Dynamic Monetary Equilibrium with
Non-Observed Economy and Shapley and
Shubik’s Price Mechanism
Labib Shami∗
April 27, 2017
Abstract
We build a general equilibrium model with inside and outside
money, heterogeneous tax evading households, government and a central bank, to demonstrate the dynamic nature of the relation between
inflation rates (as a monetary policy) and the size of the non-observed
economy (NOE) (as % of GDP). Using Shapley and Shubik’s (1977)
price mechanism we show that fiat money has positive value, monetary policy has real effects and there is a unique monetary equilibrium where formal and informal markets coexist. Moreover, interest
rates, price levels and commodity allocations are determinate and the
tractable dynamic model has a unique monetary equilibrium that can
be computed analytically. In this model, with forward-looking agents
and a government that has a long term budget constraint, there is no
inflation tax, rather, there is “inflation debt”, uncovering yet another
interpretation of the Ricardian effect. In this world the size of the
NOE depends on the past and future prices, hence an increase in the
inflation rate in any given period can increase or decrease the size of
the NOE.
∗
Department of Economics; University of Haifa. [email protected]
1
1
Introduction
Estimates of the size of non-observed economy1 around the world suggest that it is a widespread phenomenon with a potentially significant
influence (from 9-12 per cent of total economic activity in Anglo-Saxon
countries, to 20-30 per cent in southern Europe (Williams and Schneider, 2013), while in developing economies and transition countries it
is routinely around 40% (Schneider, 2008)). Yet, its interaction with
monetary policy is not well understood, in particular, the relation between the rate of inflation and the size of the NOE (Ahmed, Rosser Jr,
and Rosser, 2007; Gomis-Porqueras, Peralta-Alva, and Waller, 2014).
The central issue discussed in this study is the dynamic nature
of the relation between inflation rates (as a monetary policy) and the
size of the NOE (as % of GDP). A common empirical finding, based
on cross sectional evidence, is that inflation rates and the size of the
NOE appear to be positively correlated (Koreshkova, 2006; Aruoba,
2010; Mazhar and Méon, 2017). This finding is at odds with the
predictions of theoretical models (Gillman and Kejak, 2006; GomisPorqueras et al., 2014).
To address this issue we build a general equilibrium model with
heterogeneous tax evading households, government and a central bank.
It would be senseless to analyse monetary policy in a model where
(fiat) money has no value or where monetary policy has no real effects.
Yet, constructing such a general equilibrium model has been a challenge for decades (Kiyotaki and Wright, 1989; Dubey and Geanakoplos, 1992, 2003, 2006; Lagos and Wright, 2005; Lagos, Rocheteau, and
Wright, 2015). To assure money has a value we adopt Dubey and
Geanakoplos’s (2003) inside and outside money which appear naturally in a setting where agents bid cash to buy goods in the market,
as in Shapley and Shubik’s (1977) game.
In an otherwise frictionless economy monetary policy would have
had no real effects, if it was not for the NOE. Adding this element
into the model creates the “effect of a friction”: in this case, prices
and the stock of fiat money grow at different rates. In our first take
1
One commonly used definition for non-observed economy, which we adopt, is all economic activities, and the income derived from them, that circumvent or otherwise avoid
government regulation, taxation or observation (Schneider and Dell’Anno, 2003).
2
at this problem, agents hide a fixed amount of cash from the tax authorities, this amount is determined by the external factors: system
of enforcement, acceptable norms of behavior, etc. The agents then
can use the hidden cash to buy goods at the “black market”, which
provides any amount of goods demanded at the prevailing (official)
market price. Other things being equal, an increase in price reduces
the real size of the NOE in this model, however, to assess the full
effect of an increase in inflation, the whole path of inflation has to be
known. Thus we construct examples of equilibria where, along two
different paths, a higher inflation at any given point in time can yield
a lower or a higher NOE.
1.1
1.1.1
Related literature
Non-observed economy
In this study we use the term non-observed economy (cf. footnote
1) introduced by the United Nations System of National Accounts
(SNA) in 1993 (Calzaroni and Ronconi, 1999), which has become accepted in policy discussions within the OECD (Blades and Roberts,
2002). There exist many labels and definitions of the NOE. The terms
hidden, gray, shadow, black, informal and underground can refer to
the same concept, but, depending on the context or the author, they
can also refer to specific aspects of the NOE (for a detailed discussion
on the causal factors of the shadow economy refer to Schneider and
Buehn (2016)).
A vast array of economic policy problems (such as the analysis of
economic growth, employment and productivity; possible abuse of social insurance programs; and erosion of tax revenues) critically depend
on understanding the phenomena related to the NOE. A large NOE
is detrimental to the trust in and integrity of public institutions, and
may lead to a suboptimal design of policies and institutions (Schneider and Buehn, 2016).
Fighting tax evasion and the non-observed economy activity have
been important policy goals in OECD countries during recent decades
(OECD, 2002; Gyomai and van de Ven, 2014; Schneider, 2016). In
2002 the OECD produced an extensive handbook for measuring the
3
“non-observed economy” presenting a “systematic strategy for achieving exhaustive estimates of gross national product” taking specific
account of “activities that are missing from the basic data used to
compile the national accounts because they are underground, illegal,
informal, household production for final use, or due to deficiencies in
the basic data collection system”.
Using individual-level household-lending data across different credit
products and samples, Artavanis, Morse, and Tsoutsoura (2016) conservatively estimate that at least 26.8 billion Euros of income went
untaxed in Greece for 2009. Hence, it is no coincidence that Adjustment Programs2 for Greece, Portugal and Cyprus included strict
provisions for limiting tax evasion, as a way to improve their fiscal
position.
1.1.2
Estimates of the size of NOE
Although the non-observed economy has been investigated for a long
time, discussion regarding the appropriate methodology to assess its
scope has not come to an end yet. Measurement of the NOE is inherently difficult, not only because of the very nature of the non-observed
economic activity, but also the fact that different policy perspectives
often warrant different definitions and boundaries for the NOE.
In general, three different categories of measurement methods are
most widely used: the direct approach (a micro-economic approach
that employ surveys and questionnaires based on voluntary replies, or
tax auditing methods based on the discrepancy between income declared for tax purposes and that measured by selective checks), the
indirect approach (based on several indicators measuring discrepancies in various statistical indicators on the aggregate level, e.g., national expenditure and income statistics) and the model approach.
The latter is based on statistical models, especially the multiple indicator multiple cause (MIMIC) procedure (Weck, 1983; Frey and Weck,
2
Following the 2008 financial crisis, and faced with high levels of public debt, Portugal,
Italy, Ireland, Greece and Spain were compelled to implement harsh austerity reforms
coordinated by he European Commission (EC), the European Central Bank (ECB), and
the International Monetary Fund (IMF) (Blyth, 2013). These adjustments were followed
by deflation accompanied by an increase in real government debt.
4
1983), and the dynamic general equilibrium approach that has been
presented by Elgin, Oztunali, et al. (2012) using a two-sector dynamic
general equilibrium model to estimate the size of the shadow economy.
Their micro-founded methodology uses national income statistics and
a DGE to back out shadow economy size from the model. For more
detail please see Andrews, Sánchez, and Johansson (2011); Schneider
and Buehn (2016) and Elgin and Schneider (2016).
1.1.3
NOE and inflation in DGE models
This branch of literature, with an explicit role for the NOE, is the closest to our paper. Using a quantitative general equilibrium analysis,
Koreshkova (2006) showed how a theory of optimal taxation can rationalize government incentives to inflate in the presence of a tax-evading
sector, establishing a positive relationship between the inflation rate
and the size of the non-observed economy. The author introduced a
closed economy with no uncertainty and cash-in-advance constraint
faced by the non-observed sector, where credit is costly and transactions are made through cash in order to analyze the role of money, in
terms of an inflation tax, on this sector.
Aruoba (2010) developed a general equilibrium model with cashin-advance constraint where households optimally choose the extent
of informal activity and a benevolent government optimally chooses
policies, assuming that the institutions3 of the economy are exogenous, relying on their slow-changing nature. In addition, in order
to capture two main properties of informal activity: tax evasion and
cash intensiveness, the author used a search-based monetary model,
which is based on the structure in Lagos and Wright (2005). The
main conclusion that emerge from this study is: “better institutions
are associated with lower inflation, higher income tax rates and less
informal activity and higher levels of informal activity are associated
with lower income tax rates and higher inflation”.
In a simple model with outside money, and by using a sample of
153 developed and developing countries over the 1999-2007 period,
Mazhar and Méon (2017) empirically tested the claim that a larger
3
By institutions the author refers to the set of rules that determine how economic
activity is conducted. In his empirical analysis the author uses the rule of law indicator
to measure the quality of institutions.
5
NOE should give governments an incentive to shift revenue sources
from taxes to inflation. On the relation between the rate of inflation
and the size of the NOE, the authors concluded that the inflation rate
is an increasing function of the share of the shadow economy.
In contrast, Gomis-Porqueras et al. (2014) constructed a dynamic
general equilibrium model of tax evasion where agents choose to report
some of their income, and calibrated the model using money, interest
rate and GDP data to back out the size of the shadow economy for
a sample of countries. Using Lagos and Wright (2005) search theoretic model of money, their model generated a negative relationship
between the inflation rate and the size of the non-observed economy
(as % of the GDP).
Castillo, Montoro, et al. (2008) modeled their economy with frictions in the labor market by introducing formal and informal labor
contracts and analyzed the interaction between the two sectors and
monetary policy. They introduced informality through hiring costs
owing to labor market conditions (degree of tightness). In their model
firms in the wholesale sector are assumed to balance the high productivity in formal sector with the lower hiring costs faced by the
informal sector. The main finding of this theoretical framework is the
cyclical behavior of informal sector i.e. it expands with rising aggregate demand because of lower hiring costs. Through this channel a
link between informality, the inflation dynamics and monetary policy
is established.
Most of the papers mentioned in this section share the assumption that the NOE is different in nature from the formal economy.
These differences are such that either the goods being produced are
assumed to be different, as in Aruoba (2010), or the technologies used
to produce the goods or the means of payment required to obtain
the goods are assumed to be different as in Koreshkova (2006) and
D’Erasmo and Boedo (2012). In contrast, Anbarci, Gomis-Porqueras,
and Pivato (2012) and Gomis-Porqueras et al. (2014) consider an environment that produces a homogeneous good in different markets that
use different trading protocols, which is similar to our model.
6
1.1.4
General equilibrium with money
In the standard (Arrow-Debreu) general equilibrium model, due to
Walras law, price level can be chosen arbitrarily, i.e., prices can be denominated in any currency and its devaluation or appreciation should
have no effect, since only relative prices matter for an equilibrium allocation. Similarly, in a neo-classical model with rational expectations,
as in Lucas (1972) (building on Samuelson (1958) overlapping generations model), a publicly-announced “proportional monetary expansion
will have no [real] consequences”. Of course, in Neo-Keynesian models monetary policy does have an effect on real variables even in the
presence of rational expectations, but one has to accept price-rigidity
(for an overview see Gali (2008) and Woodford (2011)).
On the other hand, in order to employ the neo-classical framework
to analyze monetary issues, a role for money must be specified so that
the agents will wish to hold positive quantities of it. A simple approach to ”force” money use by agents is through money in utility
(MIU) models, which assumes that money yields direct utility for the
agents (Sidrauski, 1967). Another popular technique is the cash in advance (CIA) constraint, which introduced by (Clower, 1967). Later,
search theoretic models of money were developed, which assumes direct barter of commodities is costly (Kiyotaki and Wright, 1989; Lagos
and Wright, 2005). Despite all, the debate over the suitable model to
be used to analyze monetary policy still continues (Chari, Kehoe, and
McGrattan, 2009; Woodford, 2011; Lagos et al., 2015).
However, the use of the CIA constraint alone does not assure that
agents will hold money in equilibrium in order to transact (Dubey
and Geanakoplos, 1992). One way to overcome this problem is to
“force” households to sell their entire endowment of commodities for
money, as in Lucas (1980) and Lucas and Stokey (1987), or to the
government, as in Magill and Quinzii (1992). one could follow Lerner
(1947) and Heller (1974) and assume that the government is owed in
taxes precisely the sum of the cash balances of all the households, and
obliged to offer relief from taxes in exchange for money. According
to these methods, money has value because someone is willing to exchange it for a “product” which contributes to increasing the utility of
the agent. Alternatively, one can assume that the households receive
their initial endowment in fiat money that they own free and clear, the
7
outside money, as in Dubey and Geanakoplos (1992), this approach is
adopted here.
Following Gurley and Shaw (1960); Dubey and Geanakoplos (2003),
we use the notions of inside and outside money: when fiat money is
injected into the private sector in exchange for assets promising the
future delivery of money it is called inside money. Money injected
into the private sector as a transfer, or in exchange for a commodity
(which gives no claim on future repayment), is called outside money.
In a series of papers, Dubey and Geanakoplos (1992, 2003, 2006)
stress the importance of the outside money along with the competitive
banking sector (or a central bank) to generate value of holding and
transacting in money in the model. The outside money generates
the transaction role of money and the existence of the banking sector
assures that the money has value in the last period with the caveat that
in equilibrium, where money has a positive value, the gains to trade at
the initial endowment must exceed the ratio between outside money
and inside money. With inside money only, equilibrium might be
indeterminate (Drèze and Polemarchakis, 2001a,b) and money might
lose value in equilibrium giving rise to Hahn’s paradox (Hahn, 1965):
if no one is ready to accept the money “in the last period”, no one will
in any period beforehand. The latter problem can be eliminated by
working with the infinite horizon model (as suggested by Samuelson
(1958)). However for our purposes, finite horizon is an easy and clear
way to model the finite time when the long term budget constraint of
the government has to hold, or at least, the requirement that the debt
has to be limited.
1.2
Building blocks of the benchmark model
As mentioned above, and to insure a positive price for fiat money, we
adopted the notions of inside and outside money in a dynamic general
equilibrium.
There are three types of agents in the model: households, government and a central bank. Both, the central bank and the government
have an active and independent role in our model. Government is
needed to model the debt decisions and to set the inflation target4
4
Since 1990 when the Reserve Bank of New Zealand adopted an inflation targeting
8
for each period, and the central bank is committed to meet it. To
meet its target, the central bank sets the nominal interest rate and
reacts to deviations of inflation from the inflation target by changing
the monetary base (money aggregates M1). The government can run
a budget with debt when debt financing is done by issuing government bonds. The central bank may purchase government bonds on
the market, and receive interest on his investment, while financing the
purchase of government bonds can be made by printing money or by
using the savings of the households.
The agents in the model are heterogeneous: they can differ in
their utility function, in their endowments and can choose different
consumption paths, which allows us to examine the effect of large
and small players on the monetary equilibrium and the price of the
good. However, aggregate quantities are unaffected by their choices.
To avoid non-existence of equilibria, as in Dubey and Geanakoplos
(2006), we assume that the households get all of their endowment in
cash (outside money). The endowment in goods is directed entirely to
the market (and not divided between the households), and the households are obliged to pay in cash for the purchase of goods, which yields
infinite gains to trade.
Following Shapley and Shubik (1977), the mechanism for price formation is built upon the household’s and the government bids of cash
to purchase goods in the market, thus bids precede prices. In our
model, traders commit quantities of their wealth to the purchase of
the good without definite knowledge of what the per-unit price will
be. At an equilibrium this will not matter, as prices will be what the
traders expect them to be. The price in our model will be so determined that it will exactly balance the books at the market. Moreover,
because of the mechanism for price formation, all traders pay the same
price for the good, and the price paid by one trader is dependent on
the monetary actions of the others (Viner, 1932; Shubik, 1971a).
This price formation is suitable for perishable goods, like fresh
approach to monetary policy, major G7 central banks (including the Bank of England,
the Bank of Canada, and the European Central Bank) have adopted such approach as
well as central banks in both industrialized and emerging market countries, among them
Australia, Brazil, Chile, Israel, Korea, Mexico Norway, Poland, South Africa, Sweden, and
several others.
9
vegetables or fresh fish, whose supply is very inelastic in the short
run and the producers are virtually price takers. The assumption is
that quantity is predetermined by production at the market level, and
since it is not storable, price must adjust so the available quantity is
consumed. Into this model we later add the NOE by assuming that
agents can evade paying taxes on a fixed amount of their cash income
and use this cash to buy goods in a “black market”.
2
The basic model
Consider a closed economy which contains three groups of agents: government, a central bank and households N = {1, ..., N }; all have the
same discrete finite life-time span T = {1, ..., T } and indexed by t ∈ T.
Each time period is divided into two, morning and evening. Every
morning:
1. The government collects income taxes and issues one period
bonds with a face value (par value) of one unit of money, paid
to the holder at maturity. At the same time, it redeems all the
bonds issued in the previous period, plus interest. The government begins and ends its life with no debt, hence at the first
period, t = 1, it redeems no bonds and at the last period, t = T ,
it does not issue any bonds (no-ponzi condition).
2. The central bank decides on the amount and purchases government bonds, and pays back every household his savings (held by
the bank) from the previous period, plus interest. At the same
time, collects payments from its matured government bonds (including interest) from the previous period, absorbs the new savings of the households and, if required, prints money.
3. Each household receives a cash endowment (outside money) and
pays the income tax, withdraws his savings (including interest)
and collects payment from his matured government bonds (including interest) from the previous period. At the same time, he
deposits cash for savings and purchases new government bonds,
leaving some cash to purchase goods for consumption.
10
In the evening, the government and the households make cash bids to
purchase goods for consumption.
In order to support the mechanism outlined above, three markets meet
in every period t ∈ T: the money market, the asset market and the
commodity market.
2.1
The markets
At the beginning of each period, in the morning, two financial markets
open, the money market and the asset market. In the money market,
household i ∈ N can deposit cash, Sti , into (or borrow cash from) the
central bank at the nominal interest rate, Rt , set by the central bank.
The central bank stands ready to absorb any excess lending. In other
words, the demand for savings by the CB, at the nominal interest rate,
Rt , is assumed to be perfectly elastic.
At the asset market, household i ∈ N and the central bank can
purchase government bonds; Bti , Btm respectively, taking the market
price for government bonds, qt , as given. Each bond bears a nominal
interest rate, rt , set by the government.
In the second part of each period the third market opens, the commodity market, which contains a fixed amount5 , Q, of the (perishable)
consumption good (perfectly inelastic supply). Household i ∈ N and
the government bid cash, Lit and Lgt respectively, to purchase goods
for consumption from the market.
Denoting the total bid by Lt =
at each period t ∈ T will be:
g
i
i=1 Lt + Lt ,
PN
pt =
the price of the good
Lt
Q
Thus bids precede prices and pt+1 = (1 + πt+1 )pt , where πt is the
inflation rate at period t ∈ T, p1 = (1+πQ1 )I1 and I1 is the cash endowment in the first period.
5
We assume that the quantity of Q is fixed for simplicity of calculation. one can assume
that the amount of the consumption good increases each period at a given rate.
11
To complete the cash flow, at the beginning of the first period
each household i gets a cash endowment, I1i , the outside money. At
the beginning of each of the following periods the outside money is
the income from the commodity market (from the previous period),
Lt−1 = It , which is divided between households, so that each household, i ∈ N, receives the cash endowment Iti = β i It , where 0 < β i ≤ 1
is the relative weight. Here we assume, for simplicity, that the households do not see any connection between their bids and the monetary
grant they receive. The cash flow is summarized in figure 1.
savings
tax, bonds
cash income
Goods M
bonds+interest
Gov
Hh
bonds+interest
cons. purchase
consumption purchase
CB
bonds
savings+interest
Figure 1: The monetary flows in the economy each period. Each type of agent
is depicted as a vertex and the cash transactions are depicted as edges. The
fourth vertex depicts the goods market. Each cash flow equals the value of the
transaction label. Each household receives his endowments in cash from the
goods market, pays his taxes and re-allocates his savings and government
bonds portfolio. The central bank receives the households savings, prints
money and re-allocates its government bonds holdings. The government
receives its taxes, repays bonds and issues new ones. Further, each household
and the government purchase their consumption goods.
2.2
The households
The households i = 1, 2, ..., N derive their lifetime utility from the
discounted flow of private consumption:
V
i
(ci1 , ..., ciT )
=
T
X
t=1
12
ui (cit )
(1 + ρ)(t−1)
(1)
Where ρ ∈ (0, 1) is a discount factor, cit is the household’s consumption
in period t ∈ T and ui (·) is a concave, continuous, nondecreasing utility
i
function with ui (0) = 0 and limx→0 du
dx (x) = ∞. At time-period t = 1,
each household begins with a I1i = β i I1 ∈ (0, ∞) of cash (nominal
wealth).
For each t = 2, 3, ..., T − 1 the period-by-period budget constraint
faced by household i ∈ N is given by:
i
i
pt cit + qt Bti + Sti = (1 − τ )Iti + (1 + Rt−1 )St−1
+ (1 + rt−1 )Bt−1
(2)
Here, τ is the income tax rate imposed by the government. Since the
life-span is of finite duration, T , we expect that the households will use
all of the money they have on hand at the last period for consumption,
because after that it is literally worthless (Shubik, 1971b). Hence
B0i = BTi = S0i = STi = 0 and for periods t = 1, t = T we get:
p1 ci1 + q1 B1i + S1i = (1 − τ )I1i
pT ciT = (1 − τ )ITi + (1 + RT −1 )STi −1 + (1 + rT −1 )BTi −1
2.3
The government
The government allocates each period the amount of Lgt for public
consumption gt (hence Lgt = pt gt ). In addition, the government collects income taxes at a rate of τ and issues one period bonds Bt , at a
price of qt set by the market, with a face value (par value) of one unit
of money, paid to the holder at maturity. rt and τ are exogenous and
known each period. For each t = 2, 3, ..., (T − 1) the period-by-period
budget constraint faced by the government is given by:
pt gt + (1 + rt−1 )Bt−1 =
N
X
τ Iti + qt Bt
i=1
Since B0 = 0 and BT = 0 (the no-ponzi condition), at periods
t = 1, t = T we get:
p1 g1 =
N
X
τ I1i + q1 B1
i=1
pT gT + (1 + rT −1 )BT −1 =
N
X
i=1
13
τ ITi
2.4
The central bank
The central bank sets the nominal interest rate, Rt , and reacts to
deviations of inflation from the inflation target, πt∗ , by changing the
monetary base. The objective function of the central bank is:
min
T
X
(πt − πt∗ )2
t=1
For each t = 2, 3, ..., (T − 1) the period-by-period budget constraint
faced by the central bank is given by:
qt Btm
+ (1 + Rt−1 )
N
X
i
St−1
=
i=1
N
X
m
Sti + Mt + (1 + rt−1 )Bt−1
(3)
i=1
Where Mt is the amount of cash the central bank prints each period.
Since B0m = BTm = S0i = STi = 0, for periods t = 1, t = T we get:
q1 B1m
=
N
X
S1i + M1
i=1
(1 + RT −1 )
N
X
STi −1 = MT + (1 + rT −1 )BTm−1
i=1
2.5
Equilibrium
An equilibrium is a combination of quantities (Bti , Btm , Bt , Sti , Mt , Lit , Lt )
and prices (pt , qt , Rt ), such that:
(1) each household maximizes his total discounted utility (equation 1)
over his budget set (equation 2) by choosing the quantity of government bonds he will buy, Bti , his savings, Sti , and the amount of cash
he bids to buy the consumption goods, Lit ;
(2) the central bank maximizes its objective function by setting, Mt ,
the quantity of government bonds it will buy, Btm , and the nominal
interest rate, Rt ;
(3) the government covers its budget deficit by setting, Bt ;
(4) all three markets, the money market, the asset market and the
commodity market, clear each period.
14
Proposition 1. Given the path of the inflation rate targets {πt∗ }Tt=1 ,
the nominal interest rates on government bonds rt , the income tax rate
τ , the initial cash endowment I1 and the weight of each household β i
there is a unique equilibrium if the inflation targets path is consistent,
i.e., for every t ∈ T
T h
i
X
(1 + ρ)T −t (1 + πt )−1 πt = 0
(4)
t=1
If condition 4 holds then the equilibrium is characterized by the following conditions:
1. For each t = 1, 2, ..., T − 1, the “no-arbitrage” condition holds:
qt =
(1 + rt )
(1 + Rt )
And, rT = RT = qT = 0.
2. For each t = 2, 3, ..., T , the path of the outside money is:
It =
t−1
Y
(1 + πz )I1
z=1
3. For each t ∈ T and i ∈ N, we get:
cit+1 = cit
4. For each t = 1, 2, ..., T − 1, the path for the nominal interest rate
is:
(1 + ρ)(1 + πt+1 ) = (1 + Rt )
5. For each household i ∈ N, the cash bids to purchase goods for
consumption are:
Li1 = (1 + π1 )β i (1 − τ )I1
Lit
=
t
Y
(1 + πz )Li1 ,
t = 2, 3, ..., T
z=2
6. For each t ∈ T, the total cash bids made by the government and
the households are:
Lt =
t
Y
(1 + πz )I1
z=0
15
Corollary 1. A change in the path of the inflation rate targets {πt∗ }Tt=1
will not lead to a real change in the economy. In other words, monetary policy remain neutral and the government and the households will
smooth consumption over time.
For each t = 2, 3, ..., T and i ∈ N, we get:
Qt
(1 + πz )Li1
Lit
Li1
i
i
= Qz=2
= β i (1 − τ )Q
ct =
=
c
=
1
t
pt
p
1
z=2 (1 + πz )p1
Since Q is fixed, For each t = 2, 3, ..., T we get:
gt = g1
3 Tax evasion and the non-observed
economy
Further to the assumptions of the previous model, suppose that in
order to avoid paying tax, the households do not report to the Tax
Authorities on a fixed amount, D, of their income in each period.
With this money, households will be able to purchase goods at a secondary market (black market) that provides any amount according to
the price set at the main commodity market of the economy. Hence, at
the beginning of the first period each household i gets a cash endowment, I1i , the outside money, and reports I1i −D to the Tax Authorities.
At the beginning of each of the following periods the outside money
is the income from the main and the secondary commodity markets
(from the previous period), Lt−1 + N D = It , which is divided between
households, so that each household, i ∈ N, receives the cash endowment Iti = β i It , where 0 < β i ≤ 1 is the relative weight. Under these
conditions, the price of the good at each period will be::
PN
PN
g
g
i
i
i=1 Lt + Lt
i=1 Lt + Lt + N D
pt =
=
Q
Q − NpD
t
3.1
The households
The households i = 1, 2, ..., N derive their lifetime utility from the
discounted flow of private consumption:
V
i
(ci1 , ..., ciT )
=
T
X
t=1
16
ui (cit )
(1 + ρ)(t−1)
(5)
Li
Here, cit = c̄it + ĉit where c̄it = ptt and ĉit = pDt .
For each t = 2, 3, ..., T − 1 the period-by-period budget constraint
faced by household i ∈ N is given by:
i
i
pt c̄it +pt ĉit +qt Bti +Sti = (1−τ )(Iti −D)+(1+Rt−1 )St−1
+(1+rt−1 )Bt−1
+D
i
i
⇒ pt c̄it + qt Bti + Sti = (1 − τ )(Iti − D) + (1 + Rt−1 )St−1
+ (1 + rt−1 )Bt−1
(6)
i
i
i
i
Set B0 = BT = S0 = ST = 0, hence for periods t = 1 and t = T we
get:
p1 c̄i1 + q1 B1i + S1i = (1 − τ )(I1i − D)
pT c̄iT = (1 − τ )(ITi − D) + (1 + RT −1 )STi −1 + (1 + rT −1 )BTi −1
3.2
The government
For each t = 2, 3, ..., (T − 1) the period-by-period budget constraint
faced by the government is given by:
pt gt + (1 + rt−1 )Bt−1 =
N
X
τ Iti + qt Bt − τ N D
i=1
Since B0 = 0 and BT = 0 (the no-ponzi condition), at periods t = 1,
t = T we get:
N
X
p1 g1 =
τ I1i + q1 B1 − τ N D
i=1
pT gT + (1 + rT −1 )BT −1 =
N
X
τ ITi − τ N D
i=1
3.3
Equilibrium
An equilibrium is a combination of quantities (Bti , Btm , Bt , Sti , Mt , Lit )
and prices (pt , qt , Rt ), such that:
(1) each household maximizes his total discounted utility over his budget set by choosing the quantity of government bonds he will buy, Bti ,
his savings, Sti , and the amount of cash he bids to buy the consumption goods, Lit ;
(2) the central bank maximizes its objective function by setting, Mt ,
the quantity of government bonds it will buy, Btm , and the nominal
interest rate, Rt ;
17
(3) the government covers its budget deficit by setting, Bt ;
(4) all three markets, the money market, the asset market and the
commodity market, clear each period.
Proposition 2. Given the path of the inflation rate targets {πt∗ }Tt=1 ,
the nominal interest rates on government bonds rt , the income tax
rate τ , the initial cash endowment I1 , the tax evasion amount, D, and
the weight of each household β i there is a unique equilibrium if the
inflation targets path is consistent, i.e., for every j ∈ T
T h
i
X
(1 + ρ)T −t (1 + πt )−1 πt = 0
(7)
t=1
If condition 7 holds then the equilibrium is characterized by the following conditions:
1. For each t = 1, ..., T − 1, the “no-arbitrage” condition holds:
qt =
(1 + rt )
(1 + Rt )
And, rT = RT = qT = 0.
2. For each t = 1, ..., T , the path of the outside money is:
It =
t−1
Y
(1 + πz )I1
z=0
3. For each t ∈ T and i ∈ N, we get:
cit+1 = cit
4. For each t = 1, ..., T − 1, the path for the nominal interest rate
is:
(1 + ρ)(1 + πt+1 ) = (1 + Rt )
(8)
5. For each household i ∈ N, the cash bids to purchase goods for
consumption are:
PT h
Li1 = (1+π1 )(1−τ )β i I1 −(1+π1 )(1−τ )D
18
t=1
Q
−1 i
t
(1 + ρ)T −t
(1
+
π
)
z
z=1
−
PT
T −t
t=1 (1 + ρ)
PT h
−D
t=2
(1 + ρ)T −t
P
t
j=2
PT
Q
−1 i
j
πj
z=2 (1 + πz )
t=1 (1
+ ρ)T −t
And:
Lit =
t
Y
(1 + πz )Li1 +
z=2
t h
t
X
i
Y
πj (1 + πj )−1
(1 + πz ) D
j=2
z=j
For each t = 2, ..., T .
6. For each t ∈ T, the total cash bids made by the government and
the households are:
Lt =
t
Y
(1 + πz )I1 − N D
z=0
In the following example, we will demonstrate the real effect of
monetary policy as well as two opposite effects of changes in the inflation rates on the size of the NOE.
3.4
Numerical example
The economic agents in this example include 3 households, government
and an inflation-targeting central bank; all have the same discrete finite life-time span, indexed by t = 1, 2, 3. Each household i ∈ N derive
his lifetime utility from the discounted flow of private consumption,
where the time preference parameter is set to 0.05.
At time-period t = 1, each household begins with an amount
β i I1 of cash, where I1 = 10, 000, D = 1000 and β 1 = 0.5, β 2 =
0.25.
I1i =
β3 =
The government imposes income taxes at a fixed rate of τ = 0.1
and issues one period bonds Bt at a price of qt and a fixed interest
rate of r = 0.01, with a face value (par value) of one unit of money,
paid to the holder at maturity. In the second part of each period a
commodity market opens which contains a fixed amount Q = 1, 000
of the (perishable) consumption good.
By equations 7 and 8, a possible combination for the target inflation rates, set by the government, and the rates of interest, set by
19
the central bank, are: π1 = 0.010155746, π2 = 0.03, π3 = −0.04, R1 =
0.0815, R2 = 0.008.
Hence:
L11 = 3645.09, L21 = L31 = 1372.24
L1 = 7101.56
Lg1 = L1 − L11 − L21 − L31 = 711.99
p1 = 10.10, M1 = π1 I1 = 101.56
c̄11 = 360.85, c̄21 = c̄31 = 135.84, ĉ11 = ĉ21 = ĉ31 = 98.995
c11 = 459.84, c21 = c31 = 234.84, g1 = 70.48
Hence, the observed GDP is: Q̄1 = 703.02, p1 Q̄1 = 7101.56.
By budget constraint of household 1, equation 6 we get:
q1 B11 + S11 = −45.09, q1 B12 + S12 = q1 B13 + S13 = −22.55
⇒ S11 = −45.09, S12 = S13 = −22.55
P
i
By p1 g1 = N
i=1 τ I1 + q1 B1 − τ N D and q1 = 0.93 we get B1 = 10.88.
Since the households do not purchase government bonds, we get B1m =
10.88.
At period t = 2 we get:
L12 = 3784.44, L22 = L32 = 1443.41
L2 = 7404.60
Lg2 = 733.35
p2 =
L2
= 10.41, M2 = π2 I2 = 303.05
Q
c̄12 = 363.73, c̄22 = c̄32 = 138.73, ĉ12 = ĉ22 = ĉ32 = 96.11
c12 = 459.84, c22 = c32 = 234.84, g2 = 70.48
Hence, the observed GDP is: Q̄2 = 711.67, p2 Q̄2 = 7404.60.
And at period t = 3:
L13 = 3593.07, L23 = L33 = 1345.67
L3 = 6988.42
Lg3 = 704.01
20
p3 =
L3
= 9.99, M3 = π3 I3 = −416.18
Q
c̄13 = 359.72, c̄23 = c̄33 = 134.72, ĉ13 = ĉ23 = ĉ33 = 100.12
c13 = 459.84, c23 = c33 = 234.84, g3 = 70.48
Hence, the observed GDP is: Q̄3 = 699.65, p3 Q̄3 = 6988.42.
As can be seen, the government is smoothing consumption over
time and the households are smoothing their aggregate consumption
(observed plus non-observed). However, the declared household consumption is affected by the periodic inflation rate, it increases with
the increase in the inflation rate and decreases with the decline in it.
Can change in the path of the inflation rate targets {πt∗ }3t=1 lead to
a real change in the economy? In other words, does monetary policy
have a real effect on the economy?
Suppose that the combination for the target inflation rates, set by
the government, and the rates of interest, set by the central bank,
have changed to: π1 = −0.03924129, π2 = 0.09, π3 = −0.04, R1 =
0.1445, R2 = 0.008. The government and the households will still be
smoothing consumption over time, however, the government consumption will decrease, and the aggregate consumption of each household
will increase, which demonstrates a real change in the economy:
L11 = 3419.26, L21 = L31 = 1257.55
L1 = 6607.59
Lg1 = L1 − L11 − L21 − L31 = 673.23
p1 = 9.61, M1 = π1 I1 = −392.41
c̄11 = 355.89, c̄21 = c̄31 = 130.89, ĉ11 = ĉ21 = ĉ31 = 104.08
c11 = 459.98, c21 = c31 = 234.98, g1 = 70.07
Hence, the observed GDP is: Q̄1 = 687.75, p1 Q̄1 = 6607.59.
At period t = 2 we get:
L12 = 3816.99, L22 = L32 = 1460.73
L2 = 7472.27
21
First inflation rate path
t=1 t=2 t=3
0.01016 0.03 −0.04
360.84 363.73 359.72
135.84 138.73 134.72
98.995 96.11 100.12
459.84 459.84 459.84
234.84 234.84 234.84
70.48
70.48 70.48
703.02 711.67 699.65
0.2970 0.2883 0.3004
πt
c̄1t
c̄2t = c̄3t
ĉ1t = ĉ2t = ĉ3t
c1t
c2t = c3t
gt
observedQ
N OE(as%of GDP )
Second inflation rate path
t=1
t=2 t=3
−0.03924 0.09 −0.04
355.89 364.49 360.51
130.89 139.49 135.51
104.08
95.49 99.47
459.98 459.98 459.98
234.98 234.98 234.98
70.07
70.07 70.07
687.75 713.53 701.59
0.3122 0.2865 0.2984
Table 1: Monetary policy real effect
Lg2 = 733.82
p2 =
L2
= 10.47, M2 = π2 I2 = 864.68
Q
c̄12 = 364.49, c̄22 = c̄32 = 139.49, ĉ12 = ĉ22 = ĉ32 = 95.49
c12 = 459.98, c22 = c32 = 234.98, g2 = 70.07
Hence, the observed GDP is: Q̄2 = 713.53, p2 Q̄2 = 7472.27.
And at period t = 3:
L13 = 3624.31, L23 = L33 = 1362.30
L3 = 7053.38
Lg3 = 704.47
p3 =
L3
= 10.05, M3 = π3 I3 = −418.89
Q
c̄13 = 360.51, c̄23 = c̄33 = 135.51, ĉ13 = ĉ23 = ĉ33 = 99.47
c13 = 459.98, c23 = c33 = 234.98, g3 = 70.07
Hence, the observed GDP is: Q̄3 = 701.59, p3 Q̄3 = 7053.38.
In addition, and as can be seen in table 1, the decline in the inflation rate at period 1 led to an increase in the NOE (as % of GDP),
22
πt
N OE
First inflation
t=1 t=2
0.01016 0.03
0.2970 0.2883
path
t=3
−0.04
0.3004
Second inflation path
t=1
t=2 t=3
−0.03924 0.09 −0.04
0.3122 0.2865 0.2984
Third inflation
t=1
t=2
−0.06496 0.09
0.3208 0.2944
Table 2: Inflation path and the size of NOE as a % of GDP
πt
N OE
First inflation
t=1 t=2
0.01016 0.03
0.2970 0.2883
path
t=3
−0.04
0.3004
Second inflation path
t=1
t=2 t=3
−0.06496 0.09 −0.01
0.3208 0.2944 0.2973
Table 3: The impact of changing the inflation rates path on the size of NOE
(as a % of GDP)
while the increase in the inflation rate at period 2 led to a decline
in the NOE at that period. Now, suppose that the combination for
the target inflation rates and the rates of interest have changed to:
π1 = −0.06496182, π2 = 0.09, π3 = −0.01, R1 = 0.1445, R2 = 0.0395.
As illustrated in Table 2, we maintained the same increase in the rate
of inflation in the second period, but now, in contrast to the previous
result, the size of the NOE has increased. In other words, changes in
the inflation rates have an opposite direction of influence on the size of
the NOE: increasing the inflation rate in a certain period can increase,
decrease, and even not affect the size of the NOE; it all depends on
the path of the periodic inflation rates and the relative change in the
chosen inflation rate.
Let us now look at a given economy that once chooses the path of
the following inflation rates: π1 = 0.010155746, π2 = 0.03, π3 = −0.04,
and once the path of these inflation rates: π1 = −0.06496182, π2 =
0.09, π3 = −0.01. In other words, we reduced the inflation rate in the
first period and increased it in the second and third periods. The results of the NOE are presented in Table 3. Table 4 shows the direction
of change in the size of the NOE, which was to be obtained according
to the empirical approach (Positive relationship), theoretical approach
(Negative relationship) and our model.
23
path
t=3
−0.01
0.2973
Empirical
T heoretical
Current model
Sign of change
t=1 t=2 t=3
−
+
+
+
−
−
+
+
−
Table 4: The direction of change in the size of the NOE
4
Conclusion
In the first part of our study we developed a simple finite horizon
general equilibrium model, with fiat money, infinite gains to trade,
heterogeneous agents, government and a central bank. Our approach
provided a tractable dynamic model with a unique monetary equilibrium that can be computed analytically, and generated an alternative
view of inflation: inflation debt as opposed to inflation tax.
In an economy with finite number of periods and no-ponzi condition, the government can set the inflation target only for T-1 periods.
Under these conditions, if the government sets a positive inflation target in all periods but one, it follows that in that one period deflation is
inevitable. This might help to understand the emergence of deflation
in countries that face a strong pressure to decrease government debt6
in Europe (such as Greece and Spain).
However, monetary policy in the basic model without the NOE
remained neutral, as is common in the classical models. Hence, in
the second part of the paper we added to the model a non-observed
economy with tax evasion. Adding this element created the “effect
of a friction”, which allowed us to demonstrate the opposite direction
of influence that changes in the periodic inflation rate might have on
the size of the NOE. To the best of our knowledge, this is the first
theoretical model that, given the same change in the inflation rate in a
particular period, is capable to generate opposing effects on the size of
the NOE. The direction of influence depends on the path of the peri6
The phenomenon of over-indebtedness to start with and deflation following soon after,
is known in the literature since the seminal work by Fisher (1932, 1933), who coined the
term “debt deflation” (for the review of the related research see, e.g., Sau (2015)), and
further developed by Minsky (1982) and Kindleberger (2000).
24
odic inflation rates and the relative change in the chosen inflation rate.
Moreover, we showed that in this model fiat money has positive
value, a unique monetary equilibrium exists where formal and informal markets coexist and, as opposed to the “classical dichotomy”,
monetary policy is not neutral and its effects can be tracked.
In the presence of agent heterogeneity, as is true is this model,
the assumption that every agent chooses the same amount of cash
to be hidden might appear to be unrealistic, hence the next step is
to introduce endogenous tax evasion rate which depends on the enforcement mechanism in the economy, which will allow us to study
possible interactions between monetary policy and enforcement institutions. Given that the estimated size of the NOE approaches that of
a government for some countries even in the developed world, a reliable model that includes the NOE is essential for formulating sensible
fiscal and monetary policy.
25
A
Appendix: proof of proposition 1
The proof proceeds in several steps using auxiliary lemmata. First
note that given the objective of the central bank defined by a collection of inflation targets |πt∗ | < ∞, for all t ∈ T, in any equilibrium the
inflation has to satisfy |πt | < ∞ for all t ∈ T. It follows that there
is no equilibrium in which Rt < 0 for some t. Indeed, in case the
interest on savings is negative, the demand for loans by households
will be infinite, however the supply of inside money is always limited
as |πt | < ∞ for all t. So in any equilibrium Rt ≥ 0 for all t.
Qt
(1+π )I
Next, in any equilibrium pt ≤ z=1 Q z 1 < ∞, since |πz | < ∞. It
follows that the optimal consumption chosen by any household in any
period should be interior with respect to the non-negativity constraint
ct ≥ 0 as follows from the first order conditions:
u0 (cit )
− µt pt = 0,
(1 + ρ)(t−1)
µt ≥ 0
since the marginal utility at zero is infinite by assumption.
Lemma 1. In equilibrium, for each t = 1, . . . , T − 1, we get the “noarbitrage” condition:
qt =
(1 + rt )
(1 + Rt )
Proof. Consider maximization of the utility function (equation 1) of
household i ∈ N with respect to Bti and Sti . By the argument above,
the non-negativity constraint on consumption will never bind, hence
the only relevant constraint is the budget constraint, equation (2).
Hence:
u0 (cit )
(1 + rt ) pt
=
i
0
q
u (ct+1 )
t (1 + ρ) pt+1
(1 + Rt ) pt
u0 (cit )
=
i
0
(1 + ρ) pt+1
u (ct+1 )
(1 + rt )
(1 + Rt )
Note that 1 + Rt 6= 0 since Rt ≥ 0. Since in the last period households
do not save, STi = 0, and the government do not issue bonds, BT = 0,
one can set:
RT = rT = qT = 0
⇒ qt =
26
This proves statement 1 of proposition 1.
Lemma 2. Set π0 = 0. In equilibrium, for each t ∈ T, the path of the
outside money is:
t−1
Y
It =
(1 + πz )I1
z=0
Proof. Since:
pt =
pt+1 =
Lt
Q
Lt+1
Q
And:
pt+1 = (1 + πt+1 )pt
We get:
Lt+1 = (1 + πt+1 )Lt
Hence:
Lt =
t
Y
(1 + πz )L1
z=2
This proves statement 6 of proposition 1. Now, Since:
It = Lt−1
It+1 = Lt
And:
Lt = (1 + πt )Lt−1
We get, for each t = 2, .., T :
It+1 = (1 + πt )It
Hence:
It =
t−1
Y
(1 + πz )I1
z=0
This proves statement 2 of proposition 1.
Lemma 3. Set R0 = r0 = π0 = πT +1 = 0. In equilibrium, for each
i ∈ N, we get:
Li1 = (1 + π1 )β i (1 − τ )I1
27
Proof. Using the “no-arbitrage” condition and equation 2, the multiperiod budget constraint for household i ∈ N will be:
T
X
t=1
T
X (1 − τ )I i
pt cit
t
=
Qt−1
Qt−1
h=0 (1 + Rh )
h=0 (1 + Rh )
t=1
(9)
Hence, the lagrangian L1 , which is obtained from the utility function
(equation 1) and the multi-period budget constraint (equation 9), is:
L1 =
T
X
t=1
T
T
X
X
pt cit
(1 − τ )Iti
ui (cit )
−
λ
(
−
)
Q
Q
1
t−1
t−1
(1 + ρ)(t−1)
h=0 (1 + Rh )
h=0 (1 + Rh )
t=1
t=1
For each t ∈ T and i ∈ N, the first-order conditions with respect to cit
and cit+1 are:
∂L1
1
λ1 pt
=0
=
u0 (cit ) − Qt−1
i
(t−1)
∂ct
(1 + ρ)
h=0 (1 + Rh )
∂L1
1
λ1 pt+1
=0
=
u0 (cit+1 ) − Qt
i
(t)
∂ct+1
(1 + ρ)
h=0 (1 + Rh )
⇒
u0 (cit )
(1 + Rt ) pt
(1 + Rt )
1
=
=
i
0
(1 + ρ) pt+1
(1 + ρ) (1 + πt+1 )
u (ct+1 )
(10)
Where λ1 is the lagrangian multiplier and h ∈ T .
Next, since cit =
Lit
pt
=
Lit Q
Lt
=
Lit Q
g,
i
i=1 Lt +Lt
PN
the household’s objective
function and the multi-period budget constraint becomes:
Li Q
V i (Li1 , ..., LiT ) =
i
T u ( PN t j
g)
X
j=1 Lt +Lt
(1 + ρ)(t−1)
t=1
T
X
t=1
(11)
T
X (1 − τ )I i
Lit
t
=
Qt−1
Qt−1
h=0 (1 + Rh )
h=0 (1 + Rh )
t=1
(12)
Hence, the lagrangian L2 , which is obtained from the utility function
(equation 11) and the multi-period budget constraint (equation 12),
is:
Li Q
L2 =
i
T u ( PN t j
g)
X
j=1 Lt +Lt
t=1
(1 + ρ)(t−1)
T
T
X
X
(1 − τ )Iti
Lit
−λ2 (
−
)
Qt−1
Qt−1
h=0 (1 + Rh ) t=1
h=0 (1 + Rh )
t=1
28
The first-order conditions with respect to Lit and Lit+1 are:
i
∂L2
1
λ2
0 i QLt − QLt
=
u
(c
)
− Qt−1
=0
t
i
2
(t−1)
(Lt )
∂Lt
(1 + ρ)
h=0 (1 + Rh )
QLt+1 − QLit+1
∂L2
1
λ2
0 i
=0
u
(c
)
=
− Qt
t+1
i
2
(t)
(Lt+1 )
∂Lt+1
(1 + ρ)
h=0 (1 + Rh )
⇒
u0 (cit )
(1 + Rt ) Lt+1 − Lit+1
1
=
i
i
0
(1 + ρ) Lt − Lt (1 + πt+1 )2
u (ct+1 )
By equation 10, we get:
Lt+1 − Lit+1 = (1 + πt+1 )(Lt − Lit )
Since:
Lt+1 = (1 + πt+1 )Lt
We get:
Lit+1 = (1 + πt+1 )Lit
And for each t = 2, ..., T we get:
Lit =
t
Y
(1 + πz )Li1
(13)
z=2
Since for t = T we must have:
LiT = (1 − τ )ITi + (1 + RT −1 )STi −1 + (1 + rT −1 )BTi −1
By equation 12 we get:
Li1
PT −1 QT −1
Qt
i
i
i
t=1 ( h=t (1 + Rh )
z=2 (1 + πz )) + ((1 − τ )IT + (1 + RT −1 )ST −1 + (1 + rT −1 )BT −1 )
=
QT −1
(1
+
R
)
h
h=0
=
T
X
t=1
Li1
(1 − τ )Iti
Qt−1
h=0 (1 + Rh )
PT −1 QT −1
Qt
i
i
t=1 ( h=t (1 + Rh )
z=2 (1 + πz )) + ((1 + RT −1 )ST −1 + (1 + rT −1 )BT −1 )
=
QT −1
h=0 (1 + Rh )
29
=
T
−1
X
t=1
Since:
It =
(1 − τ )Iti
Qt−1
h=0 (1 + Rh )
t−1
Y
(1 + πz )I1
z=0
And:
LiT = (1 − τ )ITi + (1 + RT −1 )STi −1 + (1 + rT −1 )BTi −1 =
T
Y
(1 + πz )Li1
z=2
We get:
(1 + RT −1 )STi −1 + (1 + rT −1 )BTi −1 =
T
Y
(1 + πz )Li1 − (1 − τ )ITi
z=2
Substituting above yields:
Qt−1
PT QT
(1 + πz ))
i
t=1 ( h=t (1 + Rh )
(1 + π1 )(β i (1 − τ )I1 )
L1 = PT QT
Qtz=0
(
(1
+
R
)
(1
+
π
))
z
h
t=1
h=t
z=0
Next, Since:
cit =
cit+1 =
Lit
pt
Lit+1
pt+1
By equation 13, we get:
cit = cit+1
This proves statement 3 of proposition 1.
Hence, by equation 10, for each t = 1, ..., T − 1 we get:
(1 + ρ)(1 + πt+1 ) = (1 + Rt )
(14)
note that since Rt ≥ 0, for each t = 1, ..., T − 1 we get:
ρ
πt+1 ≥ −
1+ρ
And π1 must comply with equation 4.
This proves statement 4 of proposition 1.
Substituting 14 in:
PT QT
Qt−1
(1 + πz ))
i
t=1 ( h=t (1 + Rh )
L1 = PT QT
(1 + π1 )(β i (1 − τ )I1 )
Qz=0
t
(
(1
+
R
)
(1
+
π
))
z
h
t=1
h=t
z=0
30
Yields:
PT h
t=1
Li1 =
i
(1 + ρ)T −t (1 + πt )−1
((1 + π1 )β i (1 − τ )I1 )
PT
T
−t
t=1 (1 + ρ)
Next, using the “no-arbitrage” condition, the multi-period budget constraint for the central bank will be:
T
X
Mt
=0
h=0 (1 + Rh )
Qt−1
t=1
Substituting Mt = πt It = πt
T Y
T
X
Qt−1
z=0 (1
(1 + Rh )πt
t=1 h=t
(15)
+ πz )I1 , we get:
t−1
Y
(1 + πz )I1 = 0
z=0
By equation 14 we get:
T h
X
i
(1 + ρ)T −t (1 + πt )−1 πt = 0
(16)
t=1
Which yields:
i
j−t (1 + π )−1 π
(1
+
ρ)
t
t
t6=j
i
pij =
P h
1 + t6=j (1 + ρ)j−t (1 + πt )−1 πt
−
P
h
Hence, the government can set the inflation target for only (T − 1)
periods.
Next, by equation 16 we get:
T h
X
T h
i
i X
(1 + ρ)T −t (1 + πt )−1 =
(1 + ρ)T −t
t=1
t=1
Substituting in:
PT h
Li1 =
t=1
i
(1 + ρ)T −t (1 + πt )−1
((1 + π1 )β i (1 − τ )I1 )
PT
T
−t
t=1 (1 + ρ)
Yields:
Li1 = (1 + π1 )β i (1 − τ )I1
This proves statement 5 of proposition 1.
31
B
Appendix: proof of proposition 2
The proof proceeds in several steps using auxiliary lemmata:
Lemma 4. In equilibrium, for each t = 1, . . . , T − 1, we get the “noarbitrage” condition:
qt =
(1 + rt )
(1 + Rt )
Proof. Consider maximization of the utility function of household
i ∈ N (equation 5) with respect to Bti and Sti . The non-negativity
constraint on consumption will never bind, hence the only relevant
constraint is the budget constraint, equation (6). Hence:
u0 (cit )
(1 + rt ) pt
=
qt (1 + ρ) pt+1
u0 (cit+1 )
u0 (cit )
(1 + Rt ) pt
=
i
0
(1 + ρ) pt+1
u (ct+1 )
⇒ qt =
(1 + rt )
(1 + Rt )
Note that 1 + Rt 6= 0 since Rt ≥ 0. Since in the last period households
do not save, STi = 0, and the government do not issue bonds, BT = 0,
one can set:
RT = rT = qT = 0
This proves statement 1 of proposition 2.
Lemma 5. Set π0 = 0. In equilibrium, for each t ∈ T, the path of the
outside money is:
t−1
Y
It =
(1 + πz )I1
z=0
Proof. Since:
pt =
pt+1 =
Lt + N D
Q
Lt+1 + N D
Q
And:
pt+1 = (1 + πt+1 )pt
32
We get:
Lt+1 + N D = (1 + πt+1 )(Lt + N D)
Next, since:
It = Lt−1 + N D
It+1 = Lt + N D
And:
Lt + N D = (1 + πt )(Lt−1 + N D)
We get, for each t = 2, ..., T :
It+1 = (1 + πt )It
Thus:
It =
t−1
Y
(1 + πz )I1
z=0
This proves statement 2 of proposition 2.
Lemma 6. Set R0 = r0 = π0 = πT +1 = 0. In equilibrium, for each
i ∈ N, we get:
PT h
Li1 = (1+π1 )(1−τ )β i I1 −(1+π1 )(1−τ )D
t=1
Q
−1 i
t
(1 + ρ)T −t
(1
+
π
)
z
z=1
−
PT
T −t
t=1 (1 + ρ)
P Q
−1 i
PT h
j
t
T −t
z=2 (1 + πz )
t=2 (1 + ρ)
j=2 πj
−D
PT
T −t
t=1 (1 + ρ)
Proof. Using the “no-arbitrage” condition and equation 6, the multiperiod budget constraint for household i ∈ N will be:
T
X
T
X (1 − τ )(I i − D)
pt c̄it
t
=
Qt−1
(1
+
R
)
(1
+
Rh )
h
h=0
h=0
t=1
Qt−1
t=1
(17)
Hence, the lagrangian L3 , which is obtained from the utility function
(equation 5) and the multi-period budget constraint (equation 17), is:
L3 =
T
X
t=1
T
T
hX p (ci − ĉi )
X (1 − τ )(I i − D) i
ui (cit )
t t
t
t
−
λ
−
Qt−1
Qt−1
3
(1 + ρ)(t−1)
h=0 (1 + Rh )
h=0 (1 + Rh )
t=1
t=1
33
For each t ∈ T and i ∈ N, the first-order conditions with respect to cit
and cit+1 are as follows:
∂L3
λ3 pt
1
u0 (cit ) − Qt−1
=0
=
i
(t−1)
∂ct
(1 + ρ)
h=0 (1 + Rh )
∂L3
λ3 pt+1
1
0 i
u
(c
)
−
=0
=
Q
t+1
t
i
∂ct+1
(1 + ρ)(t)
h=0 (1 + Rh )
⇒
u0 (cit )
(1 + Rt ) pt
(1 + Rt )
1
=
=
i
0
(1 + ρ) pt+1
(1 + ρ) (1 + πt+1 )
u (ct+1 )
(18)
Where λ3 is the lagrangian multiplier and h ∈ T.
Next, since:
cit =
(Lit + D)Q
Lit + D
= PN
g
i
pt
i=1 Lt + Lt + N D
The household’s objective function and the multi-period budget constraint becomes:
V i (Li1 , ..., LiT ) =
(Lit +D)Q
)
g
i
i=1 Lt +Lt +N D
(1 + ρ)(t−1)
T ui ( PN
X
t=1
T
X
t=1
(19)
T
X (1 − τ )(I i − D)
Lit
t
=
Qt−1
Qt−1
(1
+
R
)
(1
+
Rh )
h
h=0
h=0
t=1
(20)
Hence, the lagrangian L4 , which is obtained from the utility function
(equation 19) and the multi-period budget constraint (equation 20),
is:
L4 =
(Lit +D)Q
T
)
hX
g
i
i=1 Lt +Lt +N D
−λ
4
(1 + ρ)(t−1)
t=1
T ui ( PN
X
t=1
T
X
Lit
(1 − τ )(Iti − D) i
−
Qt−1
Qt−1
h=0 (1 + Rh ) t=1
h=0 (1 + Rh )
The first-order conditions with respect to Lit and Lit+1 are:
i
∂L4
1
λ4
0 i Q(Lt + N D) − QLt − QD
=
u
(c
)
− Qt−1
=0
t
i
(Lt + N D)2
∂Lt
(1 + ρ)(t−1)
h=0 (1 + Rh )
Q(Lt+1 + N D) − QLit+1 − QD
∂L4
1
λ4
0 i
=
u
)
− Qt
(c
=0
t+1
i
2
(t)
(Lt+1 + N D)
∂Lt+1
(1 + ρ)
h=0 (1 + Rh )
34
⇒
(1 + Rt ) Lt+1 + N D − Lit+1 − D
u0 (cit )
1
=
i
i
0
(1 + ρ) Lt + N D − Lt − D (1 + πt+1 )2
u (ct+1 )
By equation 18, we get:
Lt+1 + N D − Lit+1 − D = (1 + πt+1 )(Lt + N D − Lit − D)
Since:
Lt+1 + N D = (1 + πt+1 )(Lt + N D)
we get:
Lit+1 = (1 + πt+1 )Lit + πt+1 D
Thus:
Li2 = (1 + π2 )Li1 + π2 D
h
i
Li3 = (1 + π3 )(1 + π2 )Li1 + (1 + π3 )π2 + π3 D
h
i
i
i
L4 = (1 + π4 )(1 + π3 )(1 + π2 )L1 + (1 + π4 )(1 + π3 )π2 + (1 + π4 )π3 + π4 D
..
.
LiT =
T
Y
(1 + πz )Li1 +
T h
X
z=2
πj (1 + πj )−1
T
Y
(1 + πz )
i
D
z=j
j=2
Hence, for each t = 2, ..., T we get:
Lit
=
t
Y
(1 +
z=2
πz )Li1
t h
t
X
i
Y
−1
+
πj (1 + πj )
(1 + πz ) D
j=2
(21)
z=j
And for t = T we must also have:
pT c̄iT = (1 − τ )(ITi − D) + (1 + RT −1 )STi −1 + (1 + rT −1 )BTi −1
Next, since:
cit =
cit+1 =
Lit + D
pt
Lit+1 + D
pt+1
And:
Lit+1 = (1 + πt+1 )Lit + πt+1 D
We get:
cit = cit+1
35
This proves statement 3 of proposition 2.
Hence, by equation 18, for each t = 1, ..., T − 1 we get:
(1 + ρ)(1 + πt+1 ) = (1 + Rt )
(22)
note that since Rt ≥ 0, for each t = 1, ..., T − 1 we get:
ρ
πt+1 ≥ −
1+ρ
And π1 must comply with equation 7.
This proves statement ?? of proposition 2.
Next, Using the “no-arbitrage” condition, the multi-period budget
constraint for the central bank will be:
T
X
Mt
=0
h=0 (1 + Rh )
t=1
Q
Substituting Mt = πt It = πt t−1
z=0 (1 + πz )I1 , we get:
(23)
Qt−1
T Y
T
X
(1 + Rh )πt
t=1 h=t
t−1
Y
(1 + πz )I1 = 0
z=0
By equation 22 we get:
T h
i
X
(1 + ρ)T −t (1 + πt )−1 πt = 0
t=1
h
i
j−t (1 + π )−1 π
(1
+
ρ)
t
t
t6=j
i
πj =
P h
1 + t6=j (1 + ρ)j−t (1 + πt )−1 πt
−
P
(24)
Hence, the government can set the inflation target for only (T − 1)
periods.
Q
Next, by multiplying both sides of equation 20 by Th=0 (1 + Rh ), we
get:
T h
X
t=1
Lit
T
Y
i
(1 + Rh ) =
T h
X
(1 −
t=1
h=t
τ )(Iti
− D)
T
Y
i
(1 + Rh )
(25)
h=t
Qt−1
Using It = z=0
(1 + πz )I1 , equation 22 and summing equation 21
from t = 2 to T , we get:
T +1
T h
T
T t i
i
hX
X
X
Y
πj Y
(1+πz )
=
Li1 (1+ρ)T −t
(1+πz ) +D
(1+ρ)T −t
1 + πj
t=1
z=2
t=2
36
j=2
z=j
= (1 − τ )β i I1
T h
X
(1 + ρ)T −t
t=1
−(1 − τ )D
T
Y
i
(1 + πz )(1 + πt )−1 −
z=1
T hX
(1 + ρ)T −t
t=1
TY
+1
(1 + πz )
i
z=t+1
Thus:
PT h
Li1 = (1+π1 )(1−τ )β i I1 −(1+π1 )(1−τ )D
t=1
Q
−1 i
t
(1 + ρ)T −t
(1
+
π
)
z
z=1
−
PT
T −t
t=1 (1 + ρ)
P Q
−1 i
PT h
j
t
T −t
(1
+
ρ)
π
(1
+
π
)
j
z
z=2
t=2
j=2
−D
PT
T
−t
t=1 (1 + ρ)
This proves statement 5 of proposition 2.
Lemma 7. For each t ∈ T, the total cash bids made by the government
and the households are:
Lt =
t
Y
(1 + πz )I1 − N D
z=0
Proof. Since:
Lt + N D = It+1
And:
t−1
Y
It =
(1 + πz )I1
z=0
We get:
Lt =
t
Y
(1 + πz )I1 − N D
z=0
This proves statement 6 of proposition 2.
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