition, the data are not generated by an efficient market mechanism. In such situations,
neoclassical economic theory has a very weak explanatory power. Prices are not expression of a
market equilibrium signaling consumers desire for a specific good and its relative scarcity in the
market, but is the realization of government decisions. Government rather than market behavior
should be more properly modeled in such occasions.
Let us keep in mind that our objective is to interpret the data as best we can using both
economic and econometric theory.
1.3.1.
Data Collection
Nowadays, economic data is commonly available from national statistical sources and
international sources such as FAO Statistical Data Base and the World Bank Economic
Indicators. Data are available through the internet or can be obtained upon request at low cost.
To gather the proper data we first need the appropriate specification of the demand and supply
functions. Theory is a necessary condition. What data would we collect if we do not know what
we should look for? Economic theory tells us that quantity demanded is affected by the own
price, price of close substitutes, income and demographic variables and quantity supplied varies
6
Partial Equilibrium Analysis of Policy Impacts (part I)
with the g(output price, price of other outputs, input prices, characteristics:
QD = f (P | Pj, Y, D) = f (own good price, price of other goods, income, demographic factors)
QS = g (P | Pj, Pi, O, D) = g(output price, price of other outputs, input prices, characteristics).
Note that the theory does not help regarding the choice of demographic variables and other
characteristics of the production process or the producer. What is important is that other factors
that can be included such as demographic variables, advertising effects, income distribution
indexes for the demand, and farm size, concentration indexes, decision making process
information for the production side, be exogenous, that is not determined by the model, and
relevant.
We now need to organize our information in a matrix form where each rows corresponds to an
observation and each column is a variable to be included in the model.
There are 2 type of data sets:
(1) TIME SERIES aggregate data set at the market level. Data figures can be found in the
official publications of the National Statistical Office. They are usually organized
by years, or quarters, or month. N=number of years.
(2) CROSS SECTION DATA at the individual household levels. Data come from
household expenditure and income surveys or farm/firm surveys. The level of
detail of commodities is very high. N= number of households in the sample (for a
population of 60 mil., sample size can be 25000).
Note that prices in cross section analysis are unit values given by the ratio between expenditure
and quantities. Unit values vary in relation to the household location and time of purchase. They
are household or firm specific and embed quality information. We can think at unit values as
prices that are slightly dispersed around a mean price that should be close to the nominal price
used in time series analysis for the survey year.
Time series data is commonly used to estimate aggregate demand at the market or sector level.
It describes the behavior of a representative consumer. Cross Section data allows for
disaggregate demand and policy analysis at the level of individuals. It describes the behavior of
specific households. Because of the because of the high level of commodity detail, it is necessary
to deal with the econometric problem of dealing with zero realizations. Consumers do not
consume all the goods or producers do not produce all goods. The analysis explaining why some
consumers do not consume certain goods is of great policy interest. It allows one to know the
price level at which consumers are willing to buy the good (e.g. how the government may
subsidize that good). Consider also that those who do not consume do not receive policy
benefits because their consumer surplus is of course zero.
1.3.2. Familiarization with the Data
Before starting an econometric estimation it is necessary to be familiar with the data aiming at
knowing whether the data gathered have been generated through a theoretically consistent
economic process. In essence, we would like to answer the following question:
ARE DATA RATIONAL?
If data are rational then we may expect to obtain estimates that are consistent with economic
theory.
Before estimating a demand or a supply relationship, it is important to verify whether the data
reflect rational (optimizing behavior) of the consumer or the producer using graphical analysis.
The objective is to verify, before estimation, whether the data reflect optimizing behavior of the
consumer or the producer after controlling for the effects of the other exogenous factors.
7
Training Materials
An extremely simple check to verify the rationality or optimizing behavior of the data consists in
graphing the data. Visual inspection should provide an answer to the following type of
questions: a) do price slopes slope as we expect? , b) are Engel curves (the relation between the
share consumed of a good and the logarithm of income) negatively sloped? Using graphs, we
are verifying, in a quite crude way, whether the behavior comply with the axioms of revealed
preferences and if it obeys the Slutsky law describing price and income demand effects and price
and output production effects.
This is important for at least 2 reasons:
(1) either we did not collect the data well, because, for example, the quality control of
survey data was not very precise, or,
(2) the institution that we are studying, such as the market, may not be working
properly. As an example, consider a market where prices are administered. It is
the government setting the prices, not the exchanges between producers and
consumers in a market!
Note that the theory does not help regarding the role of demographic variables and other
factors. In such cases we may use our economic intuition: e.g. as number of children increase in
the family, consumption of food is expected to increase. Or let us think at an index of inequality:
if inequality increases through time, this may have a positive effect on luxury goods and a
negative effect on necessary goods. For production, the age of the head of the farm may be
reasonably expected to be negatively correlated with productivity.
Let us apply this methodology to the construction and the analysis of the data base referring to
the rice economy in Colombia.
1.4. Description of the Aggregate Data Base (the Colombian Rice
Economy)
The data are from Scobie and Posada (1977), Fao Disappearance data and data from DANE, the
national Colombian statistical institute.
Table 6 presents the production data of the Colombian rice market both for the upland rice
sector depending on rainfalls and the irrigated sector during the period 1954-1974. In twenty
years, rainfed production decreases the area planted and yields increase relatively little if
compared with the yield growth enjoyed by an increasingly growing irrigated sector during the
same period. the supply increases also when the price decreases. Technical change decreases
the costs of production and increases productivity so that it is still remunerative to produce.
As a result, irrigated rice production increased from 58.1 percent in 1954 to 90.5 percent in 1974
as it is shown in Table 6 referring to the supply side of the rice economy. Yields and total area
planted more than doubled during the period under consideration. Inspection of table 6 further
reveals that real prices at the farm gate level increased until the mid 1960s and then declined
slightly. Despite the price decrease irrigated farmers were still investing in rice and increasing
the area planted. This is an apparent inconsistency. The new technology, in fact, allows farmers
to reduce production costs while increasing the yields, thus maintaining the relative profitability
of rice production even at lower prices. The continued adoption of new technology in the face of
falling farm prices is often referred to as Cochrane’s “agricultural treadmill.” Farmers are able to
undertake such a habitual and laborious course of action (treadmill) because farmers, being
price takers, have nothing else to do than always trying to reduce costs adopting technologies
that are also yield increasing. Note that those irrigated and better informed rice farmers, who
adopted the modern varieties in the early sixties, were also able to capture extra benefits from
rising prices. In this sense they are called “early birds” evoking those birds waking up earlier in
the morning and leaving no “feed” for the late comers.
8
Partial Equilibrium Analysis of Policy Impacts (part I)
The treadmill argument is crucial for us. It helps “rationalizing” why farmers have increased
production in spite of lower prices. In the estimation phase, it will then be important to control
for the role of technical change. Interestingly, inspecting the yield pattern we notice that yields
grew at a constant rate as a linear trend by the time modern varieties were introduced. This
evidence supports the introduction of a linear trend in the supply analysis to control for the role
of technical change. We can also note that we may reasonably expect a structural break due to
the introduction of high-yield varieties that may justify the introduction of a dummy variable in
the model.
Table 8 and related graphs show marketing margins at the wholesale and retail level and the
pattern of net exports. Marketing margins capture the costs of intermediaries, often acting as
speculators, and of the passages linking farm gate production to the consumer via the
production chain. The production and distribution of milled rice involves transport, storage,
insurance, milling, packaging, wholesaling, and retailing before it gets into the hands of the
consumers. The farm to retail margin increased from 147 percent (obtained as: (retail pricefarm price)/farm price) from 1954 to about 190 percent at the end of the period. At the
beginning of the period the Colombian rice industry was importing rice, while at the end of the
period was an exporter. The analysis of marketing margins is relevant. A reduction of the margin
is often a strict Pareto improvement in the sense that producers, consumers and the
government all gain from more efficient markets.
Table 9 reports the demand side of the rice economy. Per capita consumption increased steadily
as income levels were also increasing thanks also to greater domestic availability. The
decreasing share of rice as income increases throughout the period is in line with Engel law (an
Engel curve describes the relationship between budget share and the logarithm of income). As
prices decrease, consumption increases as required by the Slutsky law. Part of this increase can
be explained by a change in taste as can be captured by the increase from 37 percent to about 60
percent of the proportion of the urban population. These regularities of the data are important.
It means that there exists evidence supporting the hypothesis that the data set is rational. The
important implication is that we can rely on our data set as we develop our experiment. For
example, if the regression analysis does not show the correct sign, it means that the problem
does not belong to the data but to our experimental conduct. Either we are making a
programming mistake or we are not interpreting the data correctly using both economic and
econometric theory.
This data set is a standard design for the aggregate market analysis. It can be reproduced to any
market.
Now we need to apply very basic econometric techniques to estimate the demand and supply
relationships of interest.
9
Training Materials
Table 6. The Rice Market in Colombia (1954-1974): rice production
Year
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
Upland Sector
Area
Prod
(ha)
(m.t.)
111580
123600
103920
124328
119960
130210
110250
130042
124800
147779
153610
180366
160230
186770
132100
200150
154200
231310
138600
206000
178300
215000
244750
275600
236000
338600
180850
280500
150200
250600
134570
220275
121113
198248
109130
173696
103220
160524
98840
154769
95600
149830
95000
152000
Yield
(t/Ha)
1.1077
1.1964
1.0854
1.1795
1.1841
1.1742
1.1656
1.5151
1.5001
1.4863
1.2058
1.1260
1.4347
1.5510
1.6684
1.6369
1.6369
1.5916
1.5552
1.5659
1.5673
1.6000
Note: m.t.= millions of tons (t), ha=hectares
Irrigated Sector
Area
Prod
(ha)
(m.t.)
63420
171200
84070
195872
70040
212290
79750
220158
71200
232621
52190
241734
67070
263230
105000 273450
125350
353690
115400 344000
124200 385000
130000 396400
114000
341400
109850 381000
126925
535000
115890
474225
112100
554347
144380
730652
170620
882724
192020 1021102
272950 1420110
273650 1480100
Yield
(t/Ha)
2.6995
2.3299
3.0310
2.7606
3.2671
4.6318
3.9247
2.6043
2.8216
2.9809
3.0998
3.0492
2.9947
3.4684
4.2151
4.0920
4.9451
5.0606
5.1736
5.3177
5.2028
5.4087
Evolution of Production and Prices
1800000
1800
1600000
1600
1400000
1400
1200000
1200
1000000
1000
800000
800
600000
600
400000
400
200000
200
0
0
years
Prod(m.t.) of Total Area
10
Farm Real Prices$/t.
Partial Equilibrium Analysis of Policy Impacts (part I)
Evolution of Production and Prices
1800000
1800
1600000
1600
1400000
1400
1200000
1200
1000000
1000
800000
800
600000
600
400000
400
200000
200
0
0
years
Prod(m.t.) of Total Area
Farm Real Prices$/t.
Evolution of Production and Prices
1800000
1800
1600000
1600
1400000
1400
1200000
1200
1000000
1000
800000
800
600000
600
400000
400
200000
200
0
0
years
Prod(m.t.) of Total Area
Farm Real Prices$/t.
11
Training Materials
Evolution of Production and Prices
1800000
1800
1600000
1600
1400000
1400
1200000
1200
1000000
1000
800000
800
600000
600
400000
400
200000
200
0
0
years
Prod(m.t.) of Total Area
Farm Real Prices$/t.
Evolution of Production and Prices
1800000
1800
1600000
1600
1400000
1400
1200000
1200
1000000
1000
800000
800
600000
600
400000
400
200000
200
0
0
years
Prod(m.t.) of Total Area
12
Farm Real Prices$/t.
Partial Equilibrium Analysis of Policy Impacts (part I)
Total Planted Area,Upland,Irrigated
400000
350000
300000
Area (ha)
250000
200000
150000
100000
50000
0
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
Years
Area Upland Sector(ha)
Area Irrigated Sector(ha)
AreaTotal(ha)
Production of Rice
1800000
1600000
1400000
1200000
M.T
1000000
800000
600000
400000
200000
0
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
Years
Prod(m.t.) of Upland Area
Prod(m.t.) of Irrigated Area
Prod(m.t.) of Total Area
13
Training Materials
Yield of Rice
6.0000
5.0000
M.T/ HA
4.0000
3.0000
2.0000
1.0000
0.0000
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
Years
Yield of Upland (M.t/Ha)
Yield of Irrigated land (M.t/Ha)
Total Yield(M.t/Ha)
Table 7. The Rice Market in Colombia (1954-1974): the supply side
Total
Area
(ha)
175000
187990
190000
190000
196000
205800
227300
237100
279550
254000
302500
374750
350000
290700
277125
250460
233213
253510
273840
290860
368550
368650
14
Prod
(m.t.)
294800
320200
342500
350200
380400
422100
450000
473600
585000
550000
600000
672000
680000
661500
785600
694500
752595
904348
1043248
1175871
1569940
1632100
Yield
(t/Ha)
1.6846
1.7033
1.8026
1.8432
1.9408
2.0510
1.9798
1.9975
2.0926
2.1654
1.9835
1.7932
1.9429
2.2755
2.8348
2.7729
3.2271
3.5673
3.8097
4.0427
4.2598
4.4272
Production Real Prices
Irrigated
Farm
%
$/t.
58.1
1270
61.2
1284
62.0
1244
62.9
1337
61.2
1471
57.3
1375
58.5
1497
57.7
1490
60.5
1372
62.5
1321
64.2
1347
59.0
1592
50.2
1507
57.6
1418
68.1
1452
68.3
1217
73.7
1121
80.8
1044
84.6
893
86.8
978
90.5
1151
90.7
72
73
74
75
Partial Equilibrium Analysis of Policy Impacts (part I)
Evolution of Production and Prices
1800000
1800
1600000
1600
1400000
1400
1200000
1200
1000000
1000
800000
800
600000
600
400000
400
200000
200
0
0
years
Prod(m.t.) of Total Area
Farm Real Prices$/t.
Table 8. The Rice Market in Colombia (1954-1974): market margin and net exports
Real Prices
Year
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
Wholesale
$/t.
2789
2508
2687
3200
2902
2600
3281
2913
2579
2626
2928
3379
3059
2850
2780
2415
2545
2309
2089
2755
2783
Marketing Margin Exports
Retail
$/t.
3135
3135
3026
3696
3529
3071
3695
3688
3522
3012
3480
3850
3568
3259
3117
2877
2727
2735
2493
3113
3311
Farm to Retail
$/t.
%
1865
146.9
1851
144.2
1782
143.2
2359
176.4
2058
139.9
1696
123.3
2198
146.8
2198
147.5
2150
156.7
1691
128.0
2133
158.4
2258
141.8
2061
136.8
1841
129.8
1665
114.7
1660
136.4
1606
143.3
1691
162.0
1600
179.2
2135
218.3
2160
187.7
(m.t.)
0
0
0
0
0
0
0
0
4000
3000
0
0
0
0
0
16000
5000
0
3000
20000
1000
Imports
Net
Exports
(m.t.)
31000
2000
0
10000
0
0
0
39000
3000
0
0
0
0
0
0
0
0
0
0
0
0
(m.t.)
-31000
-2000
0
-10000
0
0
0
-39000
1000
3000
0
0
0
0
0
16000
5000
0
3000
20000
1000
15
Training Materials
Farm, Wholesale, Retail Price of Rice
4500
4000
3500
3000
2500
2000
1500
1000
500
0
54
55
56
57
58
59
60
61
62
63
Farm Real Prices$/t.
64
65
66
67
68
69
70
M.T/$
Wholesale($/t.)
71
72
73
74
Retail($/t.)
Marketing Margin %
250.0
200.0
%
150.0
100.0
50.0
0.0
54
55
56
57
58
59
60
61
62
63
64
years
16
65
66
67
68
69
70
71
72
73
74
Partial Equilibrium Analysis of Policy Impacts (part I)
Imports,EXports of Rice
45000
40000
35000
30000
M.T
25000
20000
15000
10000
5000
0
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
Years
Exports(m.t.)
Imports(m.t.)
Table 9. The Rice Market in Colombia (1954-1974): the demand side
Consumption
Population
Population
Per capita
consumption
Total
Urban % Urb/Tot
kg/month
325800
12486
4639
37.154
2.174
322200
13027
5073
38.942
2.061
342500
13627
5552
40.743
2.094
360200
14372
6068
42.221
2.089
380400
14892
6612
44.400
2.129
422100
15387
7171
46.604
2.286
450000
15901
7623
47.940
2.358
512600
16421
8082
49.217
2.601
584000
16918
8503
50.260
2.877
547000
17430
8945
51.320
2.615
600000
17959
9412
52.408
2.784
672000
18506
9904
53.518
3.026
680000
19074
10345
54.236
2.971
661500
19659
10805
54.962
2.804
785600
20246
11277
55.700
3.234
678500
20817
11750
56.444
2.716
747595
21360
12218
57.200
2.917
904348
21869
12659
57.886
3.446
1040248
22348
13091
58.578
3.879
1155871
22813
13523
59.278
4.222
1568940
23283
13968
59.992
5.615
Note: the mean of the rice share is: mean(w)=0.052.
Income
Rice Share
Pesos/mth
83.48
85.18
89.19
109.52
112.69
114.91
144
164.83
177.77
178.63
191.82
236.9
237.6
208.2
232.7
213.6
219.8
267.5
281.8
388.6
561.1
0.0817
0.0759
0.0711
0.0705
0.0667
0.0611
0.0605
0.0582
0.0570
0.0441
0.0505
0.0492
0.0446
0.0439
0.0433
0.0366
0.0362
0.0352
0.0343
0.0338
0.0331
17
Training Materials
Evolution of Rice Consumption and Prices
6
4.5
4
5
3.5
4
3
2.5
3
2
2
1.5
1
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21
years
Per capita Consumpt. (kg/month)
Retail Prices($/kg.)
I Engel Law: As log income rises, the share of necessities decrease
600
0.09
0.08
500
0.07
400
0.06
0.05
300
0.04
200
0.03
0.02
100
0.01
0
0
1
.9
1
2
.9
2
5
.0
2
5
.1
2
6
.2
2
5
.2
2
8
.3
2
8
.3
2
7
Log Income
Income Pesos/mth
18
Rice Share
.3
2
4
.4
5
Partial Equilibrium Analysis of Policy Impacts (part I)
Chapter 2 - Demand Analysis
The section on demand analysis is organized in an introductory and an advanced part.
In the introductory part the concepts of income and price elasticities are introduced in the
context of Engel curve analysis and then estimated using the class experiment related to the
Colombian rice economy. Own and cross price elasticities have been defined and derived both
for a time series aggregate application and for a disaggregate cross section application where a
complete demand system, in its simplest form, is estimated within the laboratory experiment.
In the advanced section, it is introduced the Almost Ideal Demand System (AIDS) which is the
most popular applied demand system. The economic interpretation of demand policy
parameters is developed using estimates of empirical applications. The welfare analysis of
consumption introduces the concepts of living standard, cost of living indexes and
compensating variation, an exact measure of consumer surplus.
The objective is to apply the experimental methodology very pragmatically simulating within a
participatory class experiment the steps that an applied economist would have to follow in
order to estimate a demand system that complies with the requirements of economic theory.
The lab experiment is the estimation of Scobie and Posada model (1977) and the application of
disaggregate demand analysis using Italian rural consumption data.
2.1.
Introductory Demand Analysis
Demand analysis can be used to understand consumer behavior alone if the interest is simply in
forecasting. Interestingly, we can build a partial equilibrium model for a public good such as a
park or a child accounting for habits, heterogeneity and tastes of the consumer. Or about a
quality characteristic or about the market for leisure time. We may apply a distributive (equity)
analysis both within society and within the family seen as a micro society.
The demand estimates can also be used to implement welfare analysis that is to use the
information on observed demand behavior to deduce utility (welfare) levels and develop exact
welfare analysis of markets, poverty and inequality measurement, the estimation of social
welfare function given by the sum of producer and consumer surpluses and to carry out taxation
analysis.
Follows a non exhaustive list of possible uses:
Study of BEHAVIOR (POSITIVE ANALYSIS).
• Demand for goods (it is specific to each good):
o Demand for Wheat flour, Bread, alcool or tobacco (theory of addiction), nutrients,
durables, cotton, public goods such as parks, or clean air.
• Analysis of structural change: habits, heterogenity or tastes? And advertisement?
• Demand for quality characteristics
• Intrahousehold distribution: adults vs children good, gender specific demand
19
Training Materials
• Demand for children
• Demand for leisure (labor supply)
Notice that these applications can be described using a partial equilibrium framework.
Implementation of WELFARE ANALYSIS
• Estimation of the Expenditure function and utility in order to derive the Compensating
Variation which is an exact measure of Consumer Surplus
• Estimates of the “Cost of Children”: demographic targeting and welfare reform.
• Poverty and Inequality: from demand to Y(u,p,d) => microsimulations on p and d!
• Social welfare functions: W=Producer Surplus
corresponding to the sum of all individual utilities
(PS)+Consumer
Surplus
(CS)
• Demand and Optimal taxation
It is relevant to be aware that demand analysis can serve both to understand behavior and to
implement welfare analysis. From a practical standpoint, our choices regarding the specification
of the model and the choice of the functional form may depend on the objective of our study. For
example, if the objective is to implement also a welfare analysis than greater accuracy is
achieved by estimating a system of demand equations rather than a single demand equation. A
system approach corresponds to a multi-market approach; a single equation approach is used
to implement a partial equilibrium framework. These concepts are going to be described below
after a brief description of some basic notions in demand analysis.
2.1.1.
Some Basic Notions
DEFINITION OF ELASTICITY
Suppose that q=f(x) is a demand (or a supply curve) where x is an exogenous variable such as
income, price, fixed factors, etc. and q is the quantity demanded or supplied. Marginal changes
such as dq /dx are in the measurement unit of q and x and so difficult to compare across
marginal changes. In contrast, the elasticity is unit free.
Elasticity: is the percentage change in the dependent variable due to a percentage change in
the independent variable.
Elasticity of q with respect to x:
Ex =
% change in q d ln q d q q dq x
=
=
=
% change in x d ln x d x x dx q
Note: The elasticity can be computed at the mean or at each data point. It can be a constant or a
function.
If the absolute value of the elasticity |Ex| >1 = elastic
If the absolute value of the elasticity 0<|Ex| <1 = inelastic.
INCOME ELASTICITY OF DEMAND
Ey =
d ln q d q q dq y marginal propensity to consume
=
=
=
d ln y d y y dy q average propensity to consume
If Ey>1, demand increases more than proportionally with income, so, the expenditure share of
this good increases as income increases. If Ey<1, the expenditure share declines as
income increases. This is the case for food.
20
Partial Equilibrium Analysis of Policy Impacts (part I)
OWN PRICE ELASTICITY OF DEMAND
Own price elasticities must be negative.
Epi =
d ln qi d qi qi dqi pi
=
=
d ln pi d pi pi dpi qi
If Ep>-1, (inelastic demand) an increase in price induces an increase in expenditure (pq) (d
(pq)/dp>0) despite a decrease in demand. The change in quantity is smaller than the
change in price.
If Ep<-1, (elastic demand) the expenditure (pq) decreases with a price increase (d (pq)/dp<0)
because the decline in demand is larger than the price increase.
Explanation:
d ( pq )
dp
=q+ p
= q(1 + Ep )
dq
dp
The same reasoning applys to factor demand in production.
CROSS PRICE ELASTICITY OF DEMAND
Epij =
dqi pj % change in qi
=
dpj qi % change in pj
If Ep ij > 0, i and j are gross substitutes
If Ep ij < 0, i and j are gross complements.
This concept is especially appropriate to a system (multi-market demand approach). Consider
the following two demand equations exhausting the full budget: food and nonfood expressed in
shares rather than in quantities:
wfood = d0 + d00 ln d +d11 ln pf + d12 ln pnf + d2 ln y
wnonfood = e0 + e00 ln d +e11 ln pf + e12 ln pnf + e2 ln y
In a single equation framework, we only have the cross effect of a price of a substitute or
complement good such as, looking at the first equation, non-food on food, but we do not know
how the price of food affects the demand for non-food and vice versa. This full matrix of effects
is possible to obtain only with a system approach. From a behavioral standpoint it is interesting
to test whether the cross-price relationships are symmetric.
CLASSIFICATION OF GOODS
Goods are defined in terms of the type of response with respect to prices and incomes. These
definitions are fundamental to interpret the results.
1) Classification in terms of price response
Giffen
Normal with inelastic demand
Normal good with elastic demand
Ep>0
0>Ep>-1
Ep<-1
21
Training Materials
2) Classification in terms of income response
Ey<0
Inferior
Necessity
0<Ey<1
Luxury
Ey>1
THE I AND II ENGEL LAWS
I Engel Law: As the log of income increases, the food share decreases.
Note: This empirical regularity linking consumption to income can be found in the data of all
societies and explains why food is an indirect measure of welfare.
Ex. Italian Household budget data 1995. Obs: 33400.
Food Share
.4
Quota Alimentare
.3
.2
.1
12
14
Log della Spesa Totale
16
18
log income
Definition. II Engel Law: As family size increases, the share of expenditure allocated to food
also increases.
Food share
.5
Quota Alimentare
.4
.3
.2
.1
13
14
15
Log della Spesa Totale
16
17
log income
The shape of the data suggests that a correct functional representation of the Engel curve is
linear in the logarithm:
wi = αi + δi ln di + βi ln y
where w=food share, d=number of children, y=income.
22
Partial Equilibrium Analysis of Policy Impacts (part I)
2.1.2. Introductory Applications
THE TIME SERIES EXERCISE: AGGREGATE MARKET DEMAND FOR RICE IN COLOMBIA
This exercise is meant for illustrating a practical method to estimating the price and income
elasticity of consumption using aggregate time series data. We use price, consumption and
income data from the data set that we assembled together for the Colombian rice economy. The
demographic variable is the percentage of urban population. It is included in the analysis to
account for tastes.
The data are transformed in logarithms both to ensure e better fit and for analytical
convenience. In a Cobb-Douglas demand specification expressed in terms of quantity, the
parameters are directly interpreted as elasticities. In the Engel type of demand functions
expressed in share form, the logarithmic specification come from the theoretical derivation of
the model.
Notation
Consider a general demand system specified using the Colombian data set:
q = f ( p, y, N, d) or w=pq/y = f ( p, y, N, d)
where:
q is quantity demanded in kg/month
p is the real retail price of rice in Pesos/Kg (the Peso is the Colombian currency)
y is total income
w is the budget share of rice: w=p q/y
N is population size
d is the urban/N ratio.
•
Specification and Choice of Functional Form
At this point of our project, we collected the time series data base and familiarized with our data.
It is time to implement the econometric analysis. In doing this, we have to make two important
choices:
1. Model Specification: What is the best model specification, that is what is the set of
relevant variables to include in the model and how should we include the selected
variables in the model? For example, income, prices and demographic effects may be
included both linearly and nonlinearly as quadratic terms. Prices, income and
demographic variables also interact with each other. Nonlinearities in demographic
effects may imply the existence of household economies of scale. A quadratic income
term may be justified by the shape of the Engel curve. A quadratic income term would
make the elasticity change across the income distribution: the same good can be a luxury
at low income levels and a necessity at high income levels.
2. Choice of the appropriate functional form: for example, what is the functional form
that interprets the data better? A double log or a semilog model or other functional
forms? It is important to emphasize that the choice of the functional form (for example
between a ad hoc single equation double log model and a semilog system of share
equations derived from utility theory ) often depends on the objectives of the research.
For partial equilibrium analysis a single equation approach may suffice. In contrast,
multimarket analysis and accurate welfare analysis requires a system approach
consistent with utility theory.
The exercise is designed to illustrate these topics through a learning by doing process.
23
Training Materials
•
Representation of preferences
First, we need to choose how to represent preferences
• “Ad hoc,” that is specified for the specific purpose at hand
Ex. linear single equation models
Ad hoc models are not based on theory: if a model is single equation cannot represent
preferences for all goods we buy, so we cannot derive the total expenditure function and utility
based on the consumption of the complete basket.
• Utility based complete demand systems are based on theory: this type of model
allows one to recover the unobservable U(x). This is necessary for welfare analysis!
•
Choice of the functional form
Ex. of a ad hoc reduced form model
ln Q = a + a1 ln d + b ln p + c ln y +d ln y*y
Ex. of a utility based structural model (system of demand equations)
wi = a0 + Σk ak ln dk + Σj bj ln pj + c ln y
for k=1,..,K; i=j=1,..,N
where wi = share of good i, pj=price of good j, dk = k-th demographic variable, y= income, K is
the number of demographic variables and N is the number of goods (and prices) included in the
system of demand equations.
•
Flexibility
In some empirical situations where the presence of nonlinearities in the data is significant, in
order to correctly interpret the data, it is useful to add more flexibility to the model:
Is the model linear in p?
Is the model linear in y?
Is the model linear in d: are there economies of scale (nonlinear demographic effects)?
Answer: let us learn from the data by graphing them! If not linear, add a quadratic income term
or nonlinearity in p or quadratic demographic effects.
Note: this is crucial to obtain estimates that agree with the theory requirements (Ex. Own price
demand elasticities must be negative!)
•
Econometric Execution
We have the option to estimate either in quantity q or in share form w. We will estimate and
analyze five specifications of an aggregate demand function:
Cobb-Douglas (univariate double log
regression)
ln q = ln A + A1 ln p = A pA1
Cobb-Douglas (multivariate)
ln q = (a0 + a00 ln d + a2 ln (y/N) ) + a1 ln p
Working-Leser (WL) Engel curve (semi-log)
w = b0 + b2 ln (y/N)
WL Engel curve with prices
w = c0 + c1 ln p + c2 ln (y/N)
WL Engel curve with prices and
demographics
w = d0 + d00 ln d +d1 ln p + d2 ln (y/N)
24
Partial Equilibrium Analysis of Policy Impacts (part I)
Note: the double-log and the semi-log are not directly comparable because the double-log and
semi-log models are not nested into each other. The dependent variables are different.
Specification 1: Cobb-Douglas (univariate double log regression)
• ln q = ln A + a1 ln p
Coefficient Standar
s
d Error
t Stat
Intercept
1.682
0.578
2.911
Prices $/t
-0.557
0.492
-1.133
R2=0.063, N. Obs. 21
This is the functional form closest to our representation of the partial equilibrium model. The
results show a clear misspecification problem. The fit is very low and the price parameter is not
statistically significantly different from zero at all levels of significance. The price here explaiins
only 6 percent of the variation observed for demand (R2=0.063).
Many relevant variables are omitted. The theory tell us that income, acting as an exogenous
shifter, should be included in the specification. Other exogenous factors can be relevant in the
sense that, if omitted, a bias may result. So, considering that our level of familiarization with the
data allows us to have confidence on our data, than it means that we can do better either by
improving the specification and/or through adopting a functional form more appropriate to
interpret correctly our data. Let us work on the specification first.
Specification 2: Cobb-Douglas (multivariate double log regression)
• ln q = a0 + a00 ln d + a1 ln p + a2 ln (y/N)
Coefficients Standar
d Error
t Stat
Intercept
1.823
0.527
3.461
urb/tot %
-1.151
0.187
-6.151
Prices $/t
-0.361
0.084
-4.296
Income/mth
0.796
0.053
14.975
R2=0.979, N. Obs. 21
Note that model 2 can be rewritten as model 1 using the following change in notation:
ln q = ln A + a1 ln p
where
A=exp( a0 + a00 ln d + a2 ln y - a2 ln N).
Model 2 controls for relevant exogenous factors such as income and demographic trends.
The set of regressors explains .979 percent of the variation observed (R2=0.979). The estimated
coefficients are all statistically significantly different from 0 at the 5 percent confidence level.
Statistical significance is analyzed using the t-distribution. The table of the t-distribution gives
the values of t which limits the interval of the acceptance region. In our case, we may look at the
table referring to a 5 percent significance level and 17 degrees of freedom (df=(n obs)-(n
Param)=21-4). For example, the t-value for the price coefficient is 2.12. This means that with tvalues less than -2.12 and greater than +2.12 the coefficients can be accepted.
Note that the t-values is equal to the ratio between the coefficients and the standard error:
Ex. Price: H0 : β=0; b - β / s = (-0.361-0)/0.084=-4.296
Then we conclude that the price coefficient is significantly different from 0 at the 5 percent
25
Training Materials
significance level. (Note that β is the population parameter).
Suppose now that we want to test whether the income coefficient is significantly different from 1
at the 5 percent significance level:
Ex. Income: H0 : β =1; b - β / s = (0.796-1)/0.053=-3.85
In the case of a t-test the null hypothesis is in general the statistical difference of the parameter
of interest with respect to 0. If the null hypothesis is accepted than the parameter is not
statistically significantly different from 0 at the selected level of confidence.
The signs comply with the theory: the own price effect is negative and the income effect is
positive. According to these estimates rice is a normal good with respect to the price effect and a
necessary good with respect to the income effect.
Note that with respect to model 1 we maintained the same functional form. The improved
specification changed the overall performance of the model.
Elasticity calculation
Ed = ∆ ln q / ∆ ln d = (∆ q / ∆ d) (d / q) = a00 = - 1.151
Ep = ∆ ln q / ∆ ln p = (∆ q / ∆ p) (p / q) = a1 = -0.361
Ey = ∆ ln q / ln y = (∆ q / ∆ y) (y / q) = a2 = 0.796
Specification 3: Working-Leser (WL) Engel curve (semi-log)
• w = b0 + b2 ln (y/N)
Coefficients Standard
Error
t Stat
Intercept
0.198
0.014
13.733
Income/mth
-0.028
0.003
-10.191
R2=0.845, N. Obs. 21 mean(w)=0.052
Elasticity calculation
w = pq / y
→
ln w = ln p + ln q - ln y
→
ln q = ln w(p,y,d) - ln p + ln y
ln q = ln (b0 + b2 ln y - b2 ln N) - ln p + ln y
So, taking the derivative we respect to y we derive the elasticity of income:
Ey = ∆ ln q / ∆ ln y = (∆ ln w / ∆ w) (∆ w / ∆ ln y) +1 = b2 / w + 1 = 0.458
where: (∆ ln w / ∆ w) = 1/w
(∆ w / ∆ ln y) = b2.
Note 1: The elasticity of a demand equation in share form is a function not a constant as we had
in the case of a double logarithmic specification. In particular, it is a function of the level
of the share. This implies that we can derive an elasticity for the poor and one for the rich
consumer. In our time series example, it means that we can compute the elasticity for the
beginning of the period when Colombia was less rich and the share of rice consumed was
higher and for the end of the period. This greater flexibility is highly desirable.
26
Partial Equilibrium Analysis of Policy Impacts (part I)
Note 2:
If b2<0 and b2>w, the good is an inferior good (Ey<0)
If b2<0 and b2<w, the good is a necessity (0<Ey<1).
If b2>0 ,the good is a luxury (Ey>1).
Specification 4: WL Engel curve with prices
• w = c0 + c1 ln p + c2 ln (y/N)
Coefficients Standard
Error
t Stat
Intercept
0.152
0.017
8.762
Prices $/t
0.034
0.009
3.532
Income/mth
-0.027
0.002
-12.100
R2=0.909, N. Obs. 21, mean(w)=0.052
Elasticity calculation
w = pq / y
ln w = ln p + ln q - ln y
ln q = ln w(p,y,d) - ln p + ln y
Ep = ∆ ln q / ∆ ln p = (∆ ln w / ∆ w) (∆ w / ∆ ln p ) - 1 = c1 / w - 1 = -0.355
Ey = ∆ ln q / ∆ ln y = (∆ ln w / ∆ w) (∆ w / ∆ ln y) +1 = c2 / w + 1 = 0.484
where: (∆ ln w / ∆ w) = 1/w
(∆ w / ∆ ln p ) = c1
(∆ w / ∆ ln y) = c2.
The interpretation of the function Ep tell us that:
if c1 <0, the good is normal with elastic demand
if c1 >0 and c1 >w, the good is a Giffen good
if c1 >0 and c1 <w, the good is normal with inelastic demand.
In our case, rice is a normal good with inelastic demand. Rice is a necessity. This evidence is in
line with the prior information that we gathered from the data. Let us see if we can do better by
controlling for taste changes due to rural-urban migration.
Specification 5: WL Engel curve with prices and demographics
• w = d0 + d00 ln d +d1 ln p + d2 ln (y/N)
Coefficients Standard
Error
t Stat
Intercept
0.389
0.025
15.801
urb/tot %
-0.088
0.009
-10.015
Prices $/t
0.021
0.004
5.302
Income/mth
-0.003
0.002
-1.397
R2=0.987, N. Obs. 21 mean(w)=0.052
27
Training Materials
Elasticity calculation
w = pq / y
→
ln w = ln p + ln q - ln y
→
ln q = ln w(p,y,d) - ln p + ln y
Ed = ∆ ln q / ∆ ln d = (∆ ln w / ∆ w) (∆ w / ∆ ln d ) = d00 / w = -1.686
Ep = ∆ ln q / ∆ ln p = (∆ ln w / ∆ w) (∆ w / ∆ ln p ) - 1 = d1 / w - 1 = -0.600
Ey = ∆ ln q / ∆ ln y = (∆ ln w / ∆ w) (∆ w / ∆ ln y) +1 = d2 / w + 1 = 0.933
where: (∆ ln w / ∆ w) = 1/w
(∆ w / ∆ ln d ) = d00
(∆ w / ∆ ln p ) = d1
(∆ w / ∆ ln y) = d2.
Note: both income and demographic effects are trends. Because of this relationship, the income
coefficient is no longer significantly different from zero at the 10 percent significance level.
Note: the results are consistent both with the expectations we built looking at the data and with
economic theory.
Next, an example applying a detailed demand analysis is presented using cross section
(household specific) data. This information allows us to move away from the representative
consumer assumption by estimating disaggregate demand parameters. Ultimately, to answer the
question: who is eating how much of what or, stated in other way, to identify the potential
beneficiaries of policy interventions.
So we leave the rice example to get back to it later when we will deal with the supply analysis.
•
Disaggregate Demand Analysis
This approach is needed for carrying out exact welfare analyses that requires a multi-equation
(multimarket) approach.
Plan of the exercise:
1) Estimate Engel Kernel regressions
2) Estimate a complete demand system equation by equation
3) restrictions from sum(share of cereals + meat + others) =1; (homogeneity)
4) test for income and demographic quadratic effects
5) role of demographic effects
6) Interpret
•
Engel curve analysis
Learning from the data. nonparametric Engel curves for Cereals, Meat and Other food items Italian Rural Household Data (ISMEA 1995). Observe the nonlinearities of cereals and meat. It
is likely that to interpret the data correctly we may need to add a quadratic income term to the
Engel curve. If we do not, we may not obtain estimates consistent with the theory.
The data set counts 1777 units of observations (households).
Next, we analyze empirically the behavioral response to income using the concept of Engel
curves describing how consumption, in terms of expenditure shares, varies with respect to the
logarithm of income. The analysis is carried out using a Gaussian kernel regression. It is a non
parametric technique that interprets the data without prior assumptions about the functional
form of the regression relationship. For this reason, this instrument is very helpful in providing
pre-estimation information about the most appropriate functional form to adopt.
28
Partial Equilibrium Analysis of Policy Impacts (part I)
An example may help clarifying the concept. If we run a linear regression, the prediction will be
of course linear. If we run a nonparametric regression, then the graph of the non parametric
regression shows if the relationship is linear or nonlinear and what type of nonlinearities.
The Engel curve nonparametric analysis is also an example of how to use graphical analysis as a
tool helping making specification decisions. If the Engel curves are nonlinear, then it can be
appropriate to include a quadratic income term in the demand function. A similar analysis can
be carried out for price and demographic effects.
Inspection of the kernel graphs presented below shows that the nonparametric Engel curves of
the Italian rural consumption of cereals, meat and other food are linear. On the basis of this
evidence, we may reasonably expect that the inclusion of a quadratic income term in the
parametric demand specification may not be significant. We will test this hypothesis at the end
of the exercise.
Engel curves of the system of Cereal (1), Meat (2) and Other Food (3) consumption (ISMEA 1995)
Kernel regression, bw = 1.8, k = 6
.208118
.207115
6.43053
Grid points
9.71057
1
Kernel regression, bw = 1.8, k = 6
.373115
.370864
6.43053
Grid points
9.71057
2
Kernel regression, bw = 1.8, k = 6
.422021
.418767
6.43053
Grid points
9.71057
3
29
Training Materials
A CROSS-SECTION APPLICATION: THE DISAGGREGATE DOMAND FOR FOOD IN ITALY
Prices in cross-section demand analysis are household specific because they are derived as
expenditures spent on a good i by household h divided by quantities consumed by household h.
We no longer have one aggregate price per year as in time-series analysis. Cross-section prices
are called unit values.
Bread & Cereals Share
1
0
0
Market Bread & Cereals Price
12459.3
However, as the graph relating the share spent on cereals and the cereal unit value shows that
prices are highly concentrated around an average price that can be compared to the aggregate
price for the survey year. Interestingly, the graph reveals that the unit value increases as the
share spent on cereal increases. This evidence supports the fact that unit values contain
information about the quality of the good.
•
Functional form:
wi = α i +
∑δ
k
ik
+ ln d k +
∑γ
j
ij
 y
ln p j + β i ln 
N
where i=j=1,..,N indexes the number of goods and prices, k=1,..,K indexes the number of
demographic variables. The above equation represents one equation of a N-system of equations.
30
Partial Equilibrium Analysis of Policy Impacts (part I)
The STATA program
#delimit;
/* ----------------------------------Estimate of Engel regressions
ISMEA data on rural consumption
Federico Perali
Cham, September 2002
--------------------------------------- */
capture clear;
capture log close;
set more off;
set mem 5m;
/* ----------------------------------- OPENING DATASET AND OUT FILE -----------------------*/
use C:\docs\Papers\papers02\siria02\fao_syria_9_02\exercises\ismea_fao1\ismea_syr.dta;
capture log using
C:\docs\Papers\papers02\siria02\fao_syria_9_02\exercises\ismea_fao1\disag_dem.log,
replace;
sum w_panal w_carni wother p_panal p_carni p_food red_glo_hh ireg area5 area3 sex edu
cprof age fsize nmales nfemales nchild spesa;
gen nord=1 if area3==1;
recode nord .=0;
gen centro=1 if area3==2;
recode centro .=0;
gen sud=1 if area3==3;
recode sud .=0;
gen spesa2=spesa*spesa;
gen nchild2=nchild*nchild;
/* take logs */
gen lp_panal=log(p_panal);
gen lp_carni=log(p_carni);
gen lp_food=log(p_food);
gen lspesa=log(spesa);
gen lspesa2=log(spesa2);
sum;
/* ------------------ Plan:------------------------------1) Estimate Engel Kernel regressions
2) Estimate a complete demand system equation by equation
3) restrictions from sum(share of cereals + meat + others) =1 (homogeneity)
4) income and demographic quadratic effects
5) role of demographic effects
6) Interpret
---------------------------------------------------------- */
/* Unit values */
graph w_panal p_panal;
/* Estimate Engel Kernel (Gaussian) regressions */
kernreg w_panal lspesa, b(1.8) k(6) np(200) saving(eng_cereal,replace) ;
31
Training Materials
kernreg w_carni lspesa, b(1.8) k(6) np(200) saving(eng_meat,replace) ;
kernreg wother lspesa, b(1.8) k(6) np(200) saving(eng_other,replace) ;
/* Complete Demand System (multimarket) equation by equation */
reg w_panal nchild age edu nord centro lp_panal lp_carni lp_food lspesa;
reg w_carni nchild age edu nord centro lp_panal lp_carni lp_food lspesa;
reg wother nchild age edu nord centro lp_panal lp_carni lp_food lspesa;
/* income and demographic quadratic effects */
reg w_panal nchild nchild2 age edu nord centro lp_panal lp_carni lp_food lspesa lspesa2;
log close;
Table 10 . The ISMEA 1995 data base on the Socioeconomic Conditions of Italian Agriculture Rural Consumption
Variable
Definition
Mean
Std. Dev.
Min
Max
w_panal
w_carni
wother
p_panal
p_carni
p_food
ireg
sex
edu
cprof
age
fsize
nmales
nfemales
nchild
spesa
nord
centro
sud
lp_panal
lp_carni
lp_food
lspesa
Cereal share
0.2077
Meat share
0.3721
Other Food share
0.4202
Price for Cereals (££/Kg)
900.53
Price for Meat (££/Kg)
6842.99
Price for Other Food
10993.91
(££/Kg)
Region (1, .., 20 regions)
11.2510
male
1.0456
education level
3.5768
professional condition
2.4513
age
51.1216
Family Size
3.4682
Number of Males
1.9387
Number of Females
1.5295
Number of Children
1.1660
Total Food Expend. (££000) 2885.77
1 if North
0.3832
1 if Center
0.2212
1 if South
0.3956
ln Price for Cereals
6.4519
ln of price for Meat
8.6938
ln of price for other food
9.2548
Ln tot Food exp
7.8201
0.1152
0.1653
0.1439
889.75
3741.05
3538.47
6.0202
0.2086
0.9443
2.8832
13.0113
1.4617
0.9669
0.9623
1.0957
1822.55
0.4863
0.4151
0.4891
0.8694
0.6858
0.3230
0.5227
Table 11. Estimates of the cereal equation (Obs: 1777, R2=0.716)
w_panal
Coef.
Std. Err.
t
P>|t|
nchild
age
edu
nord
centro
lp_panal
lp_carni
lp_food
lspesa
_cons
32
0.0016
-0.0001
0.0024
0.0067
-0.0061
0.0897
-0.0004
-0.0954
-0.0118
0.5977
0.0014
0.0001
0.0016
0.0032
0.0037
0.0017
0.0030
0.0067
0.0030
0.0544
1.1300
-0.7900
1.4900
2.1200
-1.6400
51.9500
-0.1400
-14.3000
-3.9700
10.9800
0.2590
0.4310
0.1380
0.0340
0.1020
0.0000
0.8890
0.0000
0.0000
0.0000
0.00
1
0.00
0.858
0.00
1
0.00
12459.29
0.00
42904.29
2000.00 43370.74
1.00
1.00
1.00
1.00
18.00
1.00
0.00
0.00
0.00
620.50
0.00
0.00
0.00
3.63
3.68
7.60
6.43
20
2
6
9
89
11
6
7
7
16491
1
1
1
9.43
10.67
10.68
9.71
[95% Conf.
Interval]
-0.0012
-0.0003
-0.0008
0.0005
-0.0133
0.0863
-0.0062
-0.1084
-0.0176
0.4910
0.0043
0.0001
0.0057
0.0130
0.0012
0.0931
0.0054
-0.0823
-0.0060
0.7045
Partial Equilibrium Analysis of Policy Impacts (part I)
Table 12. Estimates of the meat equation (Obs: 1777, R2=0.747)
w_carni
Coef.
Std. Err.
t
P>|t|
nchild
age
edu
nord
centro
lp_panal
lp_carni
lp_food
lspesa
_cons
-0.0039
-0.0002
0.0029
-0.0296
-0.0009
-0.0354
0.1843
-0.0421
0.0033
-0.6081
0.0019
0.0002
0.0023
0.0044
0.0051
0.0024
0.0041
0.0092
0.0041
0.0748
-2.0500
-1.4100
1.2900
-6.7800
-0.1800
-14.9200
45.1800
-4.5900
0.8100
-8.1300
0.0400
0.1590
0.1970
0.0000
0.8570
0.0000
0.0000
0.0000
0.4200
0.0000
[95% Conf.
Interval]
-0.0077
-0.0006
-0.0015
-0.0382
-0.0109
-0.0401
0.1763
-0.0600
-0.0047
-0.7547
-0.0002
0.0001
0.0073
-0.0210
0.0091
-0.0308
0.1923
-0.0241
0.0113
-0.4615
Table 13. Estimates of the other food equation (Obs: 1777, R2=0.491)
wother
Coef.
Std. Err.
t
P>|t|
[95% Conf.
Interval]
nchild
age
edu
nord
centro
lp_panal
lp_carni
lp_food
lspesa
_cons
0.0072
0.0007
0.0003
0.0339
0.0198
-0.0484
-0.1736
0.1605
0.0188
1.1986
0.0024
0.0003
-0.0054
0.0229
0.0070
-0.0543
-0.1838
0.1374
0.0085
1.0104
0.0025
0.0002
0.0029
0.0056
0.0065
0.0030
0.0052
0.0118
0.0052
0.0960
0.9600
1.5400
-1.8500
4.0800
1.0700
-17.8400
-35.1200
11.6900
1.6200
10.5300
0.3370
0.1230
0.0650
0.0000
0.2850
0.0000
0.0000
0.0000
0.1050
0.0000
-0.0025
-0.0001
-0.0110
0.0119
-0.0058
-0.0603
-0.1941
0.1144
-0.0018
0.8222
The interpretation of these results follows the same guidelines provided in the previous exercise.
It is left to the reader to compute elasticities and to interpret the statistical and economic
significance of the results.
•
Restrictions
When a complete demand system is estimated in share form, the following adding-up restriction
holds:
Σi wi =1.
Let us show how this restriction operates. Consider the following set of share demand equations
for k=1,..,K; i=j=1,..,N where K=2 and N=3 (1=cereals, 2=meat, 3=other food):
w1 = α10 + ( δ11 ln d1 + δ12 ln d2 ) + ( γ11 ln p1 + γ12 ln p2 + γ13 ln p3 ) + β1 ln y
w2 = α20 + (δ21 ln d1 + δ22 ln d2 ) + (γ21 ln p1 + γ 22 ln p2 + γ23 ln p3 ) + β2 ln y
w3 = α 30 + (δ31 ln d1 + δ32 ln d2 ) + (γ31 ln p1 +γγ32 ln p2 + γ33 ln p3 ) + β3 ln y
Notice that in the application K=5, but the line of the argument does not change. Now, the
adding-up restriction ∑i wi =1 implies the following:
w1 + w2 +w3 = (α10 + (δ11 ln d1 + δ12 ln d2 ) + (γ11 ln p1 + γ12 ln p2 + γ13 ln p3 ) + β1 ln y ) +
(α20 + (δ21 ln d1 + δ22 ln d2 ) + (γ21 ln p1 + γ22 ln p2 + γ23 ln p3 ) + β2 ln y ) +
(α30 + (δ31 ln d1 + δ32 ln d2 ) + (γ31 ln p1 + γ32 ln p2 + γ33 ln p3 ) + β3 ln y ) = 1.
Collecting terms, we obtain:
w1 + w2 + w3 = (α10 + α20 + α30 ) + (δ11 + δ21 +δ31 ) ln d1 + (δ12 + δ22 + δ32 ) ln d2 +
33
Training Materials
(γ11 + γ21 + γ31 ) ln p1 + (γ12 +γ22 + γ32 ) ln p2 + (γ13 + γ23 + γ33 ) ln p3 ) +
(β1 + β2 + β3 ) ln y = 1.
For the adding-up equality to hold, it must be that:
Σi αi =1; Σi γij = Σ βi = Σk δ k=0
Restrictions
Note, that given this set of restrictions, we may estimate only two equations and derive the
parameters of the omitted equation.
This property also guarantees the absence of what is called in the economic terminology “money
illusion”. That is, if all prices and income are increased by the same proportion, demand
remains unchanged. In other words, the homogeneity property holds and is implied by the
adding-up condition.
The homogeneity property also applies to demographic effects. If all prices, income, and
demographic effects are increased by the same proportion, demand remains unchanged.
Demographic effects do sum to 0 as well. The property provides an interesting interpretation of
demographic effects as demographic substitution effects. Suppose that the effect of the number
of children for the cereals and other food equation is positive. Because of the budget constraints
requiring that the shares sum to 1, it must be that larger families with more children buy
relatively less meat. The sign associated with the variable number of children must in fact be
negative in the meat equation.
Another implication of the homogeneity property is that the sum of all price, income and
demographic elasticities must sum to 0.
Table 14. Restrictions implied by adding up of the shares (homogeneity)
nchild
age
edu
north
centre
lp_panal lp_meat lp_food lspesa
_cons
δ
δ
δ
δ
δ
β
α
γ
γ
γ
w_cereal 0.0016 -0.0001 0.0024
0.0067 -0.0061 0.0897 -0.0004 -0.0954 -0.0118
0.5977
w_meat -0.0039 -0.0002 0.0029 -0.0296 -0.0009 -0.0354 0.1843 -0.0421 0.0033 -0.6081
w_other
0.0024 0.0003 -0.0054 0.0229
0.0070 -0.0543 -0.1838
0.1374
0.0085
1.0104
restrictio
ns
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
1.0000
mean(wcereal=0.208), mean(wmeat=0.372), mean(wother food=0.42)
•
Exercise calculation of Detailed Demand Elasticity
Table 15. Price, Income and Demographic Demand Elasticities - ISMEA Data
nchild
age
edu
north
centre
lp_panal lp_meat lp_food lspesa
Mean w
w_cereal 0.0088 -0.0231 0.0420
0.0124 -0.0065 -0.5680 -0.0020 -0.4591 0.9433
0.2077
w_meat -0.0123 -0.0317 0.0280 -0.0305 -0.0005 -0.0952 -0.5048 -0.1130 1.0088
0.3721
w_other 0.0065 0.0395 -0.0456 0.0208 0.0037 -0.1293 -0.4375 -0.6730 1.0202
0.4202
mean
1.166
51.1216
3.5768
0.3832
0.2212
The elasticities have been calculated as in the semi-log specification in share form of the timeseries exercise:
Recall: ln q = ln w(p,y,d) - ln p + ln y
Ep = ∆ ln q / ∆ ln p = (∆ ln w / ∆ w) (∆ w / ∆ ln p ) - 1 = γij / wi - 1
Ey = ∆ ln q / ∆ ln y = (∆ ln w / ∆ w) (∆ w / ∆ ln y) +1 = βi / wi + 1
•
34
A Note on the Calculation of the Demographic Elasticity
Partial Equilibrium Analysis of Policy Impacts (part I)
We need to consider two possible cases according to whether the demographic variables enters
in the demand function in the logarithms or in the anti logarithms :
1. Logarithm: ln d
Ed = ∆ ln q / ∆ ln d = (∆ ln w / ∆ w) (∆ w / ∆ ln d ) = δ ik / wi
2. Antilogarithm: d
a) ∆ ln q / ∆ d = (∆ q / ∆ d) (1 / q) = (∆ ln w / ∆ w) (∆ w / ∆ d ) = δ ik / wi
b) Ed = ∆ ln q / ∆ ln d = [(∆ q / ∆ d) (1 / q)] d = (δ ik / wi ) d.
As you can see by looking at table 10 reporting the descriptive statistics of the Italian household
survey, our case is case 2. The mean of the demographic variables are also reported at the
bottom of the table with the results.
The interpretation of these results is left to the reader.
•
Quadratic (nonlinear) demographic and income effects
Table 16. Estimates of the cereal equation with quadratic demographic and income effects (R2=.716)
w_panal Coef.
nchild
nchild2
age
edu
nord
centro
lp_panal
lp_carni
lp_food
lspesa
lspesa2
_cons
0.0053
-0.0011
-0.0001
0.0025
0.0070
-0.0061
0.0897
-0.0003
-0.0956
(dropped)
-0.0061
0.6021
Std. Err.
t
P>|t|
[95% Conf. Interval]
0.0030
0.0008
0.0001
0.0016
0.0032
0.0037
0.0017
0.0030
0.0067
1.7700
-1.4000
-0.8100
1.5300
2.2000
-1.6600
51.9500
-0.1100
-14.3400
0.0770
0.1600
0.4180
0.1260
0.0280
0.0970
0.0000
0.9090
0.0000
-0.0006
-0.0026
-0.0003
-0.0007
0.0008
-0.0134
0.0863
-0.0062
-0.1087
0.0111
0.0004
0.0001
0.0057
0.0132
0.0011
0.0931
0.0055
-0.0826
0.0015
0.0545
-4.1100
11.0500
0.0000
0.0000
-0.0091
0.4952
-0.0032
0.7090
The results show that there are no economies of scale in cereal consumption due to the presence
of extra children because the parameter associated with the quadratic demographic variable
nchild2 is not significantly different from zero. The presence of a quadratic income term lspesa2
is redundant to the point the lspesa is dropped from the estimation. This evidence favors the
hypothesis of linearity of the income effect as we were expecting from the results of the
nonparametric Engel curve analysis. Overall, the nonlinear specification adds no further
explanatory power with respect to the linear model. The R2 of the linear and nonlinear
specification of the cereal equations are virtually the same.
2.2.
•
Advanced Demand Analysis
The Consumer Maximization Problem
• Let U(x) be an ordinal utility function representing household preferences (how a
consumer trades one good for another as described by an indifference curve) increasing
and concave in the quantities x.
• In general, the consumer maximize his own utility subject to the budget (total
expenditure) constraint:
35
Training Materials

V ( p, y ) = Max x U ( x) subject to :

∑p x
i i
i

= y

where V(p,y) is the observable indirect utility function. The solution of the maximization
problem by solving the Lagrangean expression gives the Marshallian demands
x*(p,y)
Note that when we substitute the optimal choices x*(p,y) we obtain the indirect utility V(p,y).
Note further that the Max of U(x) subject to total expenditure is equivalent to minimize total
expenditure subject to a utility level u=U(x).
2.2.1. Duality Theory
A demand system derived from a known utility (preference) allows us to recover the
unobservable cost function using the theory of duality.
The EXPENDITURE function C ( p, u ) is the minimum expenditure that can be attained while
maintaining a given level of utility:
Y = C ( p, u , d ) + Min{p ' x subject toU − U ( x, d )}
The solution to this minimization problem gives the Hicksian demand functions:
Xc (p, u) (which is not observable)
Note: the expenditure function C(p,u,d) is increasing in p, u, homogenous of degree 1 and
concave in p; demands Xc (p, u) are homogenous of degree 0.
The INDIRECT UTILITY function V(p,y)
C(p,u)=Y and V(p,y) are inverse functions, where u=V.
•
Why own price elasticities must be negatively sloped?
Answer: Because the cost function is concave
DERIVATIVE RESULTS
Shephard’ s Lemma to obtain Hicksian demands from the cost function
∂C ( p, u )
c
= x i ( p, u )
∂pi
Or, in the logarithms:
∂ ln C ( p, u ) ∂C ( p, u ) p i x i p i
=
= wi
=
∂p i
∂ ln p i
C
y
Roy’s identity to obtain Marshallian demands from the indirect utility function
−
∂V ( p, y ) ∂p i
= x i * ( p, y )
∂V ( p, y ) ∂y
Note: the price elasticities [( ∂ x/ ∂ p) p/x] are the matrix of second derivatives of the cost
function multiplied by p/x. The component ( ∂ 2 C(u,p) / ∂ p2) = ( ∂ xc/ ∂ p) is a symmetric
negative matrix. This explains why the own price elasticities must be negative!
•
The Slutsky equation: price (substitution) and income effects
The reader may also refer above were substitution and income effects have been studied using a
36
Partial Equilibrium Analysis of Policy Impacts (part I)
graphical approach. The illustration here is analytical.
By duality theory, at the optimum, holds the following equality:
Xc (p,u) = X*(p,y)=X*(p,C(p,u))
Non observable
Observable
By differentiating with respect to p:
∂x c ( p, u ) ∂x * ( p, y ) ∂x * ( p, y ) ∂C ( p, u )
+
=
∂y
∂p i
∂p i
∂p i
or, using the derivative property (Shephard’s Lemma):
Compensated
non compensated
∂x c ( p, u ) ∂x * ( p, y ) ∂x * ( p, y )
x * ( p, y )
=
+
∂p i
∂p i
∂y
Price
Income effect
(Non observable)
Observable
which is the Slutsky matrix of first derivatives of the Hicksian quantities (= the elasticities if we
take logarithms, and derive the matrix of second derivatives of the cost function). As before, it
must be symmetric and negative (that is, all the elements in the diagonal must be negative).
Note: the hicksian quantities are compensated by the income effect. Hicksian effects are not
directly observable but can be estimated as the sum of the observable Marshallian price effect
and the income elasticity.
The total effect of a price change can be decomposed in two components:
(1) the income effect, which is the change in quantity demanded resulting from
a change in income holding prices constant;
(2) the substitution effect, which is the change in quantity demanded resulting
exclusively from a relative price change ( a change in terms at which one product
can be exchanged for another) after compensating the consumer for the change
in real income.
Intuitively, when a price changes, everything else held constant, the real purchasing power of
the or real income of the consumer also changes. For example, if a product price falls, real
income rises to maintain indirect utility constant and vice versa.
2.2.2. Empirical Implementation: The Almost Ideal Demand System (AIDS)
Let us choose a general representation of preferences (They are unobservable since utility
cannot be observed directly, but we estimate observable demands and then we recover C(u,p)
and U(x), V(p,y) by going back to the preferences we derived demand from)! So,
Y=C(u,p,d)=(A(p)uB(p))D(p,d)
where u is a utility level, p are prices, and d are demographic variables. In the logarithms:
lnC(u,p,d)=(lnA(p)+B(p) lnu)+lnD(p,d)
We need first to specify a functional form also for the index functions A(p) and B(p):
∑α
ln A( p ) = α 0 + .5
i
i
ln p i +
∑∑ γ
i
ij
ln p i ln p j
j
37
Training Materials
B( p) = β 0
∏ p β , and ln D( p, d ) = ∑ β
i
i
i
ln p i ln d k
i
i
So, let us derive the demand in share form w= (p x)/y by taking the derivative with respect to
prices of ln C(u,p,d) by using the derivative property (Shephard’s Lemma):
∂ ln C (u , p, d )
= wi
∂ ln p i
and after substitution for u, we obtain:
wi = α i +
∑δ
k
ln d k +
k
∑γ
j
ij
 y 

ln p j + β i ln
 A( p ) 
which is the AIDS system of demand equations.
Observe that they are like Engel functions with the inclusion of prices!.
•
Estimation
The AIDS specification provides the basis for an econometric estimation of the demand
parameters
Note: the specification of ln A(p) makes the model non linear. We can linearize the model using
Stone’s approximation:
ln A(p) ≈ Σ w i ln p i = P
i
Stochastic Specification:
wi = α i +
∑δ
k
ln d k +
k
∑γ
j
ij
 y 
 + ei
ln p j + β i ln
 A( p ) 
where e is a random variable (error) with mean zero and finite variance.
•
Theoretical Restrictions
The structure about the shape of C(u,p,d) (must be concave) and the requirement of
homogeneity 1 in prices implies the following restrictions:
∑α
i
= 1;
i
∑γ
ij
=
i
∑γ
ij
j
=
∑β
i
= 0,
i
∑δ
j
=0
j
and symmetry: γij = γji .
Note 1. The theoretical restrictions implied by consumer theory: homogeneity and symmetry
can be tested using standard statistical tests.
Note 2. Because by definition the shares sum to 1, the dependent variables are linearly
dependent, so the variance of the error e is singular. Hence, when estimating we must
drop one equation. The parameters from the equation dropped can be recovered from
the restrictions. The parameter estimates are invariant to the equation dropped if
maximum likelihood is used.
•
Elasticities in the AIDS model
Matrix of Own price elasticities
E ii = −1 +
38
γ ij
wi
− βi
Partial Equilibrium Analysis of Policy Impacts (part I)
Matrix of Cross price elasticities
E ij =
γ ij
wi
−
βi
wi
wj
Vector of income elasticities
ηi = 1 +
βi
wi
Note: if β < 0 => the good is a NECESSITY
If β > 0 => the good is a LUXURY
Trick to derive the elasticities: Try it!
Let w=pq/y → ln w = ln p + ln q - ln y → ln q = ln w -ln p + ln y.
Then,
∂ ln q ∂ (ln w − ln p − ln y )
=
∂ ln p
∂ ln p
price elasticity
∂ ln q ∂ (ln w − ln p − ln y )
=
∂ ln y
∂ ln y
income elasticity
2.2.3. Cost of Living Indexes and Compensating Variations
The Cost of living index measures the relative costs of reaching a given standard of living under
two different situations.
The most convenient scale with which to measure welfare is the expenditure necessary at
constant prices to maintain the various welfare levels being considered.
These concepts which use money to measure changes in welfare are limited to the measurement
of quantities and prices that arise in the market. So, we do not consider goods that are
important for consumers’ well being but are not purchased through the market. Examples are
health care, natural areas, clean air, or the smile of a child in a household.
Definition. The Cost of Living Index: A cost of living index (CLI) is the ratio of the
minimum expenditure necessary to reach the reference indifference curve at the two
sets of prices.
Hence, if ur is the label of the indifference curve taken as reference, the true CLI is:
P(u, p1,p0) = C(u, p1) / C(u,p0)
•
The Compensating Variation
Since the Hicksian demand functions are the derivatives of the cost function, integration gives
the difference in costs of reaching the same indifference curves at two different price vectors.
This is an exact measure of consumer surplus.
Instead of using index numbers based on ratios, we have compensating variations (CV) in terms
of differences expressed as money measures rather than pure numbers. This is usefully to
compare welfare effects of government policies.
Definition. Compensating Variation: CV is the minimum amount by which consumers
should be compensated after a price change in order to be as well off as before the
change:
39
Training Materials
CV = C(u, p1) - C(u,p0)
The GAUSSX program that follows implements all the above concepts applied to a time series
data base of consumption data for the United States. As an exercise interprets the results.
2.2.4. A GAUSSX program for advanced demand analysis
CREATE (a) 1950 1992 ;
? We are interested in analyzing US Consumption behavior. So, we obtained
? time series data on aggregate consumption during the period 1950-1992.
? The data includes price (pi) and quantity (Qi) information on 3
? commodity groups: 1=food; 2= durable goods; and 3= non durable goods.
LOAD YEAR =
1950 1951 1952 1953
1960 1961 1962 1963
1970 1971 1972 1973
1980 1981 1982 1983
1990 1991 1992 ;
1954
1964
1974
1984
1955
1965
1975
1985
1956
1966
1976
1986
1957 1958 1959
1967 1968 1969
1977 1978 1979
1987 1988 1989
LOAD P1=
25.4 28.2
30.0 30.4
39.2 40.4
86.8 93.6
132.4 136.3
28.7 28.3
30.6 31.1
42.1 48.2
97.4 99.4
137.9 ;
28.2 27.8 28.0 28.9 30.2 29.7
31.5 32.2 33.8 34.1 35.3 37.1
55.1 59.8 61.6 65.5 72.0 79.9
103.2 105.6 109.0 113.5 108.2 125.1
LOAD P2=
34.9 37.9
38.1 38.1
44.1 46.0
83.0 89.6
113.4 116.0
38.0 37.7
38.5 38.6
46.9 48.1
95.1 99.8
118.6;
36.8
39.0
51.5
105.1
36.1 36.1 37.2 37.8 38.4
38.8 38.9 39.4 40.7 42.2
57.4 60.9 64.4 68.6 75.4
106.8 106.6 108.2 110.4 112.2
29.7
33.2
54.0
102.5
29.5 29.9 30.9 31.7 31.5
33.8 35.1 35.7 37.1 38.9
58.3 60.5 64.0 68.6 77.2
104.8 103.5 107.5 111.8 118.2
LOAD P3=
27.0 29.5 29.8 29.7
32.0 32.2 32.5 32.9
40.8 42.1 43.5 47.5
87.6 95.2 97.8 99.7
126.0 130.3 132.8;
LOAD Q1 =
2.122 2.152
2.753 2.789
3.625 3.651
3.937 3.924
4.568 4.559
2.233 2.311 2.368 2.467
2.846 2.877 3.003 3.136
3.764 3.653 3.595 3.653
3.963 4.086 4.168 4.271
4.595;
2.550
3.224
3.831
4.374
2.598
3.293
3.906
4.411
2.579 2.717
3.444 3.517
3.897 3.917
4.514 4.517
LOAD Q2 =
0.882 0.797 0.771 0.867 0.871 1.077 1.057 1.067 0.984 1.114
1.141 1.100 1.220 1.341 1.456 1.636 1.760 1.7922 1.990 2.042
1.934 2.113 2.360 2.579 2.388 2.339 2.627 2.835 2.949 2.840
2.560 2.550 2.486 2.755 3.024 3.304 3.654 3.731 3.959 4.094
4.0942 4.002 3.914;
LOAD Q3 =
3.637 3.699 3.849 3.966 4.028 4.227 4.373 4.436 4.470 4.714
4.782 4.889 5.040 5.147 5.414 5.677 5.940 6.076 6.334 6.483
40
Partial Equilibrium Analysis of Policy Impacts (part I)
6.627 6.729 7.016 7.148 7.051 7.136 7.468 7.663 7.893 7.944
7.795 7.817 7.896 8.202 8.517 8.772 9.200 9.405 9.604 9.725
9.718 9.397 9.220;
? Data generation section
GENR E1 = P1.*Q1;
GENR E2 = P2.*Q2;
GENR E3 = P3.*Q3;
GENR II = E1+E2+E3;
GENR W1= E1./II;
GENR W2= E2./II;
GENR W3= E3./II;
GENR LNII=LN(II);
GENR LNP1=LN(P1);
? expenditure on food
? expenditure on durables
? expenditure on other goods
? Total expenditure
? food share
? durables share
? other goods share
? ln of tot expenditure
? ln of p1
? This is the Stone's Price Index to linearize the model
GENR LP = W1.*LN(P1)+W2.*LN(P2)+W3.*LN(P3);
? Descriptive statistics of all the data
cova(d) P1 P2 P3 Q1 Q2 Q3 W1 W2 W3 E1 E2 E3 II LP ;
? Let us learn from the data non parametrically,
? that is, without choosing a functional form for f: w=f(p,y,d)
h = 0;
genr w1s = sortc(w1,1);
? we sort the data to order it for plotting
genr w1shat = npe(w1s,w1s,h); ? we smooth the series
plot (p,d) w1s w1shat;
? we plot the food share and food price
h=0;
genr p1s = sortc(p1,1);
genr p1shat = npe(p1s,p1s,h);
plot (p,d) p1s p1shat;
? I. non parametric relation (npr) between w and p
npr (d,p) w1 lnp1;
? command asking for NonParametric Regression
replic = 50;
window = 2;
oplist = cv direct;
? or cv print (fourier default)
title = "Non parametric w and ln p relation";
forcst yhatwp;
? nonparametric forecast
oplist = nocv;
graph (p,h,m) yhatwp lnp1;
? II. relation between w and y (The Engel relation)
npr (d,p) w1 lnii;
? command asking for NonParametric Regression
replic = 50;
window = 2;
oplist = cv direct;
? or cv print (fourier default)
title = "Non parametric Engel curve";
forcst yhateng;
? nonparametric forecast
oplist = nocv;
41
Training Materials
graph (p,h,m) yhateng lnii;
? Estimation of a complete demand system
PARAM a0 a1 a2 a3 a11 a12 a13 a21 a22 a23 a31 a32 a33 b1 b2 b3 ;
? Unrestricted model
FRML eq1a W1= a1+ a11*LN(P1) + a12*LN(P2) + a13*LN(P3)
+b1*(LN(II)-LP);
FRML eq2a W2= a2+ a21*LN(P1) + a22*LN(P2) + a23*LN(P3)
+b2*(LN(II)-LP);
NLS (i,d) eq1a eq2a ;
LLU = LLF ;
? Homogeneity restriction (a13=-a11-a12) imposed
FRML eq1b W1= a1+ a11*LN(P1) + a12*LN(P2) + (-a11-a12)*LN(P3)
+ b1*(LN(II) -LP);
FRML eq2b W2= a2+ a21*LN(P1) + a22*LN(P2) + (-a21-a22)*LN(P3)
+ b2*(LN(II)-LP);
NLS (i,d) eq1b eq2b ;
LLH = LLF ;
? Symmetry restriction (a12=a21) imposed
FRML eq1c W1= a1+ a11*LN(P1) + a12*LN(P2) + a13*LN(P3)
+b1*(LN(II)-LP);
FRML eq2c W2= a2+ a12*LN(P1) + a22*LN(P2) + a23*LN(P3)
+b2*(LN(II)-LP);
NLS (i,d) eq1c eq2c ;
LLS = LLF ;
? Both homogeneity and symmetry restrictions imposed
FRML eq1d W1= a1+ a11*LN(P1) + a12*LN(P2) + (-a11-a12)*LN(P3)
+b1*(LN(II)-LP);
FRML eq2d W2= a2+ a12*LN(P1) + a22*LN(P2) + (-a21-a22)*LN(P3)
+b2*(LN(II)-LP);
NLS (i,d) eq1d eq2d ;
LLHS = LLF ;
? Hyppothesis testing
TEST
TEST
TEST
TEST
LLU
LLU
LLU
LLH
LLH ; METHOD = LRT ; order = 2;
LLS ; METHOD = LRT ; order = 1;
LLHS ; METHOD = LRT ; order = 3;
LLHS ; METHOD = LRT ; order = 1;
? recovery of the parameters of the omitted equation
? (see your note: implementation)
a3 = (1 -a1 -a2); ? from the adding up to 1 of the shares
b3= (-b1 -b2);
a31= (-a11 -a12);
? from homog of degree 0 in prices
a32= (-a21 -a22);
42
Partial Equilibrium Analysis of Policy Impacts (part I)
a33= (-a13 -a23);
? Computation of the std errors of the omitted parameters
? ANALYZ (p,d,v) eqp1 eqp2 eqp3 eqp4 eqp5;
cova (p,d) w1 w2 w3;
fetch w1 w2 w3 II p1 p2 p3;
? Computation of the elasticities at the data means
w1m=meanc(w1);
w2m=meanc(w2);
w3m=meanc(w3);
ey1=1+b1/w1m;
ey2=1+b2/w2m;
ey3=1+b3/w3m;
e11=-1+a11/w1m-b1;
e22=-1+a22/w2m-b2;
e33=-1+a33/w3m-b3;
? income elasticities
? own price elasticities
e12=a12/w1m-b1*(w2m/w1m); ? cross price elasticities
e13=a13/w1m-b1*(w3m/w1m);
e21=a12/w2m-b1*(w1m/w2m);
e23=a23/w2m-b1*(w3m/w2m);
e31=a31/w3m-b1*(w1m/w3m);
e32=a32/w3m-b1*(w2m/w3m);
eyy=ey1|ey2|ey3;
@@ "vector of income elasticities" eyy;
mat_e=e11~e12~e13|e21~e22~e23|e31~e32~e33;
@@ "matrix of price elasticities" mat_e;
? The Slutsky equation in elasticity form: matrix of compensated elasticities
? That is, compensated by the income effect!
? e11_c=e11+ey1*w1;
? compensated price elasticity
? e12_c=e12+ey1*w2; ....
? w1*e11_c;
? own term of the Slutsky equation
? The recovery of the cost function
A_p=exp( a1*ln(p1)+a2*ln(p2)+a3*ln(p3)+
1/2*( a11*ln(p1).*ln(p1)+a12*ln(p1).*ln(p2)+a31*ln(p1).*ln(p3)+
a12*ln(p2).*ln(p1)+a22*ln(p2).*ln(p2)+a32*ln(p2).*ln(p3)+
a31*ln(p3).*ln(p1)+a32*ln(p3).*ln(p2)+a33*ln(p3).*ln(p3) ) );
B_p=(p1^b1).*(p2^b2).*(p3^b3);
lnu=(ln(meanc(II))-ln(meanc(A_P)))/meanc(B_p);
? the mean level of u
Costf=exp(ln(A_p)+B_p*lnu);
@@ "The mean of total expenditure=income" meanc(costf);
? The cost of living
43
Training Materials
Output
Estimated Elasticities and Welfare Analysis
Vector of income elasticities
ey1 0.7554 Food
ey2 1.6398 Durables
ey3 0.9164 Other goods
Matrix of Marshallian own and cross price elasticities
Food Durables Other goods
Food
-0.483 -0.104 -0.184
Durables
-0.124 -0.553 -0.174
Other goods
-0.121 -0.110 -0.663
Note 1: all own price elasticities in the diagonal are negative as required by the theory!!
Note 2: goods can be defined as
Complements: if the cross price elasticity is < 0.
(Ex. coffee and sugar).
Substitutes: if the cross price elasticity is > 0.
(Ex. butter and margarine).
The mean of total expenditure=income
C(u,p) = Y = 847.04417807
The cost of living index of 1951/1950
C(u,p_1951)/C(u,p_1950) = 1.09099598
The compensating variation
CV=C(u,p_1951)-C(u,p_1950) = 37.11242740
44
Partial Equilibrium Analysis of Policy Impacts (part I)
Chapter 3 - Supply Response
3.1.
Introductory Supply Analysis
The supply response for crops can be studied for yield, area, or output in the short or in the long
run.
Definition. Short run: producers can alter only the level of variable factors. Land and
household labor supply, for example, cannot be changed.
Definition. Long Run: producers can vary all factors of production and producers and
resources can enter or leave the industry.
The elasticity of supply response is smaller in the short run since fixed factors are not a decision
variable. Long run elasticities of supply response can be very high since fixed factors become
increasingly variable and can be reallocated. Further, if output grows as a consequence of new
entrant firms or farms, then an expansion of supply can be achieved without large price rises
and the supply curve can be quite flat.
In general, the elasticity of yield is smaller and more unstable than the elasticity of area and, of
course, of output which is the sum of the two.
•
Risk and Uncertainty
Farmers operate under uncertainty with respect to yields, and prices of inputs and outputs.
Failing to recognize this fact in the modeling strategy may result in inadequate analysis.
The perception of uncertainty is subjective. For this reason, when uncertainty is taken into
account, the utility of profit rather than profit itself, is the object of interest. Therefore, in un
uncertain environment where outcomes realize with a probability of occurrence, the producer
maximize expected utility. Utility describes the farmer’s attitude to the variability of outcomes
(risk).
In general, the farmer in developing countries is risk averse. The farmer is willing to pay in
order to avoiding risk. If farmer is risk averse and the product price is uncertain, then a smaller
level of output is produced than under perfect certainty. The farmer does not equate marginal
cost to (average) price but produce at a point where marginal cost is less than that price. The
difference is the equivalent amount of money the farmer is willing to pay to “buy” certainty.
3.1.1.
Approaches to the estimation of supply response
There are two approaches to the estimation of supply response:
A) Indirect Structural Form Approach
we first estimate the structure (a profit or a cost function), then we derive the input
demand and output supply response -This approach is more theoretically rigorous but
fails to take into account the partial adjustment in production and the mechanism used
by farmers in forming expectations.
45
Training Materials
B) Direct Reduced Form Approach
Direct Estimation of the Supply Response including Partial Adjustment and
Expectations Formation -The Nerlovian Model
We study method B.
The Nerlovian Models of Supply Response
How do agricultural producers make decisions?
In agriculture the OBSERVED PRICES are known after production has occurred, while planting
decisions are based on the prices farmers EXPECT to prevail later at harvest time. Because of
this time lag, it is crucial to model the formation of expectations in the analysis of agricultural
supply response.
Similarly, OBSERVED QUANTITIES may differ from the DESIRED ones because of the
adjustment lags of variable factors. When a price changes, it may take several years (or never)
before farmers can reach their desired production patterns given the new price setting.
The specification of these adjustment lags is the essence of Marc Nerlove’s (University of
Maryland) model.
•
The Nerlove specification
Let us suppose that we are interested in the supply response of a DESIRED area to be allocated
to a crop at time t. The relation of interest can be expressed as a function of expected relative
prices and exogenous shifters as follows:
qtd = α1 + α2 pte + α3zt + ut
where:
qtd = desired cultivated area in period t
pte = a set of expected relative prices including the p of the crop, competing crops, and
factor prices
zt = set of exogenous shifter (analogous to demographic information in demand analysis)
such as weather, political factors, farmers education, technical change
ut = accounts for unobserved random factors; it has an expected value of zero
α2 = is the long run coefficient (or elasticity in a double log model) of supply response.
Notation: the d superscript means desired, the e superscript means expected.
Note: both expected prices and desired quantities are not observable.
•
The AREA adjustment
Full adjustment of the DESIRED allocation of land may not be complete in the short run.
Therefore, the ACTUAL adjustment in area will only be a fraction δ of the desired adjustment:
qt -qt-1 = δ ( qtd -qt-1 ) + vt,
where:
qt = actual area planted of the crop,
δ = PARTIAL ADJUSTMENT coeffic. with 0 ≤ δ ≤ 1,
vt = spheric random error (with mean zero and finite variance).
•
The PRICE adjustment
The farmer’s expected price at harvest time cannot be observed. So, we have to formally
46
Partial Equilibrium Analysis of Policy Impacts (part I)
describe how decision makers form expectations based on the knowledge of actual and past
prices and other observable information. We may think that farmers maintain in their memory
the magnitude of the mistake they made the previous period and LEARN by adjusting the
difference between actual and expected p in t-1 by a fraction γ:
pte -pt-1e = γ ( pt-1 -pt-1e ) + wt,
with 0 ≤ γ ≤ 1, or
pte = γ pt-1 + (1 - γ ) pt-1e + wt,
where:
pt-1 is the price at the time decisions are made, γ is the ADAPTIVE expectation coefficient, and
wt is a spheric random error, that is with zero expectation. An alternative interpretation of this
learning process is that the expected price pte is the weighted sum of all past prices of which
farmers have memory with a geometrically declining set of weights:
pte = γ
∞
∑ (1-γ)
i-1
pt-i .
i
Note: for i=1, .., 3, pte = γ pt-1 +γ (1 - γ ) pt-2 + γ (1 - γ )2 pt-3.
•
The REDUCED form
Since pte and qtd are not observable, we need to derive an estimable expression. Take the pte
equation expressed as the weighted sum of all past prices and insert in qtd. Then insert the qtd
equation into the partial area adjustment equation qt. Rearranging we obtain the following
ESTIMABLE reduced form:
qt = π1 +π2 pt-1 + π3 qt-1 + π4 qt-2 +π5 zt +π6 zt-1 + et,
where:
π1 = α1 δ γ,
π2 = α2 δ γ, short run coefficient (elasticity) of supply response,
π3 = (1-δ)+(1-γ),
π4 = -(1-δ)(1-γ),
π5 = α3 δ,
π6 = α3 δ(1-γ), and
et = vt-(1-γ)vt-1+δut-δ(1-γ)ut-1+α2δwt.
The reduced form is overidentified: there are 6 reduced form coeff. π and 5 structural
parameters α1, α2, α3, γ, and δ. So for a unique solution we need an extra restriction:
π62 -π4π52 + π3 π5 π6 = 0.
•
Derivation of the reduced form
Objective: Estimate Short run and Long run supply elasticities
Structural Form (non estimable):
1)
qtd=α1+α2 pte+α3zt+ut;
α2 = long run elasticity
The desired area equation as a PARTIAL ADJUSTMENT process:
2)
qt -qt-1 = δ ( qtd -qt-1 ) +vt
The expected price equation as an ADAPTIVE process:
47
Training Materials
3)
pte = γ
∞
∑ (1 - γ )
i-1
pt-i
i
Estimable Reduced Form obtained after substitution of (1) and (3) into (2):
qt = π1 +π2 pt-1 + π3 qt-1 + π4 qt-2 +π5 zt +π6 zt-1 + et ; π2 = short Run elasticity
•
Estimation and Recovery of the Structural Form
The model should be estimated by Maximum Likelihood techniques correcting for serial
correlations in the error term. The structural coefficients can be recovered using the following
set of equalities:
δ2 + (π3 - 2) δ + 1 - π3 - π4 = 0, ≥ δ=1- π3 using –b ± ( b2-4ac)½ /2a and π4=0
γ= 1 + π4/(1-δ),
α1 = π1 / δγ,
α2 = π2 / δγ,
the long run coef. (elasticity) of supply response,
α3 = π5 / δ.
Note that the short run price response π2<α2= π2 / δγ , the long run price response since both δ
and γ are less than 1, as expected!
•
Simplified models with either no Partial Adjustment or no Price Expectations
The Nerlove models admits several estimable versions depending on the economic and
institutional situations that we are studying. These situations help model identification (the
possibility to uniquely determine the parameters of the structural form) because we have either
no partial adjustment δ = 1 or no expectation formation γ=1.
In some cases, areas are fully adjusted in a year span, implying qtd=qt and δ = 1. In cases when
administered prices are announced at planting time so are known with certainty by farmers
implying pte=pt-1 and γ = 1.
Note: when γ = 1, the model is exactly identified. In all models, though, with either γ = 1 or δ = 1,
the long run supply elasticity of supply response is
α2 = π2 / δγ= π2 / (1- π3)
Note: in all these models pt-1, qt-1 are exogenous variables.
•
Critique of the Nerlovian Model
Nerlove’s adaptive expectations are based on the history of past prices with weights declining
geometrically over time:
pte = f ( past prices ) = γ
∞
∑ (1 - γ )
i-1
pt-i
i
This approach has been criticized because:
A. Price weights are ad hoc instead of being determined optimally or actually
corresponding to the length of farmers’ memory;
B. Price predictions do not take fully account of the information available to farmers:
1) price predictions are formed using structural market information, that is
information on both supply and demand;
2) farmers may use available forecasts about prices and about exogenous variables
48
Partial Equilibrium Analysis of Policy Impacts (part I)
affecting the process such as rain;
3) farmers may take into account anticipated policy changes affecting price
formation: Lucas’ critique (Lucas, 1976).
The Rational Expectation Model
Rational expectations (Muth, 1961) reproduce the process of formation of expectations based on
both sides of market information. We assume that farmers do not make decisions only on past
prices, but forecasts are based on
a) knowledge of a structural model of price determination, that is they think as if they
understood the market,
b) exogenous forecasts of the independent variables in the model, and
c) expectations about the policy instruments in the model. Rational expectations thus
use the model prediction’s of the endogenous variables, including PRICES, to form
expectations:
pte = f ( model prediction I exogenous variables forecasts and expected policy changes ).
•
Critique of the Rational Expectation Model
In general, the approach relies too heavily on rational behavior in forming expectations and on
an easy access to information.
A.
Agents may not use all the information that is potentially available to them, because
acquiring it is costly (Farmers’ organization play an important role in distributing
information and forming farmers in interpreting it!). Sometimes, though they are fully
informed still do not act rationally (in the sense, as expected) because they may face
constraints not revealed in market behavior.
B.
Agents may not use the information as intelligently as the model. They do not know the
model, but interpret the market signal subjectively. So the same market signal may be
interpreted differently by farmers. After all, they have incomplete information. At the
same time, farmers subjective predictions may perform better than the model ....
C.
Agents may not know how to forecast the exogenous variables and policy changes.
Empirically rational expectations has not proved its superiority to ad hoc specifications such as
the Nerlovian adaptive hypothesis. But adaptive expectations are too naive. We may improve
our formal understanding of the formation of expectations if we can account for how farmers
think about the future, the cost of accessing information and their ability to process it, quality
and expected benefits from the use of information.
3.2.
An Introductory Exercise
3.2.1. The Colombian Rice Economy: Supply side
Purpose of the Exercise:
Obtain an unbiased estimate of the price elasticity of supply to complete the market model of the
Colombian rice industry.
Farmers do not respond only to prices. The state of technology, the natural, economic and
institutional environment should be taken into account as well. Yield and price (of both outputs
and inputs) uncertainty are also very important. Rational farmers are in general risk averse and
may be very prudent towards innovations that may increase uncertainty. Small farmers are
more sensitive to risk with respect to rich farmers. In the case of Colombia, rice producers in
49
Training Materials
rainfed areas where more cautious in adopting because the modern varieties were not
appropriate for their environment, and therefore more risky, and experienced a more difficult
access to information. Let us see what the data are going to tell us about our conjectures.
In our data base we do not have production or price information about competing crops such as
corn or sorghum or other crops. Therefore, we will first estimate a model with no behavioral
assumptions with just price information. We then add more information about technical change
and the proportion of irrigated area. Our specification exercise for the selected double-log
functional form, that was maintained the same for all models, continues by adopting a Nerlovian
specification. We then estimate the same specification both for the rainfed and irrigated
production. The interpretation and comparison of the results are left to the reader as an
exercise.
Plan of the exercise:
Cobb-Douglas (univariate double
log regression)
ln q = ln B + B1 ln p = B pB1
Cobb-Douglas with technical change
and other shifters
ln q = (b0 + b01 ln %Irr + b02 DT) + b1 ln p
Nerlove model: Dynamic CobbDouglas with technical change and
other shifters
ln q= (b0 + b01 ln %Irr + b02 DT) + b1 ln p(-1)+ b2 ln q(1)
where D T is a linear trend capturing the effect of technical change and %Irr is the proportion of
irrigated rice production. Now, let us refer back to our Colombian data base.
Regression of Rice Production | Price (Total Production)
Let us specify a double logarithmic specification:
• ln q = ln B + B1 ln p = B pB1
Coefficien Standard
Error
ts
t Stat
Intercept 24.0887
4.1898
5.7494
Price
0.5848
-2.5779
-1.5076
R2=0.259
The price explains only 26 percent of the observed variation of supply (R2=0.259). The price
coefficient is significant, but has the wrong sign. The elasticity (-1.51) is quite high.
The negative sign of the price coefficient is not surprising if we remind the “treadmill” effect that
we found exploring the data. Quantity produced was indeed increasing in spite of decreasing
prices. We need to control for other effects such as technical change and the differential rate of
adoption between rainfed and irrigated producers.
As we learned from the data, we can now improve our specification by adding the variable
describing the effect of linear technical change and of other shifters such as the proportion of
irrigated production.
Regression of Rice Production | Price controlling for Technical Change
Recalling our observations about the characteristic of technical change when we were
familiarizing with the data, we found reasonable to represent technical change as a linear trend,
that is, increasing at a constant rate.
50
Partial Equilibrium Analysis of Policy Impacts (part I)
Technology acted also as a structural break. When modern varieties were introduced, the
technology previously used to produce rice changed radically especially in the irrigated sector.
To account for the effect of the adoption of new varieties after 1966, we may construct a dummy
taking the value of zero before 1966, and 1 after 1966 capturing the effect of the adoption of new
varieties after 1966. The variable used in the regression analysis DT as a proxy for technical
change is the interaction of the dummy and the trend because, as a matter of fact, technical
change occurred only after the institutional change also took place, that is the creation of the
Colombian National Research Institute and of CIAT that had a key role in making the modern
varieties available to farmers, and, of course, after the adoption. Therefore, the variable DT is a
vector with elements = 0 until 1966, and sequence from 1 to 8 with increments of 1 from 1967 to
1974.
• ln q = (b0 + b01 ln %Irr + b02 DT ) + b1 ln p
Coefficien Standard
Error
ts
t Stat
Intercept
4.5534
6.7955
0.6701
%Irrig
-0.4359
0.9566
-0.4557
Year=tech 0.2317
0.0516
4.4950
Price
0.5780
2.4542
1.4185
R2=0.814
The model with technical change explains a much greater variation than the previous model
does and the sign of the price coefficient is now coherent with the theory. This is another
example supporting the importance of the pre-estimation phase that requires a thorough
knowledge of the market and of the collected data.
The elasticity associated to the shifter corresponding to the proportion irrigated is not
significantly different from zero. In the context of the present specification, technology is highly
significant. The price elasticity of supply (1.42) has the correct sign, is significant and quite
elastic.
The Nerlove Model:
Regression of Rice Production | lag(prod), lag(price) while controlling for
Technical Change
• ln q= (b0 + b01 ln %Irr + b02 DT) + b1 ln p(-1)+ b2 ln q(-1)
Coefficien Standard
Error
ts
t Stat
Intercept
0.7888
3.8753
0.2036
%Irrig
0.1623
0.4895
0.3316
Year=tech 0.0128
0.0336
0.3823
Prod(-1)
0.9265
0.1143
8.1023
Price(-1)
-0.0604
0.3405
-0.1775
R2=0.966
The Nerlove model explains a greater proportion of the supply variation if compared to the
previous specifications. Technical change and the proportion of irrigated rice production are not
significantly different from zero. The effect of technical change is absorbed by the effect of the
change in previous year production. The signs comply with the theory.
51
Training Materials
Short and Long Run Supply Elasticities
Short Run Price Elasticity: (b1) = -0.06
Long Run Price Elasticity: b1/(1-b2) = -0.0604/(1-0.9265) = -0.8218.
In line with our expectations, the long run supply elasticity is higher than the short run elasticity
which is, in our example, not significantly different from zero. However, the sign of the long run
elasticity is wrong. We may hypothesize that this is the result of aggregate behavior. So, let us
see what the separate analysis of rainfed and irrigated production tell us. First, a simple
simulation.
Suppose now that the Colombian government would like to increase the price of rice by 10
percent in 1975. What would you expect the production to be in 1975, 1976 and 1977?
Year
1974
1975
1976
1977
Index of price
change
100
110
110
110
% Expected
changes in prod
100
99.396
108.660
117.925
Rice price
1151
1266.1
1266.1
1266.1
Rice production
1569940
1560451
1695587
1999513
Note that the price effect works only in the first year, when it is replaced by the past production
impulse.
The Nerlove Model for Rainfed and Irrigated Colombian Rice Producers
Rainfed Producers
Coefficien Standard
Error
ts
Intercept
0.4184
t Stat
2.2974
0.1821
Year=tech -0.0098
0.0229
-0.4274
Prod(-1)
0.7971
0.1329
5.9985
Price(-1)
0.2892
0.4478
0.6459
R2=0.866
Irrigated Producers
Coefficien Standard
Error
ts
Intercept
t Stat
2.4781
2.2703
1.0915
Year=tech 0.0610
0.0335
1.8233
Prod(-1)
0.7898
0.1314
6.0121
Price(-1)
0.0290
0.3359
0.0862
R2=0.965
Home exercise: Compute the short run and long run price elasticity of supply and the impact
of technical change for both rainfed and irrigated rice production. Interpret, compare and
contrast your results across upland, rainfed and total production.
52
Partial Equilibrium Analysis of Policy Impacts (part I)
Price Elasticity of Supply of Agricultural Products in Syria
Product
Period
R2
Wheat
1961-72
Short run
Price el.
0.64
Barley
1961-72
0.27
0.40
.50
Maize
1947-60
0.51
0.69
.84
Millet
1961-72
1.21
1.60
-
Potatoes
1950-60
0.65
1.30
.87
Source: Scandizzo and Bruce, 1980.
Long run
Price el.
3.23
.57
%2
2
3.2.2. Estimation of a Nerlovian Supply Response Model for groundnuts in Senegal
First, let us share some background information about the economic and institutional situation
of the Senegales Groundnut Market 1960-1988. Groundnuts represent the major source of
income for Senegalese farmers and the main source of export earnings for the country. The
government administers the price and extracts the positive difference between world price and
producer price as government revenue. (So, we do not to explain how farmers form
expectations about groundnut prices, since the government announces the price before planting
time)!
The dramatic decline of world peanuts prices in the 1980s, also coinciding with the adoption of a
structural adjustment program, forced the government to subsidize the sector.
Our main questions are:
What is the impact on farmer from the removal of this costly subsidy?
What is an acceptable combination between market AND state?
What is the impact of structural adjustment?
Further information:
millet market is free and farmers do not know the price at planting time (so, we need to
model how farmers form expected prices for millet).
We use a lagged price to model expectations. Similarly for rainfall.
Note: This is very important for both a statistically and economically sound specification of the
model!!!
53
Training Materials
The data set
Year
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
54
Area in
groundnuts
(1,000 ha)
1323
1230
1233
1185
1055
1016
1017
1064
1091
1097
1051
1060
1087
1280
1020
1201
1175
1079
970
995
1097
1216
1148
925
873
750
789
823
787
Current prices
Groundnuts
(CFA/mt)
Millet
20500
22000
21500
21500
20600
20600
21000
18000
18000
21200
23100
23100
25500
41500
41500
41500
41500
41500
41500
41500
50000
60000
60000
60000
75000
75000
90000
90000
70000
22000
22500
23000
25000
25000
25800
22000
23000
23000
24000
23000
29000
22000
37000
35000
42000
55000
65000
45000
45000
45000
55000
75000
85000
95000
100000
100000
110000
115000
Rainfall
(mm)
817
685
609
699
830
660
897
886
457
841
496
745
428
461
556
801
573
437
637
666
418
573
553
337
492
546
735
809
500
Cons.
price
index
0.95
1.00
1.07
1.14
1.21
1.26
1.28
1.32
1.31
1.37
1.42
1.51
1.60
1.69
1.95
2.37
2.51
2.67
2.87
3.11
3.52
3.73
4.37
4.88
5.46
6.16
6.54
6.82
6.54
Real prices
Groundnuts
(1961 CFA/mt)
21652
22000
20101
18873
17039
16401
16355
13678
13730
15520
16256
15288
15977
24513
21282
17533
16508
15520
14445
13344
14209
16100
13730
12303
13741
12175
13761
13191
10700
Millet
23236
22500
21503
21945
20678
20541
17134
17477
17544
17570
16186
19193
13784
21855
17949
17744
21877
24308
15663
14469
12788
14759
17162
17429
17406
16234
15291
16122
17579
Partial Equilibrium Analysis of Policy Impacts (part I)
Data Preparation
Year
Area in
groundnuts
(1,000 ha)
Lag area in
groundnuts
t
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
ln At
7.19
7.11
7.12
7.08
6.96
6.92
6.92
6.97
6.99
7.00
6.96
6.97
6.99
7.15
6.93
7.09
7.07
6.98
6.88
6.90
7.00
7.10
7.05
6.83
6.77
6.62
6.67
6.71
6.67
ln At-1
7.19
7.11
7.12
7.08
6.96
6.92
6.92
6.97
6.99
7.00
6.96
6.97
6.99
7.15
6.93
7.09
7.07
6.98
6.88
6.90
7.00
7.10
7.05
6.83
6.77
6.62
6.67
6.71
Prices (1961 CFA) Last Rainfall Previous Agricultural
structural
last year
three
year
adjustment
years
1979 - 88
mean
Groundnuts
Millet
rainfall
ln Pg,t
ln Pm,t-1 ln Rt-1 ln Rt-(1-3)
DUM
9.98
0
10.00
10.05
6.71
0
9.91
10.02
6.53
0
9.85
9.98
6.41
6.56
0
9.74
10.00
6.55
6.50
0
9.71
9.94
6.72
6.57
0
9.70
9.93
6.49
6.59
0
9.52
9.75
6.80
6.68
0
9.53
9.77
6.79
6.70
0
9.65
9.77
6.12
6.62
0
9.70
9.77
6.73
6.59
0
9.63
9.69
6.21
6.39
0
9.68
9.86
6.61
6.54
0
10.11
9.53
6.06
6.32
0
9.97
9.99
6.13
6.30
0
9.77
9.80
6.32
6.18
0
9.71
9.78
6.69
6.41
0
9.65
9.99
6.35
6.47
0
9.58
10.10
6.08
6.40
0
9.50
9.66
6.46
6.31
1
9.56
9.58
6.50
6.36
1
9.69
9.46
6.04
6.35
1
9.53
9.60
6.35
6.31
1
9.42
9.75
6.32
6.24
1
9.53
9.77
5.82
6.19
1
9.41
9.76
6.20
6.13
1
9.53
9.69
6.30
6.13
1
9.49
9.63
6.60
6.38
1
9.28
9.69
6.70
6.55
1
55
Training Materials
Evolution of Acreage and Price of Groundnut
11.00
10.50
10.00
9.50
ln At
9.00
ln Pg,t
8.50
Lineare (ln At)
Log. (ln Pg,t)
8.00
Lineare (ln Pg,t)
7.50
7.00
6.50
19
88
19
86
19
84
19
82
19
80
19
78
19
76
19
74
19
72
19
70
19
68
19
66
19
64
19
62
19
60
6.00
Years
Evolution of Groundnut Price
10.20
10.00
9.80
ln(Pgt)
9.60
Serie1
9.40
9.20
9.00
8.80
1
2
3
4
5
6
7
8
9
10
11 12
13 14
15 16 17
years
56
18 19
20 21 22
23 24
25 26 27
28 29
Partial Equilibrium Analysis of Policy Impacts (part I)
Acreage supply
7.30
7.20
7.10
7.00
ln(At)
6.90
Acreage supply
6.80
6.70
6.60
6.50
6.40
19
88
19
86
19
84
19
82
19
80
19
78
19
76
19
74
19
72
19
70
19
68
19
66
19
64
19
62
19
60
6.30
years
Acreage and price
7.30
7.20
7.10
7.00
ln(At)
ln At
6.90
Log. (ln At)
Log. (ln At)
6.80
6.70
6.60
6.50
9.20
9.30
9.40
9.50
9.60
9.70
9.80
9.90
10.00
10.10
10.20
ln(Pgt)
57
Training Materials
Acreage Supply Response Equation for groundnuts in Sub-Saharan Africa
Regression Results: Estimates
(See Excel file 4supply fed2)
Dependent
Lag area
ln At-1
variable Constant
Prices (1961 CFA)
Rainfall
Structural
adjustment
1979 - 88
R2
Current Millet ln last year Three-year Additive On price
ln Rt-1 average ln DUM DUM*ln adjusted
groundnut Pm,t-1
Rt-(1-3)
Pgt
ln Pgt
Short Run
elast
0.349
-0.341
0.091
0.821
3.865
-3.878
1.814
0.789
1. lnAt
1.623
1.804
0.675
5.387
2. lnAt
1.668
1.642
0.621
4.951
0.338
3.708
-0.368
-4.125
3. lnAt
3.852
2.997
0.645
5.552
0.252
2.692
-0.423
-4.771
0.052
1.064
4. lnAt
4.136
2.889
0.648
5.531
0.239
2.425
-0.441
-4.699
0.053
1.075
5. lnAt
4.293
3.330
0.595
5.539
0.177
1.925
-0.372
-4.201
0.075
1.653
0.202
2.420
0.825
0.792
-0.087
-2.268
-4.866
-2.488
0.855
0.822
-0.009
-2.161
0.852
0.818
0.502
2.442
0.886
0.853
Note1: t-statistics in italic; adjusted R2 in italic = 1-[T/(T-K)](1-R2), where T is the number of
observations and K the number of exogenous variables.
Note2: 1) since groundnut price is administered, then gamma=1, so delta=1-PI3=1-b1=>Long
run el=pi2/(1-b1) (Hint: watch your definiton of pi3 in your notes).
2) when there is the interaction term DUM*ln Pgt, then the price elasticity=coeff(Pgt)
+coeff(DUM*ln Pgt) when DUM=1.
58
Partial Equilibrium Analysis of Policy Impacts (part I)
Acreage Supply Response Equation for groundnuts in Sub-Saharan Africa
Regression Results: Elasticities
(See Excel file 4supply fed2)
Elasticities (at mean values) w.r.t.
Groundnut price
Millet price
Short
Long
Short
pi2/(gam*delta)
0.349
0.464
-0.341
0.338
0.327
-0.368
0.252
0.561
-0.423
0.239
0.650
-0.441
0.177
1.678
-0.372
Model
1
2
3
4
5
Long
-1.050
-0.971
-1.190
-1.250
-0.920
Actual and Estimated area of groundnuts: Model 1
7.3
7.2
7.1
7
Ln(At)
6.9
Y actual
6.8
Y predicted
6.7
6.6
6.5
6.4
6.3
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Years
59
Training Materials
Simulation and price policy analysis
1.5% increase/year
Estimated area in groundnuts
60
Model 1
(1,000 ha)
best model 5
1323
1266
1213
1202
1152
1088
1072
1055
1071
1116
1108
1094
1071
1200
1191
1055
1090
1048
989
964
982
1052
1043
972
900
847
804
812
791
1323
1274
1207
1189
1147
1093
1062
1063
1079
1083
1114
1067
1070
1162
1160
1036
1095
1033
959
991
1050
1176
1087
949
908
815
833
835
727
Estimated area in groundnuts
under alternative policy
High price
High price of
No structural
with struct. adjustment(1,
groundnuts
adj. (1,000 ha)
new Pg,t
000 ha)
(CFA/mt)
20808
22330
1215
1215
21823
1158
1158
21823
1169
1169
20909
1136
1136
20909
1098
1098
21315
1061
1061
18270
1131
1131
18270
1153
1153
21518
1144
1144
23447
1220
1220
23447
1178
1178
25883
1166
1166
42123
1400
1400
42123
1306
1306
42123
1246
1246
42123
1417
1417
42123
1262
1262
42123
1130
1130
42123
1170
1285
50750
1309
1393
60900
1542
1542
60900
1469
1591
60900
1267
1450
76125
1174
1270
76125
1098
1264
91350
1140
1233
91350
1203
1329
71050
1049
1287
Partial Equilibrium Analysis of Policy Impacts (part I)
Chapter 4 - The Market Model
We are now ready to combine the two sides of the market to find an equilibrium price. First, it is
instructive to review some basic notions of market equilibrium and analysis.
4.1.
Causality in Economic Analysis
Let us think at the familiar supply and demand model. The price P0 is an equilibrium price
because at this price all buyers are able to buy as much as they like and all suppliers are able to
sell as much as they like. No other price will satisfy these conditions.
At a price above P0, for example, sellers could not sell as much as they desire. Each can remedy
the problem by lowering the price a bit. Buyers will discover sellers are receptive to price
discounts. The result is downward pressure on price and the pressure will continue as long as
price remains above P0.
This analysis of demand and supply is of limited interest. The real significance of the model
resides in a set of hypotheses not explicitly shown in the classic diagram as to variables that
expand or contract demand and supply and therefore CAUSE price and quantity to change
called shifters.
Demand shifters: population, per capita income, prices of related goods and consumer
preferences.
Supply shifters: Number of sellers, input prices, weather, and technology.
4.1.1.
Exogenous and Endogenous Variables in a Model
Definition. Exogenous Variable: Variables are EXOGENOUS to a model or theory if their
values are determined by processes not described by the model.
Example: Demand and supply shifters.
Note: whereas changes in exogenous variables CAUSE changes in endogenous variables,
changes in endogenous variables have NO EFFECT on exogenous variables. CAUSALITY flows
in one direction. For example, weather can greatly influence grain prices BUT grain prices have
no perceptible impact on weather .....
Definition. Endogenous Variables: Variables are said to be ENDOGENOUS if their values
are determined jointly by processes described by the theory (a model).
Example: Price and quantities are endogenous variables because jointly determined by the
model.
Note: the value of one does not imply the value of the other. It would be nonsensical to ask
whether an increase in price would CAUSE quantity to fall ..... Price might rise because of a
decrease in supply, in which case quantity falls.
61
Training Materials
The interest in POLICY ANALYSIS is to ask how a change in an exogenous variable affects the
values and the direction of change of an endogenous variable.
To ask how a change in one endogenous variable affects the value of another endogenous
variable is a WRONG question.
A simple algebraic representation of a demand-supply model
The well-known graphical description of a market can be represented with a simple set of
equations:
Qd = β0 -β1 Pd
Demand
Qs = γ0 + γ1 Ps
Supply
Qs = Qd = Q
Equilibrium Condition
Pd = P s = P
Equilibrium Condition
This system consists of 6 equations and 6 endogenous variables: Qd, Qs, Pd, Ps, Q, P.
Definition. Equilibrium Condition: a market is in equilibrium when the excess demand is
zero (Qd -Qs )=0, that is when the market is cleared.
At the equilibrium, there is no tendency to change. For this reason, the analysis of what the
equilibrium state is like is referred to as comparative statics analysis.
Note: the variables Q and P are introduced to allow expressing the model in equilibrium
values only, as shown below:
•
Q = β0 -β1 P
Demand
Q = γ0 + γ1 P
Supply
Solution of the Market Model
In order to obtain the previous model in equilibrium values, we have used the equilibrium
conditions to rid the system of four endogenous variables that is Qd, Qs, Pd, Ps. In general, a
standard algebraic procedure is to reduce the size of a system of equations by eliminating
variables through substitution (that is, getting rid of variables through giving up equations).
If this process is carried out to its logical conclusion, we can find the solution of a consistent
system of equations. So, solving the equilibrium model for the solution values of P and Q yields:
P=
Q=
β0 − γ 0
β1 + γ 1
β 0γ 1 + β 1γ 0
β1 + γ 1
These are the equilibrium values for Q and P as in the graphical representation.
4.2.
A Structural Market Model with Exogenous Variables
Let us insert exogenous shifters in the model! Suppose that:
Z1 = population (expanding demand) , and
Z2 = technological change (expanding supply), then we have the:
STRUCTURAL MODEL
Q = β0 -β1 P + β2 Z1
62
Demand
Partial Equilibrium Analysis of Policy Impacts (part I)
Q = γ0 + γ1 P + γ2 Z2
Supply
Solving the system for P and Q, we obtain the:
REDUCED FORM MODEL
P=
Q=
1
(β 0 − γ 0 + β 2 Z 1 − γ 2 Z 2 )
β1 + γ 1
1
(β 0γ 1 + β 1γ 0 + γ 2 β 2 Z1 + β 1γ 2 Z 2 )
β1 + γ 1
4.2.1. Structural and Reduced Form Models
Question: How do we do POLICY analysis?
Definition. Structural Model: The distinguishing feature of a STRUCTURAL MODEL is
that AT LEAST ONE of its equations contains two or more endogenous variables.
Note: both equations of our structural model contain two endogenous variables.
Definition. Reduced Form Model: The distinguishing feature of a REDUCED FORM
MODEL is that each equation has NO MORE than ONE endogenous variables.
Note: All the exogenous variables are on the right hand side.
Structural models are expressions of economic theories. They provide simplified description of
real world processes such as market, consumption or production behavior.
Reduced models are of great use to derive the hypothesis that are derived from the theory. Only
once the reduced form equations have been derived, one for each endogenous variable, the
theorist can do the POLICY ANALYSIS by finding the partial derivatives describing the impact
of a change in the exogenous variables on the endogenous variables!
Determining whether the partial derivatives are positive, negative or equal to zero is the essence
of qualitative policy analysis!
•
Causality
An example
Suppose we let demand increase because of population growth. As a result equilibrium price
and quantities increase. Our hypothesis, then, asserts that population growth CAUSES P and Q
to rise. From the solution of our structural model it is apparent that an increase in Z1
(population) also causes P and Q to increase. Similarly, an increase in technological progress
(Z2) uses P to fall and Q to rise (just looking at our reduced form model).
The graphic and algebraic approaches lead to the same result!
The advantage of the graphic approach is that it allows for easy derivation of hypothesis when
the structural model is simple and consists of a few equations.
The advantage of the algebraic approach is that it allows derivation of hypotheses in case of
complex structural models.
In our reduced form model, it is clear that changes in P and Q are CAUSED by Z1 and Z2.
What role is played by the structural parameters β and γ?
An example: suppose that β1 is very close to zero (not statistically and economically significantly
different from zero). That is to say that the demand for output is perfectly vertical or inelastic.
Now, let technological change shift supply from S0 to S1. The effect of technical change is to
63
Training Materials
decrease P but Q does not change at all! So, the importance of β1 is that it conditions then price
and quantity effects of technical change.
If it were true world wide that the demand for food is highly inelastic, then technological
progress would not affect output very much at least for very short periods of time. It would
mainly tend of lower food prices.
$
Structural parameters are important in explaining real world events because
they condition the effects of changes in exogenous variables.
This explains why it is important to have reliable estimates of structural parameters such as the
elasticities of demand and supply.
Market examples: Demand and Supply Shift
Demand shift: a population increase
P
S
P1
P0
D1
D0
Q0
Q1
Q
Supply shift: technological progress with rigid demand
P
D
S0
S1
P0
P1
Q0
64
Q
Partial Equilibrium Analysis of Policy Impacts (part I)
4.3.
The Colombian Rice Economy: the market model
We choose the double-log specification for both demand and supply because this is the
specification selected by Scobie and Posada (1977) in our reference article. The summary of our
results follows.
Demand Side
• ln q = a0 + a00 ln d + a1 ln p + a2 ln (y/N)
Coefficients Standar
d Error
t Stat
Intercept
1.823
0.527
3.461
urb/tot %
-1.151
0.187
-6.151
Prices $/t
-0.361
0.084
-4.296
Income/mth
0.796
0.053
14.975
R2=0.979, N. Obs. 21
Let
ln A= a0 + a00 ln d + a2 ln (y/N)
So,
A = exp(ln A) and
mean(A)= 4.402944
Therefore, the exponential specification is:
QD = A P η
with the associated inverse demand
P = (1/A QD ) 1/ η
Supply side with technical change
• ln q = (b0 + b01 ln %Irr + b02 DT ) + b1 ln p
Coefficien Standard
Error
ts
t Stat
Intercept
4.5534
6.7955
0.6701
%Irrig
-0.4359
0.9566
-0.4557
Year=tech 0.2317
0.0516
4.4950
Price
0.5780
2.4542
1.4185
R2=0.814
with technical change
ln Bt = b0 + b01 ln %Irr + b02 DT and
mean(exp(Bt)) = 2.727727
without technical change
ln B = b0 + b01 ln %Irr and mean(exp(B)) = 1.540489
65
Training Materials
The Market
Finally, our estimated market model for the Colombian rice industry is:
without technical change
with technical change
QD = A P
= 4.4 P -0.36
QD = A P
= 4.4 P -0.36
QS = B P
= 1.5 P 1.42
QS = Bt P
= 2.7 P 1.42
QD = QS
QD = QS
The graph is presented in the following page.
Exercise: Compute the equilibrium for both the situation with and without technical change
and check the analytical solution with the graphical solution shown in the graph.
Interpret.
Solution: For convenience, let us linearize the model taking logarithms:
ln QD = ln A + η ln P
ln QS = ln B + β ln P
Then, let us use the market clearing condition to obtain the analytical solution for the
equilibrium price Pe :
ln A + η ln P = ln B + β ln P
→
ln A - ln B = ln P ( β - η )
ln Pe = ( ln A - ln B ) / ( β - η ) = ln A/B / ( β - η )
Pe = exp ( ln A/B / ( β - η ) )
So, to find the equilibrium price Pe without technical change:
1.83
Pent = exp ( ln A/B / ( β - η ) ) = exp ( ln 1.63 / 1.78 ) = exp (1.08 / 1.78 ) = exp( 0.6 ) =
and with technical change:
1.32
Pet = exp ( ln A/Bt / ( β - η ) ) = exp ( ln 2.93 / 1.78 ) = exp (0.49 / 1.78 ) = exp ( 0.27 ) =
To find the equilibrium quantity Qe corresponding to Pe substitute ln Pe into either equation of
the system:
ln Qe = ln A + η (ln A/B / ( β - η ) ) = (ln A ( β - η ) + η ln A - η ln B) / ( β - η ) =
= (β ln A - η ln B) / ( β - η ).
4.4.
Welfare Analysis of Technical Change
In the market model with technical change consumers are the main gainers by visually
appreciating the change in Consumer Surplus (∆CS=1+2=area below the demand curve
delimited by the price change before and after technical change has occurred). The gain to
consumers increase as the demand is more inelastic. The effect on producers is small
(∆PS=(4+3)-(1+3)(area above the supply curve after technical change)-(area above the supply
curve before technical change)). The Net Social Gains (NSG=2+4=area below the demand curve
comprised between the before and after technical change supply curve) are captured mostly by
66
Partial Equilibrium Analysis of Policy Impacts (part I)
consumers. If demand is completely inelastic, only consumers gain because NSG=∆CS. For this
reason, there is nothing surprising if consumers invest through tax transfers to the government
on agricultural research.
If demand is infinitely elastic (parallel to the horizontal axis), because rice is a tradable
commodity on open global markets or because the government intervenes in the market buying
and stocking all increases in production at a target price, then technical change is neutral to
consumers. Then, producers gain the full NSG= ∆PS=2. If the good is tradable, the country
gains from increased exports and foreign earnings. In this case, farmers should be very
supportive of technical change when the good is tradable. If the good is non tradable, farmers
may still gain if the government supports the price.
The measurement of the welfare gains from technological change, as shown in the article by
Scobie and Posada, is important because, by dividing the welfare gains from research by the
social costs of generating innovations, we obtain the rate of return from investment in
agricultural research.
Technical change for non tradable
goods
∆CS=1+2; ∆PS=(4+3)-1+3);
NSG≈∆CS=2+4
D
P
Technical change for tradable goods
or price support
∆CS=0; ∆PS=2; NSG=∆PS=2
P
S0
S0
S1
S1
P0
P0
1
2
1
P1
2
3
4
Q0
Q
Q
P
The Colombian Rice Market
80
70
60
50
40
30
20
10
0
Series1
Series2
Series3
1
2
3
4
5
6
7
8
9
10
Q
67
Partial Equilibrium Analysis of Policy Impacts (part I)
References
Berndt, E. (1996): “The practice of Econometrics. Classics and Contemporary,” Addison-Wesley
Publishing Company, Reading, Mass., USA.
Colman, D. and T. Young “Principles of Agricultural Economics. Market and Prices in Less
Developed Countries ,” Cambridge University Press, 1989.
Deaton, A. and J. Muellbauer (1980): “Economics of Consumer Behavior,” Cambridge
University Press, Cambridge.
Deaton, A. (1997): “The Analysis of Household Surveys. A Micro-econometric Approach to
Development Policy Analysis,” The John Hopkins University Press, Baltimore, USA.
De Janvry, A. and E. Saudolet (1995): “Quantitative Development Policy Analysis ,“ The John
Hopkins University Press, Baltimore, USA..
Helmberger, P. and J.P. Chavas (1997): “Economics of Agriculture: Production, Marketing and
Prices,” Addison-Wesley Publishing Company, Reading, Mass., USA.
FAO/TCAS ID8 (1992): Agricultural Policies Analysis. Exercises. Rome.
Johansson, Per-Olov (1991): “An Introduction to Modern Welfare Economics,” Cambridge
University press, Cambridge.
Rivas, L., J. Garcia, C. Serè, T. Jarvis, and L.R. Sanint (1999): “Economic Surplus Analysis
Model”, Centro Internacional de Agricultura Tropical (CIAT), Cali, Colombia.
Scobie, G. and R. Posada (1977): “The Impact of High-Yielding Rice Varieties in Latin America,”
Centro Internacional de Agricultura Tropical (CIAT), Cali, Colombia, Series JE-01.
69