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Chapter 5
A ClosedEconomy
One-Period
Macroeconomic
Model
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Chapter 5 Topics
• Construct closed-economy one-period
macroeconomic model, which has: (i)
representative consumer; (ii) representative
firm; (iii) government.
• Introduce the government.
• Economic efficiency and Pareto optimality.
• Experiments: Increases in government
spending and total factor productivity.
• Consider a distorting tax on wage income
and study the Laffer curve.
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Closed-Economy One-Period
Macroeconomic Model
There are three different actors in this
economy:
• the representative consumer who stands in
for the many consumers in the economy
who sell labour and buy consumption
goods.
• the representative firm that stands in for
the many firms in the economy that buy
labour and sell consumption goods.
• the government
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The Government
• In this one-period closed economy model the
behaviour of the government is quite simple.
• In this model, the government wants to purchase a
given quantity of consumption goods, G, and
finances these purchases by taxing the
representative consumer, which is denoted by T.
• In practice, governments provide many different
goods and services, including roads and bridges,
national defence, air traffic control, and education.
The Government
• Which goods and services the government should
provide is subject to both political and economic
debate.
• But economists generally agree that the government
has a special role to play in providing public goods,
which have two special charactristics: nonrivalness in
consumption and nonexcludability in the benefits of
consumption, that are difficult or impossible for the
private sector to provide.
• Example of a public good: National defence
The Government
• To keep things simple, for now in the model we will
not be specific about the public-ggods nature of
government expenditures.
• What we want to capture here is that government
spending uses up resources, and we will model this
by assuming that government spending simply
involves taking goods from the private sector.
• Output is produced, and the government purchases
an exogenous amount G of this output, with the
remainder consumed by the representative
consumer.
The Government
• An exogenous variable is determined outside the model,
while an endogenous variable is determined by the model
itself.
• Government spending is exogenous in our model, as we are
assuming that government spending is independent of what
happens in the rest of the economy.
• The government must abide by the government budget
constraint, which we write as
G = T,
or government purchases (G) equal to taxes (T), in real terms.
The Government – Fiscal Policy
• Introducing the government in this way allows
us to study some basic effects of fiscal policy.
• In general, fiscal policy refers to the
government’s choices over its expenditures,
taxes, transfers, and borrowing.
• In the current one-period model, the
government cannot borrow to finance
government expenditures, since there is no
future in which to repay its debt.
The Government – Fiscal Policy
• The government does not tax more than it spends, as
this would imply that the government would
foolishly throw goods away.
• The government budget deficit, which is G – T here,
is always zero.
• Thus, only elements of fiscal policy we will study in
Chapter 5 are the setting of government purchases,
G, and the macroeconomic effects of changing G.
• In Chapter 8, we will explore what happens when the
government run deficits and surpluses.
Competitive Equilibrium
• We have to now understand how consistency
is obtained in the actions of all three
economic agents.
• Mathematically, a macroeconomic model
takes the exogenous variables and
determines values for the endogenous
variables, as outlined in Figure 5.1.
A Model Takes Exogenous Variables
and Determines Endogenous Variables
(Figure 5.1)
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Competitive Equilibrium
• Exogenous variables in the model: G, z and K.
• Endogenous variables: C, Ns, Nd, T, Y and w
Competitive Equilibrium
• Representative consumer optimizes
given market prices.
• Representative firm optimizes given
market prices.
• The labor market clears.
• The government budget constraint is
satisfied, or G = T.
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Income-Expenditure Identitity
In a competitive equilibrium, the incomeexpenditure identity is satisfied, so
Y=C+G
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The Production Function
• We will work with the this simple
macroeconomic model in graphical form.
• Start with representing the production
function in graphical form.
• In a competitive equilibrium, Nd = Ns = N and
we will refer to N as employment.
• Production function: Y=zF(K,N)
(5.4)
• We graph the production function in Figure
5.2(a) for a given capital stock K.
The Production Function
• The maximum output that could be produced in this
economy is Y* in Figure 5.2(a).
• In equilibrium, N = h – l.
• Substituting for N in the production function (5.4),
we get
Y = zF(K, h-l),
(5.5)
which is relationship between output Y and leisure l,
given exogenous variables z and K.
• This relationship is graphed in Figure 5.2(b) and we
get a mirror image of the production function in
Figure 5.2(a).
The Production Function and the
Production Possibilities Frontier
(Figure 5.2)
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The Production Function
• Note that since the slope of the production function in Figure
5.2(a) is MPN, the marginal product of labour, the slope of the
relationship in Figure 5.2(b) is – MPN, since this relationship is
just the mirror image of the production function.
• Since in equilibrium C = Y – G, from the income-expenditure
identity, from (5.5) we get
C = zF(K, h-l) – G,
which is a relationship between C and l, given the exogenous
variables z, K and G.
• This relationship, graphed in Figure 5.2(c), is just the
relationship in Figure 5.2(b) shifted down by the amount of G.
Production Possibilities Frontier (PPF)
• The relationship in Figure 5.2(c) is called a
production possibilities frontier (PPF).
• It describes what technological possibilities
are for the economy as a whole, in terms of
the production of consumption goods and
leisure.
• All of the points in the shaded area inside the
PPF and on the PPF in Figure 5.2(c) are
technologically possible in this economy.
Production Possibilities Frontier (PPF)
• The PPF captures the tradeoff between leisure
and consumption that the available
production technology makes available to the
representative consumer in the economy.
• Note that the points on the PPF on line AB are
not feasible for this economy, as consumption
is negative.
• Only the points on the PPF on line DB are
feasible.
Production Possibilities Frontier (PPF)
• As in Figure 5.2(b), the slope of the PPF in Figure
5.2(c) is –MPN.
• The negative of the slope of the PPF is called the
marginal rate of transformation.
• The marginal rate of transformation is the rate at
which one good can be converted technologically
into another; in this case, the marginal rate of
transformation is the rate at which leisure can be
converted in the economy into consumption goods
through work.
• MRTl,c = MPN = -(slope of the PPF)
Competitive Equilibrium
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Key Properties of a Competitive
Equilibrium
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Pareto Optimality
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Key Properties of a Pareto Optimum
• In this model, the competitive equilibrium
and the Pareto optimum are identical, as
the marginal rate of substitution is equal
to the marginal rate of transformation.
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First and Second Welfare Theorems
• These theorems apply to any
macroeconomic model
• First Welfare Theorem: Under certain
conditions, a competitive equilibrium is
Pareto optimal.
• Second Welfare Theorem: Under certain
conditions, a Pareto optimum is a
competitive equilibrium.
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Sources of Social Inefficiencies
•
There are three reasons why a competitive
equilibrium could fail to be Pareto-optimal.
1. Externalities
2. Distorting taxes
3. Monopoly Power
Using the Second Welfare Theorem to
Determine a Competitive Equilibrium
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Effects of an Increase in G
• Essentially a pure income effect
• C decreases, l decreases, Y increases, w
falls
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Equilibrium Effects of an Increase in
Government Spending
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World War II Increase in G
• Very large increase in G
• Y increases, C decreases by a small amount
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GDP, Consumption, and Government
Expenditures
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Effects of an Increase in z (or an
increase in K)
• PPF shifts out, and becomes steeper –
income and substitution effects are
involved.
• C increases, l may increase or decrease, Y
increases, w increases
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Increase in Total Factor Productivity
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Competitive Equilibrium Effects of an
Increase in Total Factor Productivity
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Income and Substitution Effects of an
Increase in Total Factor Productivity
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Deviations from Trend in Real GDP and the
Solow Residual
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The Relative Price of Energy and the
Solow Residual
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A Simplified Model with a
Proportional Income Tax
• Use the model to study the incentive
effects of the income tax, and to derive
the “Laffer curve.”
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Production function without capital
• Labor is the only input, but there is still
constant returns to scale (linear
production function).
Y = zNd
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Production Possibilities Frontier
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Consumer’s budget constraint
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Profits for the firm
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The consumer’s budget constraint in
equilibrium
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The Production Possibilities Frontier in the
Simplified Model
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The Labor Demand Curve in the
Simplified Model
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Competitive Equilibrium in the Simplified
Model with a Proportional Tax on Labor
Income
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Revenue for the government given
the tax rate t
• REV = tz[h-l(t)]
• G = tz[h-l(t)]
(5.12)
A Laffer Curve
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There Can Be Two Competitive
Equilibria
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