MACROECONOMICS I, Phd Programme

Lecture 1: The Centralized Economy

Sarantis Kalyvitis

1

Modern Macroeconomics

Dynamic general equilibrium (DGE) macroeconomics: The aggregate

economy with microeconomic foundations (no ad hoc assumptions)

DGE models have emerged in recent years as the latest step in the

development of macroeconomics from its origins in the work of

Keynes in the 1930s

DGE models can cover all aspects of macroeconomics, e.g.:

¾ neoclassical and (new-)Keynesian models

¾ fiscal and monetary policy with rigorous implications

¾ asset pricing

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What is a DGE model?

A system that reflects the collective decisions of rational individuals

over a range of variables that relate to both the present and the future.

Individual decisions are then coordinated through markets to produce

the macroeconomy.

The economy is viewed as being in continuous equilibrium in the

sense that, given the information available, people make decisions

that appear to be optimal for them, and so do not make persistent

mistakes.

Behavior is said to be rational: any errors are attributed to

information gaps, such as unanticipated shocks to the economy.

3

Short-run and long-run equilibria

The economy is assumed to always be in short-run equilibrium.

The long run, or the steady state, is a mathematical property of the

macroeconomic model that describes its path when all past shocks

have fully worked through the system.

¾ static equilibrium: all variables are constant

¾ growth equilibrium: in the absence of shocks, there is no tendency

for the economy to depart from a given path in which the main

macroeconomic aggregates grow at the same rate

The (short or long run) equilibrium is general because all variables

are assumed to be simultaneously in equilibrium -not just some of

them or a particular market, known as partial equilibrium).

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Individual decisions

Individual decisions are assumed to be based on maximizing the

discounted sum of current and future expected welfare subject to

preferences and four constraints:

¾ budget or resource constraints

¾ endowments

¾ available technology

¾ information.

DGE macroeconomics is intertemporal: whether to consume today,

or save today in order to consume in the future.

This entails being able to transfer today’s income for future use, or

future income for today’s use.

5

Achieving equilibrium in DGE models

Transfers may be achieved by holding financial assets or by

borrowing against future income.

The different decisions are reconciled through the economy-wide

market system, and by market prices (including asset prices).

Stock (not only flows) are important

The focus of modern macroeconomics is, therefore, on:

¾ the responses of individuals to shocks

¾ how these are likely to affect multiple markets simultaneously,

both in the present and in the future.

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Decisions by agents

Three main types of decisions are taken by economic agents:

¾ goods and services

¾ labor

¾ assets: physical assets (capital stock, durables, housing, etc.) or

financial assets (money, bonds, and equity)

Decisions of individuals are made according to its type:

¾ Household: consumption, labor supply, and asset

¾ Firm: supply of goods and services, labor demand, investment,

productive and financial capital, and the use of profits.

¾ Government: spending, taxes, transfers, base money, public debt.

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Forward-looking decisions

Current decisions are affected by expectations about the future.

We need intertemporal dynamic optimization, in which people are

treated as rationally processing current information about the future

when making their decisions; this is linked to rationality.

Traditional macroeconomics: myopic decision making in which

mistakes, even when realized, were often persistent.

Rational expectations: people do not make persistent mistakes once

they are identified.

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Information and mistakes

RE do not imply that people have more or less complete knowledge -

the mistakes are largely the result of shocks or unanticipated

information gaps.

In RE it is sufficient to suppose that mistakes are not repeated.

As forward-looking decisions must be based on expectations of the

future, they may be incorrect; decisions that seem correct ex ante may

not therefore be correct ex post.

The economy is in disequilibrium only as a result of this type of

mistake.

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A note on macroeconomic modelling

All models are abstractions from reality

Macroeconomic models are highly stylized

Assumptions are crucial

We proceed from simple to more complex models

We try to verify the robustness of the results and relax or add

assumptions as necessary

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The Ramsey model: Centralized economy

National income identity:

Capital accumulation:

Production function:

,

,

Inada conditions:

Constant population N :

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The economy-wide resource constraint

Combining the structural equations:

(economy-wide resource constraint)

Given an initial stock of capital, kt (the endowment), the economy

must choose its preferred level of consumption for period t, namely

ct, and capital at the start of period t +1, namely kt+1.

This can be shown to be equivalent to choosing consumption for

periods t, t+1, t+2, . . . , with the preferred levels of capital, output,

investment, and savings for each period obtained from the model.

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Preferences in the economy

Consumption at period t:

Consumption should be sustainable: in each period investment is

required to maintain the capital stock and to produce next period’s

output.

In effect, we are assuming that the aim is to maximize consumption

in each period.

With no distinction being made between current and future

consumption, the problem has been converted from one with a very

short-term objective to one with a very long-term objective.

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Long-run consumption

In the long run kt+1=kt:

Consumption in the long run is output less that part of output

required to replace depreciated capital in order to keep the stock of

capital constant.

In the long run the only investment undertaken is that to replace

depreciated capital; the output that remains can be consumed.

Choose k to max c:

,

Capital should be accumulated up to the point where its marginal

product equals the depreciation rate.

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Maximum level of consumption

, with ∆k=0 in the long run

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Maximum level of output and the ‘golden rule’

k# is the ‘golden rule’ capital stock.

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Stability and the ‘golden rule’

Due to the constraint that the capital stock is constant, c# is

sustainable indefinitely if there are no disturbances to the economy.

If there are disturbances, the economy becomes dynamically unstable

at {c#, k#}.

To maintain c#, it would be necessary to consume some of the

existing k (∆k < 0), and k would no longer be constant. Future output

would be even smaller and attempts to maintain consumption at c#

would cause further decreases in the capital stock.

The economy can reduce its consumption temporally and divert

output to rebuilding the capital stock to a level that restores the

original equilibrium.

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Optimal determination of consumption

Intertemporal discounting of utility

Discount factor:

The problem is to maximize Vt subject the economy-wide resource

constraint.

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Lagrangian and FOCs

Lagrange equation:

First-order conditions with respect to ct+s , kt+s (kt is predetermined!)

Transversality condition:

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The Euler equation

Eliminating the Lagrange multiplier from the FOCs:

(Euler equation)

The Euler equation is the fundamental dynamic equation in

intertemporal optimization problems in economics with dynamic

constraints.

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Interpretation of the Euler equation

Two-period utility:

reduction by dct = gain in dct+1:

current utility loss + future utility gain = 0

By the resource constraint:

21

Intertemporal Production Possibility Frontier

Concave relation between ct+1 and ct:

22

Intertemporal PPF: graphical illustration

Equilibrium:

23

Static equilibrium

This is different from the ‘golden rule’ due to discounting.

The required capital return is higher and thus the capital stock is

lower in the steady state.

24

Optimal long-run consumption

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Dynamics of optimal solution

Dynamic equations:

Linearization of the Euler equation:

26

Graphical illustration of dynamics for c and k

Consumption Capital

27

Phase diagram

28

The effect of a positive technology shock

29

Endogenous labor supply

Up to now, the labor supply by households is assumed to be

exogenous (vertical labor supply).

We can introduce an upward sloping labor curve by assuming that

labor supply depends on the wage rate through the utility function.

Higher leisure (less labor) increases utility, like consumption.

The household needs to be compensated by a higher wage rate to

offer more units of labor.

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Leisure in the utility function

There is one unit of time divided between labor, n, and leisure, l:

nt + lt = 1

Instantaneous utility function: U = U(nt , lt )

with Uc > 0, Ul > 0, Ucc ≤ 0, Ull ≤ 0

Production function: F = F(nt , kt )

with Fn > 0, Fk > 0, Fnn ≤ 0, Fkk ≤ 0, Fkn ≥ 0, Inada conditions

Resource constraint:

31

Utility maximization with endogenous labor supply

Lagrange:

w.r.t.

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Optimality conditions with endogenous labor supply

Euler equation:

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Role of leisure in optimal behavior

Consider giving up dlt = -dnt < 0 units of leisure.

Loss of utility is Ul,t dlt < 0 (left-hand side).

This is compensated by an increase in utility due to producing extra

output of Fn,t dnt

= -Fn,t dlt .

When consumed, each unit of output gives extra Uc,t in utility,

implying a total increase in utility of -Uc,tFn,t dlt > 0 (right-hand side

when dlt = -1).

34

Steady state with leisure

Capital return:

Constant returns to scale:

since

we get:

Real wage rate:

In the steady state rt = θ:

35

Comparison with exogenous supply model

Exogenous supply: nt = 1

Implied wage rate:

Equilibrium:

The solutions for consumption and capital do not change

36

Investment

The basic model obtains the optimal levels of consumption and

capital, which implies a theory of net investment.

Following a permanent change in the steady state capital stock, the

adjustment to the new equilibrium is gradual along the saddle path.

The adjustment path implies an optimal level of investment each

period, which will differ each period until the steady state is reached,

where investment is only replacing depreciated capital.

Investment adjusts instantaneously to the optimal level.

In practice, however, due to costs of installation, optimal investment

adjusts slowly.

37

Adjustment costs for investment

Additional resource costs for investment:

Resource constraint:

Lagrange:

38

Optimality with adjustment costs for investment

By FOC for investment:

Tobin’s q:

39

Interpretation of q

An extra unit of capital raises output, and hence consumption and

utility

λ : marginal benefit in terms of the utility of sacrificing a unit of

current consumption in order to have an extra unit of investment, and

hence the extra capital.

µ : marginal benefit in terms of utility of an extra unit of investment.

q : benefit from investment per unit of benefit from capital.

Expressing utility in terms of units of output, q can also be

interpreted as the ratio of the market value of one unit of investment

to its cost.

40

Steady-state value of q

In the long run:

Investment demand:

Long-run value of q:

41

Solution for capital

FOCs:

By the long-run value of q:

In the absence of costs of installation, φ = 0:

qt = 1 and Fk = θ+δ (basic closed-economy model)

Fk > θ+δ implies a lower capital stock because installation costs

reduce the resources available for consumption and investment.

42

Short-run solution with investment adjustment costs

Linear approximation:

Deviations from equilibrium at steady-state values:

Forward-looking solution for q:

qt is the present value of the extra output produced by undertaking

one more unit of investment.

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Interpretation of q and measurement problems

The greater the value of q, the more investment will be undertaken in

period t.

Since the price of one unit of investment is 1, qt -1 is the increase in

the implied value of the firm.

Although qt can be interpreted as the ratio of the market value of one

unit of investment to its cost, it is often estimated by the ratio of the

market value of a firm to its book value.

This implies using the average value of current and past investment

instead of the marginal value of new investment.

Hayashi (1982) proved that this can hold under strict conditions.

44

Dynamics of k and q

Capital dynamics:

Dynamics of q and k:

Approximation around the steady state:

Negative relation between k and q:

45

Graphical analysis of dynamics between k and q

46

Effects of a productivity increase

A productivity increase raises qt so that qt > 1. This induces a rise in

investment above its normal replacement level δk.

Initially, kt remains unchanged and so the economy moves to point B.

New investment increases the capital stock each period until the

economy reaches its new long-run equilibrium at C by moving along

the saddle path from B.

At this point qt is restored to its long-run equilibrium level of one and

the equilibrium capital stock, output, and consumption are

permanently higher.

47

Time-to-build investment

Kydland and Prescott (1982): investment expenditures recorded at

time t are the result of decisions to invest is t made earlier.

A proportion φi of recorded investment in period t is investment starts

made in period (t-i):

Α proportion φi of investment undertaken in period t is installed and

ready for use as part of the capital stock by period (t+i):

where

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