Chapter 4

The Lucas Model

The present-value interpretation of the monetary model underscores the idea that we should expect the exchange rate to behave like the prices of other assets–such as stocks and bonds. This is one of that model’s attractive features. One of its unattractive features is that the model is ad hoc in the sense that the money demand functions upon which it rests were not shown to arise explicitly from decisions of optimizing agents. Lucas’s [95] neoclassical model of exchange rate determination gives a rigorous theoretical framework for pricing foreign exchange and other assets in a ßexible price environment and is not subject to this criticism. It is a dynamic general equilibrium model of an endowment economy with complete markets where the fundamental determinants of the exchange rate are the same as those in the monetary model.

The economic environment for dynamic general equilibrium analysis needs to be speciÞed in some detail. To make this task manageable, we will begin by modeling the real part of the economy that operates under a barter system. We will obtain a solution for the real exchange rate and real stock-pricing formulae. This perfect-markets real general equilibrium model is sometimes referred to as an Arrow [3]—Debreu [34] model because it can be mapped into their static general equilibrium framework. We know that the Arrow—Debreu competitive equilibrium yields a Pareto Optimum. Why is this connection useful? Because it tells us that we can understand the behavior of the market economy by solving for the social optimum and it is typically more straightforward to obtain the social optimum than to directly solve for the market

105

106	CHAPTER 4. THE LUCAS MODEL

equilibrium.

In order to study the exchange rate, we need to have a monetary economy. The problem is that there is no role for Þat money in the Arrow—Debreu environment. The way that Lucas gets around this problem is to require people to use money when they buy goods. This

(76) requirement is called a ‘cash-in-advance’ constraint and is a popular strategy for introducing money in general equilibrium along the lines of the transactions motive for holding money. A second popular strategy that puts money in the utility function will be developed in chapter 9.

The models we will study in this chapter and in chapter 5 have no market imperfections and exhibit no nominal rigidities. Market participants have complete information and rational expectations. Why study such a perfect world? First, we have a better idea for solving frictionless and perfect-markets models so it is a good idea to start in familiar territory. Naturally, these models of idealized economies will not fully explain the real world. So we want to view these models as providing a benchmark against which to measure progress. If and when the data ‘reject’ these models, take one should note the manner in which they are rejected to guide the appropriate extensions and reÞnements to the theory.

There is a good deal of notation for the model which is summarized in Table 4.1.

4.1The Barter Economy

Consider two countries each inhabited by a large number of individuals who have identical utility functions and identical wealth. People may believe that they are individuals but the respond in the same way to changes in incentives. Because people are so similar you can normalize the constant populations of each country to 1 and model the people in each country by the actions of a single representative agent (household) in Lucas model. This is the simplest way to aggregate across individuals so that we can model macroeconomic behavior.

‘Firms’ in each country are pure endowment streams that generate a homogeneous nonstorable country-speciÞc good using no labor or

4.1. THE BARTER ECONOMY

107

capital inputs. Some people like to think of these Þrms as fruit trees. You can also normalize the number of Þrms in each country to 1. xt is the exogenous domestic output and yt is the exogenous foreign output. The evolution of output is given by xt = gtxt−1 at home and by yt = gt yt−1 abroad where gt and gt are random gross rates of change that evolve according to a stochastic process that is known by agents. Each Þrm issues one perfectly divisible share of common stock which is traded in a competitive stock market. The Þrms pay out all of their output as dividends to shareholders. Dividends form the sole source of support for individuals. We will let xt be the numeraire good and qt be the price of yt in terms of xt. et is the ex-dividend market value of the domestic Þrm and et is the ex-dividend market value of the foreign Þrm.

The domestic agent consumes cxt units of the home good, cyt units of the foreign good and holds ωxt shares of the domestic Þrm and ωyt shares of the foreign Þrm. Similarly, the foreign agent consumes cxt, units of the home good, cyt units of the foreign good and holds ωxt shares of the domestic Þrm and ωyt shares of the foreign Þrm.

The domestic agent brings into period t wealth valued at

Wt = ωxt−1(xt + et) + ωyt−1(qtyt + et ),

(4.1)

where xt + et and qtyt + et are the with-dividend value of the home and foreign Þrms. The individual then allocates current wealth towards new share purchases etωxt + et ωyt , and consumption cxt + qtcyt

Wt = etωxt + e ωy		+ cxt + qtcy .	(4.2)
t	t	t
Equating (4.1) to (4.2) gives the consolidated budget constraint
cxt + qtcyt + etωxt + et ωyt = ωxt−1(xt + et) + ωyt−1(qtyt + et ).			(4.3)

Let u(cxt, cyt) be current period utility and 0 < β < 1 be the subjective discount factor. The domestic agent’s problem then is to choose se-

quences of consumption and stock purchases, {cxt+j, cyt+j, ωxt+j, ωyt+j}∞j=0, to maximize expected lifetime utility

Et		jX	,	(4.4)
		∞ βju(cxt+j, cyt+j)
		=0

108	CHAPTER 4. THE LUCAS MODEL

subject to (4.3).

You can transform the constrained optimum problem into an unconstrained optimum problem by substituting cxt from (4.3) into (4.4). The objective function becomes

u(ωxt−1(xt + et) + ωyt−1(qtyt + et ) − etωxt − et ωyt − qtcyt , cyt )
+Et[βu(ωxt(xt+1 + et+1) + ωyt(qt+1yt+1 + et+1)	(4.5)
−et+1ωxt+1 − et+1ωyt+1 − qt+1cyt+1 , cyt+1 )] + · · ·

Let u1(cxt, cyt) = ∂u(cxt, cyt)/∂cxt be the marginal utility of x-consumption and u2(cxt, cyt) = ∂u(cxt, cyt)/∂cyt be the marginal utility of y-consumption. Di erentiating (4.5) with respect to cyt, ωxt, and ωyt, setting the result

(77) to zero and rearranging yields the Euler equations

cyt :	qtu1(cxt, cyt) = u2(cxt, cyt),		(4.6)
ωxt :	etu1(cxt, cyt) = βEt[u1(cxt+1, cyt+1)(xt+1 + et+1)],		(4.7)
ωyt :	e u1(cxt, cyt) = βEt[u1(cxt+1, cyt+1)(qt+1yt+1	+ e	)]. (4.8)
	t	t+1

These equations must hold if the agent is behaving optimally. (4.6) is the standard intratemporal optimality condition that equates the relative price between x and y to their marginal rate of substitution. Reallocating consumption by adding a unit of cy increases utility by u2(·). This is Þnanced by giving up qt units of cx, each unit of which costs u1(·) units of utility for a total utility cost of qtu1(·). If the individual is behaving optimally, no such reallocations of the consumption plan yields a net gain in utility.

(4.7) is the intertemporal Euler equation for purchases of the domestic equity. The left side is the utility cost of the marginal purchase of domestic equity. To buy incremental shares of the domestic Þrm, it costs the individual et units of cx, each unit of which lowers utility by u1(cxt, cyt). The right hand side of (4.7) is the utility expected to be derived from the payo of the marginal investment. If the individual is behaving optimally, no such reallocations between consumption and saving can yield a net increase in utility. An analogous interpretation holds for intertemporal reallocations of consumption and purchases of the foreign equity in (4.8).

4.1. THE BARTER ECONOMY

109

The foreign agent has the same utility function and faces the analogous problem to maximize

Et

jX

, cyt+j) ,

(4.9)

∞ βju(cxt+j

=0

subject to

c

+ qtc

+ etω + e ω

= ω

(xt

+ et) + ω

(qtyt

+ e ).

(4.10)

xt

yt

xt

t yt

xt−1

yt−1

t

The analogous set of Euler equations for the foreign individual are

c

:

qtu1

(c

, c

) = u2(c

, c ),

(4.11)

yt

xt

yt

xt

yt

ω

:

etu1

(c

, c

) = βEt[u1(c

, c

)(xt+1 + et+1)],

(4.12)

xt

xt

yt

xt+1

yt+1

ω

:

e u1(c

, c

) = βEt[u1(c

, c

)(qt+1yt+1

+ e

)].(4.13)

yt

t

xt

yt

xt+1

yt+1

t+1

A set of four adding up constraints on outstanding equity shares and the exhaustion of output in home and foreign consumption complete the speciÞcation of the barter model

ωxt + ω	=	1,	(4.14)
xt
ωyt + ωyt	=	1,	(4.15)
cxt + c	=	xt,	(4.16)
xt
cyt + c	=	yt.	(4.17)
yt

Digression on the social optimum. You can solve the model by grinding out the equilibrium, but the complete markets and competitive setting makes available a ‘backdoor’ solution strategy of solving the problem confronting a Þctitious social planner. The stochastic dynamic barter economy can conceptually be reformulated in terms of a static competitive general equilibrium model—the properties of which are well known. The reformulation goes like this.

We want to narrow the deÞnition of a ‘good’ so that it is deÞned precisely by its characteristics (whether it is an x−good or a y−good), the date of its delivery (t), and the state of the world when it is delivered (xt, yt). Suppose that there are only two possible values for xt (yt) in

110	CHAPTER 4. THE LUCAS MODEL

each period–a high value xh(yh) and a low value x`(y`). Then there are 4 possible states of the world (xh, yh), (xh, y`), (x`, yh), and (x`, y`). ‘Good 1’ is x delivered at t = 0 in state 1. ‘Good 2’ is x delivered at t = 0 in state 2, ‘good 8’ is y delivered at t = 1 in state 4, and so on. In this way, all possible future outcomes are completely spelled out. The reformulation of what constitutes a good corresponds to a complete system of forward markets. Instead of waiting for nature to reveal itself over time, we can have people meet and contract for all future trades today (Domestic agents agree to sell so many units of x to foreign agents at t = 2 if state 3 occurs in exchange for q2 units of y, and so on.) After trades in future contingencies have been contracted, we allow time to evolve. People in the economy simply fulÞll their contractual obligations and make no further decisions. The point is that the dynamic economy has been reformulated as a static general equilibrium model.

Since the solution to the social planner’s problem is a Pareto optimal allocation and you know by the fundamental theorems of welfare economics that the Pareto Optimum supports a competitive equilibrium, it follows that the solution to the planner’s problem will also describe the equilibrium for the market economy.1

We will let the social planner attach a weight of φ to the home individual and 1 − φ to the foreign individual. The planner’s problem is to allocate the x and y endowments optimally between the domestic and foreign individuals each period by maximizing

∞	hφu(cxt+j, cyt+j) + (1 − φ)u(cxt+j, cyt+j)i ,
X
Et j=0 βj		(4.18)

subject to the resource constraints (4.16) and (4.17). Since the goods are not storable, the planner’s problem reduces to the timeless problem of maximizing

φu(cxt, cyt) + (1 − φ)u(cxt, cyt),

1Under certain regularity conditions that are satisÞed in the relatively simple environments considered here, the results from welfare economics that we need are, i) A competitive equilibrium yields a Pareto Optimum, and ii) Any Pareto Optimum can be replicated by a competitive equilibrium.

4.1. THE BARTER ECONOMY

111

subject to (4.16) and (4.17). The Euler equations for this problem are

φu1(cxt, cyt)	=	(1 − φ)u1(cxt	, cyt	),	(4.19)
φu2(cxt, cyt)	=	(1 − φ)u2(cxt	, cyt	).	(4.20)

(4.19) and (4.20) are the optimal or e cient risk-sharing conditions. Risk-sharing is e cient when consumption is allocated so that the marginal utility of the home individual is proportional, and therefore perfectly correlated, to the marginal utility of the foreign individual. Because individuals enjoy consuming both goods and the utility function is concave, it is optimal for the planner to split the available x and y between the home and foreign individuals according to the relative importance of the individuals to the planner.

The weight φ can be interpreted as a measure of the size of the home country in the market version of the world economy. Since we assumed at the outset that agents have equal wealth, we will let both agents be equally important to the planner and set φ = 1/2. Then the Pareto optimal allocation is to split the available output of x and y equally

cxt = c	=	xt	,	and cyt = c	=	yt	.
xt	2			yt	2

Having determined the optimal quantities, to get the market solution we look for the competitive equilibrium that supports this Pareto optimum.

The market equilibrium. If agents owned only their own country’s Þrms, individuals would be exposed to idiosyncratic country-speciÞc risk that they would prefer to avoid. The risk facing the home agent is that the home Þrm experiences a bad year with low output of x when the foreign Þrm experiences a good year with high output of y. One way to insure against this risk is to hold a diversiÞed portfolio of assets.

A diversiÞcation plan that perfectly insures against country-speciÞc risk and which replicates the social optimum is for each agent to hold stock in half of each country’s output.2 The stock portfolio that achieves

2Agents cannot insure against world-wide macroeconomic risk (simultaneously low xt and yt).

112	CHAPTER 4. THE LUCAS MODEL

complete insurance of idiosyncratic risk is for each individual to own half of the domestic Þrm and half of the foreign Þrm3

	ωxt = ωxt = ωyt = ωyt =	1	.	(4.21)
		2

We call this a ‘pooling’ equilibrium because the implicit insurance scheme at work is that agents agree in advance that they will pool their risk by sharing the realized output equally.

The solution under constant relative-risk aversion utility. Let’s adopt a particular functional form for the utility function to get explicit solutions. We’ll let the period utility function be constant relative-risk aversion in Ct = cθxtc1yt−θ, a Cobb-Douglas index of the two goods

	Ct1−γ
u(cx, cy) =			.	(4.22)
	1 − γ
Then		θCt1−γ
u1(cxt, cyt) =				,
		cxt

u2(cxt, cyt) = (1 − θ)Ct1−γ .

cyt

and the Euler equations (4.6)—(4.13) become

qt

et

xt et

qtyt

=	1 − θ			xt		,
	θ	"		yt
= βEt					Ct+1
					C
		µ				t
= βEt		"			Ct+1
					C
		µ				t

							(4.23)
(1−γ)	Ã1		e		!# ,
¶		+	t+1				(4.24)
			x
	Ã1	+ q		t+1		!# .	(4.25)
(1−γ)				t y
				e +1
¶			t+1			t+1

From (4.23) the real exchange rate qt is determined by relative output levels. (4.24) and (4.25) are stochastic di erence equations in the ‘price- (79) dividend’ ratios et/xt and et /(qtyt). If you iterate forward on them as

3Actually, Cole and Obstfeld [31]) showed that trade in goods alone are su cient to achieve e cient risk sharing in the present model. These issues are dealt with in the end-of-chapter problems.