Page 172, no 4
“The law of diminishing returns is at best a short run phenomenon”.
Discuss.
In the production analysis, a time factor is involved because time is
always required when moving from one production process to
another. In the production process time is divided into short run and
long run. The short run is the period that is too short for a firm to
change is production outlay. The long run is the period that is long
enough for a firm to change its production outlay. In the long run all
factors of production are varied i.e. flexible while in the short run
some factors are fixed while others can be varied. The short run and
long run situation varies between firms depending on the size or
scale of the firm. Short run of a firm can be the long run of another
firm.
The law of diminishing returns is also known as principle of
diminishing marginal productivity and law of variable proportions.
The law states that if one input in the production of a commodity in
increased while other inputs are held fixed, a point will eventually be
reached at which additions of the input yields progressively smaller,
or diminishing, increases in output.
This decrease in the marginal output of a production process as the
amount of a single factor of production is marginally increased while
the quantity other factors remain constant. This law doesn’t imply
that increasing the quantity of a variable factor will decrease total
production i.e. give rise to negative returns but the marginal increase
in output starts to fall. This law only gives rise to negative returns
when the when the quantity of the variable factor input exceeds the
fixed factor input.
An example can be seen in ABUAD BAKERY. ABUAD bakery employs
about 15 workers and the productivity is 250 loaves of bread each
day. the increase in the number of workers available to do the job
causes the production in the bakery to increase. If the manager
increases in the number of workers continuously, it would get to a
point where the increase in the productivity of output in the
production process is reduce due to the fact the workers may think
that they are many workers employed so if they work or not doesn’t
really matter . In all of these processes producing an additional unit
of output per unit of time will eventually cause increasingly more,
due to inputs being used less effectively according to the classical
economists the successive diminishment of output in relations to the
increase in the input is as a result of low quality of inputs that are
used while the neoclassical assume that each unit of the variable
factor i.e. labour is identical. However I the diminishing returns can
be said to be as a result of disruptions of the entire productive
process as additional units of a variable factor (labour) are added to
a fixed input (land).
This is the law guiding production in the short run.
For this law to holds, there are some assumptions which include
1.
2.
3.
4.
5.
Units of capital and labour are used as variable factor inputs.
The prices of the factors do not change.
All units of variable factor are equally efficient.
There is no change in technique of production
The combination of factors of production has crossed the level
optimum point.
6. There is no change in the fixed factor of production.
The effect of this law is usually felt more in the agricultural sector,
however when a firm is too large and supervision is inefficient,
this law sets in in the industry.
Page 172, no 7
What is an isocost? How can we derive one? Using the tools of
isocost and isoquant, show the firms output maximizing decision
subject to a cost constraint.
The isocost curve is also known as the factor cost curve. It is a
curve that shows all the input bundles that can be purchased at a
specific cost. An isocost is a curve that shows the locus of the
various combinations of two variable factor inputs-labours and
capital-each of which costs the producer the same amount of
money. The isocost curve portrays the various alternative
combinations of factor inputs which a firm can purchase, given the
prices of inputs and the firms budgeted expenditure(C) on inputs.
Although, the isocost curve is similar to the consumer budget line
in the analysis of theory of consumer behaviour, the use of isocost
pertains to cost minimization as opposed to utility maximization in
the theory of consumer behaviour. The equation of the isocost
line’s generally defined by:
T.C= X.Px + Y.Py
Where:
T.C represents the budgeted expenditure
X.Px represents the total expenditure on input X
Y.Py represents the total expenditure on input Y
It could also be denoted as: 𝐶 = 𝑤𝐿 + 𝑟𝐾
Where C represents the total cost outlay
r represents rate of rental of machine (capital)
K represents the rate of capital use
w represents the wages for labour
L represents the rate of labour use
𝐶 = 𝑤𝐿 + 𝑟𝐾
Make 𝐾 the subject of formula,
𝐾=
𝐶−𝑤𝐿
𝑟
𝜕𝐾
𝑤
=−
𝜕𝐿
𝑟
The slope of the isocost is given as: −
𝑤
𝑟
The aim or objective of the firm is to maximise profit by making use of the combination of factors
which the cost is the least. The firm can maximise its profits either by maximising the level of output
for a given cost or minimising the cost of producing a given output. The firm’s output maximizing
level is the point at which the slope of the isoquant is tangential to the slope of the isocost i.e. the
output level where the slope of the isoquant and the slope of the isocost are equal.
The slope of the isoquant is defined as the marginal rate of technical substitution The MRTS is the
rate at which an input is substituted for another input while maintaining a constant level of output. .
It depicts the movement along the isoquant.
Given; ∆𝐾. 𝑀𝑃𝑘 = −∆𝐿. 𝑀𝑃𝑙
Divide through by ∆𝐿. 𝑀𝑃𝑘
=
∆𝐾. 𝑀𝑃𝑘 −∆𝐿. 𝑀𝑃𝑙
=
∆𝐿. 𝑀𝑃𝑘
∆𝐿. 𝑀𝑃𝑘
Neglecting the sign, this becomes:
Δ𝐾
Δ𝐿
𝑀𝑃
= 𝑀𝑃 𝑙
𝑘
∴ 𝑀𝑅𝑇𝑆𝑙𝑘 =
Δ𝐾 𝑀𝑃𝑙
=
Δ𝐿 𝑀𝑃𝑘
The firms output maximizing decision subject to a cost constraint is the point of tangency between
the slope of the isocost and isoquant.
That is:
∴
𝑀𝑃𝑙
𝑤
𝑊
𝑟
=
𝑀𝑃
= 𝑀𝑃 𝑙
𝑀𝑃𝑘
𝑟
𝑘
Showing that the marginal product per naira should be the same for all inputs i.e.
equimarginal principle.
Page 172, no 8
Given the hypothetical productivity schedules for “Go-getters Ltd”.
Units of
10
20
30
40
50
labour(L)
12
11
9
6
2
𝑀𝑃𝑙
Units of
50
40
30
20
10
capital(K)
4
7
12
𝑀𝑃𝑘
If the price of a unit of labour is N1 and the price of a unit of capital is N2, given that the firm has a
budget of N100 and decides to use all on the employment of both labour and capital, determine the
combinations, of labour and capital that will maximise the firm’s output subject to cost constraint
and represent your answer in a diagram.
𝐶 = 𝑤𝐿 + 𝑟𝐾
Price of a unit of labour; w=N1
Price of a unit of capital; r= N2
Budget/constraint; C= N100
100=L + 2K
Page 80, no 3
Distinguish between substitution and income effect of an increase in the price of rice in household’s
demand.
The substitution effect is a price change that alters the slope of the budget constraint but leaves the
consumer o the same indifference curve. In considering the effect of an increase the price of rice in
household’s demand, the substitution effect emphasises that the rational consumer is induce to buy
lesser quantity of rice because rice has become relatively more expensive than the other
commodities whose prices has remained constant i.e. a rational consumer prefers to substitute the
purchase of rice for commodities whose prices have remained constant hence relatively cheaper to
that of rice.
The income effect is the phenomenon observed through the changes in consumption patterns due
to changes in purchasing power. This can occur from changes in income, price changes or currency
fluctuations. It reveals the change in quantity demanded brought by a change in real income. The
income effect differs depending on the type of good.
In case of a normal good, an increase the in the price of the good would cause decreases the
consumer’s ability to purchase the good and decrease in the price of the good would cause increases
the consumer’s ability to purchase more of the commodity. In case of an inferior good, an increase
in the price of the commodity would cause an increase in the consumer’s ability to purchase the
commodity while a decrease in its price implies a decrease in the purchase of the good.
The income effect emphasises that as the price of rice increases, leaving the prices of other
commodities constant, the consumer’s ability to purchase more rice falls hence he purchases lesser
quantities because of the decrease his real income which the change in price.
Page 80, no5
What is an Engel curve? Explain the behaviour of such curves in the case of (i) a normal good (ii) an
inferior good.
An Engel curve describes how household expenditure on a particular good or service varies with
household income. There are two varieties of Engel Curves. Budget share Engel Curves describe how
the proportion of household income spent on a good varies with income. Alternatively, Engel curves
can also describe how real expenditure varies with household income. The best-known single result
from the article is Engel's law which states that the poorer a family is, the larger the budget share it
spends on nourishment.
In microeconomics, an Engel curve shows how the quantity demanded of a good or service changes
as the consumer's income level changes. In order to be consistent with the standard model of utilitymaximization, Engel curves must possess certain properties. For example, Gorman (1981) proved
that a system of Engel curves must have a matrix of coefficients with rank three (or less) in order to
be consistent with utility maximization.
When considering a system of Engel curves, the adding-up theorem also dictates that the sum of all
total expenditure elasticities, when weighted by the corresponding budget share, must add up to
unity. This rules out the possibility of saturation being a general property of Engel Curves across all
goods as this would imply that the income elasticity of all goods approaches zero starting from a
certain level of income. The adding-up restriction stems from the assumption that consumption
always takes place at the upper boundary of the household's opportunity set, which is only fulfilled if
the household cannot completely satisfy all its wants within the boundaries of the opportunity set.
In microeconomics Engel curves are used for equivalence scale calculations and related welfare
comparisons, and determine properties of demand systems such as agreeability and rank. Engel
curves have also been used to study how the changing industrial composition of growing economies
are linked to the changes in the composition of household demand.
Engel curves are also of great relevance in the measurement of inflation, and tax policy. Engel curve
in Economics is a curve that shows/describes how a person's quantity demanded for a particular
good or service varies with as the income level changes. Graphically, the Engel curve is represented
in the first-quadrant of the Cartesian coordinate system. Income is shown on the Y-axis and the
quantity demanded for the selected good or service is shown on the X-axis.
The Engel Curve tracks the consumption of a Good X as an individual’s income changes. Income is
plotted on the x-axis and the quantity of Good X consumed is plotted on the y-axis. The curve that
follows the amount of Good X consumed as income increases plots the Engel Curve.
The slope of the Engel Curve also tells us whether or not the good is a normal good or
inferior good. If the slope of the curve is positive, the good is a normal good because consumption
increases as income is increased. If the slope of the curve is negative, the good is an inferior good
because consumption decreases as income is increased.
The Engel Curve can be derived from the Income Expansion Path. Each budget constraint in
the Income Expansion Path provides the income. The amounts of Good X consumed at the points of
consumers’ optimum on the budget constraint provide the quantity of Good X consumed at those
income levels.
Income
100
150
200
250
Quantity
Of Good X
10
20
30
45
Figure 2.0
This shows an Income Expansion Path for goods X and Y with four points of
consumer’s optimum shown. Good Y is a numeracies (priced at 1) and the relevant points for
forming an Engel Curve are in the table to the right of the Income Expansion Path.
The information from the Income Expansion Path (IEP) can produce two different Engel
Curves, one for Good X and one for Good Y. Each would use the income values provided by the
budget constraints, and the variables respective quantity values.
Figure 2.1
The Engel Curve is formed plotting the quantities of Good X consumed at the varying
incomes presented in the Income Expansion Path in figure 2.0. As income increases, the quantity of
Good X continues to increase. Good X is a normal good.
An example of a good that has an Engel curve with both normal good and inferior good
segments is a grilled cheese sandwich. At lower quantities, an average low-income consumer would
want to consume more as their income increases, but eventually, the consumer will reach an income
where grilled cheese becomes an inferior good. The consumer’s income would then be at a level
where they desire less grilled cheese as their income increases. A possible explanation would be that
the consumer has replaced low-cost grilled cheese sandwiches with a higher cost food because their
income reached a point where they could afford a more diverse diet.
Figure 2.3
An example of an Engel Curve with both normal good and inferior good segments.
After income increases pass 100, Good X shifts to being an inferior good.
In summary, the behavior of an Engel curve under the case of a Normal and an Inferior goods can be
deduce as follows;
- For normal goods, the Engel curve has a positive gradient. That is, as income increases, the quantity
demanded increases. Amongst normal goods, there are two possibilities. Although the Engel curve
remains upward sloping in both cases, it bends toward the y-axis for necessities and towards the xaxis for luxury goods.
- For inferior goods, the Engel curve has a negative gradient. That means that as the consumer has
more income, they will buy less of the inferior good because they are able to purchase better goods.
- For goods with Marshallian demand function (specifies what the consumer would buy in each price
and wealth situation, assuming it perfectly solves the utility maximization problem - "how should I
spend my money in order to maximize my utility?") generated by a utility in Gorman polar form (a
functional form for indirect utility functions in economics.), the Engel curve has a constant slope.
According to Engel’s studies, as the income of a family increases, the proportion of its income spent
on necessities such as food falls and that spent on luxuries (consisting of industrial goods and
services) increases. In other words, the poor families spend relatively large proportion of their
income on necessities, whereas rich families spend a relatively a large part of their income on
luxuries. This change in the pattern of consumption expenditure (that is, decline in the proportion of
income spent on food and other necessities and increase in the proportion of income spent on
luxuries) with the rise in income of the families has been called Engel’s law.
Page 189, no 2
Fill the gaps in the table below and plot the graphs of the corresponding cost curves.
Q
STC
TFC
TVC
SAC
AFC
SMC
10
1000
500
500
20
1200
500
30
40
1450
500
850
50
60
1850
1100
𝑆𝑇𝐶 = 𝑇𝐹𝐶 + 𝑇𝑉𝐶
𝑇𝐹𝐶 = 𝑆𝑇𝐶 − 𝑇𝑉𝐶
𝑇𝑉𝐶 = 𝑆𝑇𝐶 − 𝑇𝐹𝐶
𝑆𝐴𝐶 =
𝑆𝑇𝐶
𝑄
𝐴𝐹𝐶 =
𝑇𝐹𝐶
𝑄
𝑆𝑀𝐶 =
Δ𝑆𝑇𝐶
Δ𝑄
Q
STC
TFC
TVC
SAC
AFC
SMC
10
1000
500
500
100
50
20
1200
500
700
60
25
20
30
1350
500
850
45
16.67
15
40
1450
500
950
36.25
12.5
10
50
1600
500
1100
32
10
15
60
1850
500
2350
30.83
8.33
25
189 Question 3
a) 60 to 100units
It is clear that between 60 to 100 units, the firm was experiencing an increasing return to scale since
output was increased at an increasing rate.
b) 100 to 140 units
As inputs increased beyond 50 to 60, there were constant returns to scale in output. The outputs
were having a constant rate.
c) 210 to 230 units
From 210 to 230 units, diminishing returns to scale sets in. here, outputs was diminishing at a
decreasing rate.
Page 188, exercise 5,10, 13, 15
5. Given a production
1
𝑄 = 4𝑥 + 2𝑥 2 − 𝑥 3
3
Demonstrate rigorously that
𝑀𝑃𝑥 = 𝐴𝑃𝑥 When 𝐴𝑃𝑥 is at its maximum.
𝑄 = 𝑓(𝑥)
𝜕𝑄
= 4 + 4𝑥 − 𝑥 2
𝜕𝑥
𝑀𝑃𝑥 =
𝐴𝑃𝑥 =
1
4𝑥 + 2𝑥 2 − 3 𝑥 3
𝑥
1
𝐴𝑃𝑥 = 4 + 2𝑥 − 𝑥 2
3
When 𝐴𝑃𝑥 is at maximum,
𝜕𝐴𝑃𝑥
=0
𝜕𝑥
𝜕𝐴𝑃𝑥
2
=2− 𝑥
𝜕𝑥
3
𝜕𝐴𝑃𝑥
=0
𝜕𝑥
2
2− 𝑥 =0
3
2
𝑥=2
3
multiply through by 3
2𝑥 = 6
Divide through by 2
𝑥=3
𝑀𝑃𝑥 =
𝜕𝑄
𝜕𝑥
= 4 + 4𝑥 − 𝑥 2
substitute 𝑥 = 3 into 𝑀𝑃𝑥
𝑀𝑃𝑥 = 4 + 4(3) − 32
= 4 + 12 − 9
=7
1
𝐴𝑃𝑥 = 4 + 2𝑥 − 𝑥 2
3
Substitute 𝑥 = 3 into 𝐴𝑃𝑥
1
𝐴𝑃𝑥 = 4 + 2(3) − (3)2
3
=4+6−3
=7
∴ 𝑀𝑃𝑥 = 𝐴𝑃𝑥 𝑤ℎ𝑒𝑛 𝐴𝑃𝑥 𝑖𝑠 𝑎𝑡 𝑚𝑎𝑥𝑖𝑚𝑢𝑚
10. From the following production and cost data, determine the optimum resource combination.
(i) 𝑄 = 𝑋1 2 + 10𝑋1 𝑋2 + 𝑋2 2
𝑇𝐶 = 𝑁500; 𝑃𝑋1 = 𝑁5; 𝑃𝑋2 = 𝑁20
The related budget equation is given as;
𝑇𝐶 = 𝑋1 𝑃𝑋1 + 𝑋2 𝑃𝑋2
500 = 5𝑋1 + 20𝑋2
The Lagragian function to be maximised is;
𝑍 = (𝑋1 𝑋2 ) + 𝜆(𝑇𝐶 − 𝑋1 . 𝑃𝑋1 − 𝑋2 . 𝑃𝑋2 )
The associated Lagragian function is;
𝑍 = (𝑋1 2 + 10𝑋1 𝑋2 + 𝑋2 2 ) + 𝜆(500 − 5𝑋1 − 20𝑋2 )
Differentiate 𝑍 with respect to 𝑋1 , 𝑋2 𝑎𝑛𝑑 𝜆 and setting it equal to zero
𝜕𝑍
= 2𝑋1 + 10𝑋2 − 5𝜆 = 0
𝜕𝑋1
𝜕𝑍
= 10𝑋1 + 2𝑋2 − 20𝜆 = 0
𝜕𝑋2
𝜕𝑍
= 500 − 5𝑋1 − 20𝑋2 = 0
𝜕𝜆
2𝑋1 + 10𝑋2 − 5𝜆 = 0
(1)
10𝑋1 + 2𝑋2 − 20𝜆 = 0
(2)
5𝑋1 + 20𝑋2 = 500
(3)
10𝑋1 + 40𝑋2 = 1000
(4)
Substitution method
Multiply (3) by 2
Subtracting (2) from (4)
38𝑋2 + 20𝜆 = 1000
Make 𝜆 the subject of formula from (5) above
(5)
𝜆=
1000 − 38𝑋2
20
𝜆 = 50 − 1.9𝑋2
(6)
Substitute (6) in (1) and (2)
2𝑋1 + 10𝑋2 − 5(50 − 1.9𝑋2 ) = 0
2𝑋1 + 10𝑋2 − 250 + 9.5𝑋2 = 0
2𝑋1 + 19.5𝑋2 = 250
(7)
10𝑋1 + 2𝑋2 − 20(50 − 1.9𝑋2 ) = 0
10𝑋1 + 2𝑋2 − 1000 + 38𝑋2 = 0
10𝑋1 + 40𝑋2 = 1000
(8)
Compare (7) and (8)
2𝑋1 + 19.5𝑋2 = 250
(7) × 5
10𝑋1 + 40𝑋2 = 1000
(8) × 1
10𝑋1 + 97.5𝑋2 = 1250
10𝑋1 + 40𝑋2 = 1000
(9)
(10)
Subtract (10) from (9)
57.5𝑋2 = 250
𝑋2 = 4.34
Substitute 𝑋2 = 4.34 into (10) above
10𝑋1 + 40(4.34) = 1000
10𝑋1 + 173.6 = 1000
10𝑋1 = 1000 − 173.6
10𝑋1 = 826.4
𝑋1 = 82.64
𝑋1 = 82.64, 𝑋2 = 4.34
(ii) 𝑄 = 150𝑋1 + 180𝑋2 − 4𝑋1 2 − 2𝑋2 2
𝑇𝐶 = 𝑁1200; 𝑃𝑋1 = 𝑁10; 𝑃𝑋2 = 𝑁20
The related budget equation is given as;
𝑇𝐶 = 𝑋1 𝑃𝑋1 + 𝑋2 𝑃𝑋2
1200 = 10𝑋1 + 20𝑋2
The Lagragian function to be maximised is;
𝑍 = (𝑋1 𝑋2 ) + 𝜆(𝑇𝐶 − 𝑋1 . 𝑃𝑋1 − 𝑋2 . 𝑃𝑋2 )
𝑍 = (150𝑋1 + 180𝑋2 − 4𝑋 1 2 − 2𝑋2 2 ) + 𝜆(1200 − 10𝑋1 − 20𝑋2 )
Differentiate 𝑍 with respect to 𝑋1 , 𝑋2 𝑎𝑛𝑑 𝜆 and setting it equal to zero
𝜕𝑍
= 150 − 8𝑋1 − 10𝜆 = 0
𝜕𝑋1
𝜕𝑍
= 180 − 4𝑋2 − 20𝜆 = 0
𝜕𝑋2
𝜕𝑍
= 1200 − 10𝑋1 − 20𝑋2 = 0
𝜕𝜆
8𝑋1 + 10𝜆 = 150
4𝑋2 + 20𝜆 = 180
(1)
(2)
10𝑋1 + 20𝑋2 = 1200
(3)
Solution by substitution
Compare (2) and (3)
4𝑋2 + 20𝜆 = 180
(2) × 5
10𝑋1 + 20𝑋2 = 1200
(3) × 1
20𝑋2 + 100𝜆 = 900
(4)
10𝑋1 + 20𝑋2 = 1200
(5)
Subtract (4) from (5)
(6)
10𝑋1 − 100𝜆 = 300
Make 𝜆 the subject of formula from (6) above
𝜆=
10𝑋1 − 300
100
𝜆 = 0.1𝑋1 − 3
(7)
Substitute (7) in (1) and (2) above
8𝑋1 + 10(0.1𝑋1 − 3) = 150
(1)
8𝑋1 + 𝑋1 − 30 = 150
9𝑋1 = 150 + 30
9𝑋1 = 180
180
= 20
9
4𝑋2 + 20(0.1𝑋1 − 3) = 180
𝑋1 =
4𝑋2 + 2𝑋1 − 60 = 180
4𝑋2 + 2𝑋1 = 180 + 60
4𝑋2 + 2𝑋1 = 240
(2)
Substitute 𝑋1 = 20 in the above equation
4𝑋2 + 2(20) = 240
4𝑋2 + 40 = 240
4𝑋2 = 240 − 40
4𝑋2 = 200
𝑋2 =
200
= 50
4
𝑋1 = 20 , 𝑋2 = 50
(iii)𝑄 = 30𝑋1 + 44𝑋2 − 2𝑋1 𝑋2
𝑃𝑋1 = 𝑁4; 𝑃𝑋2 = 𝑁5; 𝑇𝐶 = 𝑁2000
The related budget equation is given as;
𝑇𝐶 = 𝑋1 𝑃𝑋1 + 𝑋2 𝑃𝑋2
2000 = 4𝑋1 + 5𝑋2
The Lagragian function to be maximised is;
𝑍 = (𝑋1 𝑋2 ) + 𝜆(𝑇𝐶 − 𝑋1 . 𝑃𝑋1 − 𝑋2 . 𝑃𝑋2 )
𝑍 = (30𝑋1 + 44𝑋2 − 2𝑋1 𝑋2 ) + 𝜆(2000 − 4𝑋1 − 5𝑋2 )
Differentiate 𝑍 with respect to 𝑋1 , 𝑋2 𝑎𝑛𝑑 𝜆 and setting it equal to zero
𝜕𝑍
= 30 − 2𝑋2 − 4𝜆 = 0
𝜕𝑋1
𝜕𝑍
= 44 − 2𝑋1 − 5𝜆 = 0
𝜕𝑋2
𝜕𝑍
= 2000 − 4𝑋1 − 5𝑋2 = 0
𝜕𝜆
2𝑋2 + 4𝜆 = 30
(1)
2𝑋1 + 5𝜆 = 44
(2)
4𝑋1 + 5𝑋2 = 2000
(3)
Solve using substitution method
2𝑋1 + 5𝜆 = 44
(2) × 2
4𝑋1 + 5𝑋2 = 2000
(3) × 1
4𝑋1 + 10𝜆 = 88
(4)
4𝑋1 + 5𝑋2 = 2000
(5)
5𝑋2 − 10𝜆 = 1912
(6)
Subtract (4) from (5)
From (6), make 𝜆 the subject of formula
𝜆=
5𝑋2 − 1912
10
𝜆 = 0.5𝑋2 − 191.2
Substitute 𝜆 = 0.5𝑋2 − 191.2 in (1) and (2)
2𝑋2 + 4𝜆 = 30
(1)
2𝑋2 + 4(0.5𝑋2 − 191.2) = 30
2𝑋2 + 2𝑋2 − 764.8 = 30
4𝑋2 − 764.8 = 30
4𝑋2 = 30 + 764.8
4𝑋2 = 794.8
𝑋2 = 198.7
2𝑋1 + 5𝜆 = 44
(2)
2𝑋1 + 5(0.5𝑋2 − 191.2 ) = 44
2𝑋1 + 2.5𝑋2 − 956 = 44
2𝑋1 + 2.5𝑋2 = 44 + 956
2𝑋1 + 2.5𝑋2 = 1000
Substitute 𝑋2 = 198.7
2𝑋1 + 2.5(198.7) = 1000
2𝑋1 + 496.75 = 1000
2𝑋1 = 1000 − 496.75
2𝑋1 = 503.25
𝑋1 = 251.625
𝑋1 = 251.625, 𝑋2 = 198.7
15. Given a production𝑄 = 𝑓(𝐿, 𝐾), demonstrate rigorously that output elasticities 𝑤𝐿 =
𝑤𝐾 =
𝑀𝑃𝐾
𝐴𝑃𝐾
𝑀𝑃𝐿
𝐴𝑃𝐿
and
respectively.
For a production𝑄 = 𝑓(𝐿, 𝐾) the output elasticity of input 𝐿 𝑜𝑟 𝐾,denoted by 𝜔𝐿 𝑜𝑟 𝜔𝐾 may be
defined as the proportionate rate of change in total output 𝑄 with respect to 𝜔𝐿 𝑜𝑟 𝜔𝐾 .
OR
𝜔𝐿 =
𝜕 log 𝑄 𝐿 𝜕𝑄 𝑀𝑃𝐿
=
=
𝜕 log 𝐿 𝑄 𝜕𝐿
𝐴𝑃𝐿
𝜔𝐾 =
𝜕 log 𝑄 𝐾 𝜕𝑄 𝑀𝑃𝐾
=
=
𝜕 log 𝐾 𝑄 𝜕𝐾 𝐴𝑃𝐾
𝜔𝐿 =
𝜕 ln 𝑄
𝜕 ln 𝐿
∂ln 𝑄 =
𝜕𝑄
𝑄
∂ln 𝐿 =
𝜕𝐿
𝐿
Divide both:
𝜕𝑄 𝜕𝐿 𝜕𝑄 𝐿
÷
=
×
𝑄
𝐿
𝑄 𝜕𝐿
𝜕𝑄 𝐿
×
𝜕𝐿 𝑄
𝑀𝑃𝐿 =
𝜕𝑄
𝜕𝐿
𝐴𝑃𝐿 =
𝑄
𝐿
𝐿
1
1
= =
𝑄 𝑄 𝐴𝑃𝐿
𝐿
𝜕𝑄 𝐿
1
𝑀𝑃𝐿
× = 𝑀𝑃𝐿 ×
=
𝜕𝐿 𝑄
𝐴𝑃𝐿 𝐴𝑃𝐿
𝜔𝐾 =
𝜕 ln 𝑄
𝜕 ln 𝐾
∂ln 𝑄 =
𝜕𝑄
𝑄
𝜕 ln 𝐾 =
𝜕𝐿
𝐾
Divide both:
𝜕𝑄 𝜕𝐾 𝜕𝑄 𝐾
÷
=
×
𝑄
𝐾
𝑄 𝜕𝐾
𝜕𝑄 𝐾
×
𝜕𝐾 𝑄
𝑀𝑃𝐾 =
𝜕𝑄
𝜕𝐾
𝐴𝑃𝐾 =
𝑄
𝐾
𝐾
1
1
= =
𝑄
𝑄
𝐴𝑃𝐾
𝐾
𝜕𝑄 𝐾
1
𝑀𝑃𝐾
× = 𝑀𝑃𝐾 ×
=
𝜕𝐾 𝑄
𝐴𝑃𝐾 𝐴𝑃𝐾
Page 188, Q1
1. Explain the measuring of expansion path of a firm. What might cause the expansion path of
a firm to change?
Expansion path is the locus of all optimal combinations of two inputs of a firm. it is formed by
finding the producer’s optimum budget constraint (isocost) as well as the producers optimum input
combination (isoquant) for the two inputs. The points at which isocost line and isoquant curve are
tangential to each other in the expansion path. An expansion path of the firm can change when
output or input price ratio changes. For instance the directors of a fir decide to spend more on one
input at the expense of the other due to reduction in price of one, the expansion path will shift
towards the input which is being increased in quanty.
OUTPUT
EXPANTION PATH
ISOQUANT
INPUT ISOCOST
Q2 page 188
Diminishing marginal productivity and return to scale have to do with the behavior of the fir
when inputs are increased.
In the case of diminishing marginal productivity, as a successive or additional unit of input is
added, all things being equal it will yield a lower and positive out put
Return to scale could either be constant increasing or decreasing. Constant return to scale is a
situation where output increases at the same proportion as change in input. Increasing return to
scale occurs when output increases more than the proportional increase in change in input.
Decreasing return to scale occurs when output increases it does increase as the same
proportional rate as change in input it is usually lesser than the change in input.
Diminishing marginal productivity deals with increase in one input while the other is held
constant while return to scale deals with increase in all inputs of a firm.
Page 225, Problem 6
How are the short-run average cost curve and long run average cost curves related?
A cost curve is a graph that shows the various costs incurred in the producing an output as a
function of the total quantity produced of that output. These curves are used by firms or
producers to find the optimal production point or output such that either the firm is a profit
maximiser or cost minimiser he is able to achieve his aim. There are various cost curves some of
which applies to the long run while others to the short run. In the short-run some factors of
production are fixed while others are varied, hence there are three major relevant cost namely
Total Fixed Cost (TFC), Total Variable Cost (TVC) and Total Cost (TC). Other costs considered in
the short-run include Average Fixed Cost (AFC), Average Variable Cost (AVC), Average Total Cost
(ATC) and the Short-run Marginal Cost (SMC). In the long-run all the factors of production are
varied hence the costs are not classified as variable costs or fixed costs. The Long-run Marginal
Cost (LMC) and the Long-run Average Costs (LAC) are the relevant costs in the long run.
The Average Cost curve (AC) is defined as the total cost incurred per unit of output produced. It
is the Total Cost (TC) divided by the quantity of output produced (Q). It is denoted as: 𝐴𝐶 =
𝑇𝐶
.
𝑄
Graphically the average cost curves are ‘U’ shaped because the left half of the ‘U’ accounts for
fixed costs that are usually incurred even before takes place while the right half of the ‘U’ is as a
result of diseconomies of scale that that occur at a high level of production or output i.e. cost is
increasing a higher rate than output.
The short-run costs are assumed to be fixed because there are some costs incurred in getting the
factors of production that are fixed because of the fixed nature of some factors of production.
The SAC shows the minimum possible cost of producing at each output level when variable
factors are operated in a cost minimising manner. In the long-run all costs are variable because
managers can change all input levels.