International Macroeconomics and Finance: Theory and Empirical Methods Phần 4 pdf

4.1. THE BARTER ECONOMY 107

capital inputs. Some people like to think of these firms as fruit trees.

You can also normalize the number of firms in each country to 1. xt

is the exogenous domestic output and yt is the exogenous foreign out￾put. The evolution of output is given by xt = gtxt−1 at home and by

yt = g∗

t yt−1 abroad where gt and g∗

t are random gross rates of change

that evolve according to a stochastic process that is known by agents.

Each firm issues one perfectly divisible share of common stock which

is traded in a competitive stock market. The firms 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 firm and e∗

t is the ex-dividend market value of the foreign

firm.

The domestic agent consumes cxt units of the home good, cyt units

of the foreign good and holds ωxt shares of the domestic firm and ωyt

shares of the foreign firm. Similarly, the foreign agent consumes c∗

xt,

units of the home good, c∗

yt units of the foreign good and holds ω∗

xt

shares of the domestic firm and ω∗

yt shares of the foreign firm.

The domestic agent brings into period t wealth valued at

Wt = ωxt−1(xt + et) + ωyt−1(qtyt + e∗

t ), (4.1)

where xt +et and qtyt +e∗

t are the with-dividend value of the home and

foreign firms. The individual then allocates current wealth towards new

share purchases etωxt + e∗

tωyt , and consumption cxt + qtcyt

Wt = etωxt + e∗

tωyt + cxt + qtcyt . (4.2)

Equating (4.1) to (4.2) gives the consolidated budget constraint

cxt + qtcyt + etωxt + e∗

tωyt = ωxt−1(xt + et) + ωyt−1(qtyt + e∗

t ). (4.3)

Let u(cxt, cyt) be current period utility and 0 < β < 1 be the subjec￾tive discount factor. The domestic agentís 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



X∞

j=0

βj

u(cxt+j , cyt+j)



 , (4.4)

108 CHAPTER 4. THE LUCAS MODEL

subject to (4.3).

You can transform the constrained optimum problem into an un￾constrained optimum problem by substituting cxt from (4.3) into (4.4).

The objective function becomes

u(ωxt−1(xt + et) + ωyt−1(qtyt + e∗

t ) − etωxt − e∗

tωyt − qtcyt , cyt )

+Et[βu(ωxt(xt+1 + et+1) + ωyt(qt+1yt+1 + e∗

t+1)

−et+1ωxt+1 − e∗

t+1ωyt+1 − qt+1cyt+1 , cyt+1 )] + ···

(4.5)

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.

Differentiating (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∗

t u1(cxt, cyt) = βEt[u1(cxt+1, cyt+1)(qt+1yt+1 + e∗

t+1)]. (4.8)

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 financed 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 indi￾vidual 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 do￾mestic equity. The left side is the utility cost of the marginal purchase

of domestic equity. To buy incremental shares of the domestic firm, 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 payoff 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 anal￾ogous problem to maximize

Et



X∞

j=0

βj

u(c∗

xt+j , c∗

yt+j)



 , (4.9)

subject to

c∗

xt + qtc∗

yt + etω∗

xt + e∗

tω∗

yt = ω∗

xt−1(xt + et) + ω∗

yt−1(qtyt + e∗

t ). (4.10)

The analogous set of Euler equations for the foreign individual are

c∗

yt : qtu1(c∗

xt, c∗

yt) = u2(c∗

xt, c∗

yt), (4.11)

ω∗

xt : etu1(c∗

xt, c∗

yt) = βEt[u1(c∗

xt+1, c∗

yt+1)(xt+1 + et+1)], (4.12)

ω∗

yt : e∗

t u1(c∗

xt, c∗

yt) = βEt[u1(c∗

xt+1, c∗

yt+1)(qt+1yt+1 + e∗

t+1)].(4.13)

A set of four adding up constraints on outstanding equity shares and

the exhaustion of output in home and foreign consumption complete

the specification of the barter model

ωxt + ω∗

xt = 1, (4.14)

ωyt + ω∗

yt = 1, (4.15)

cxt + c∗

xt = xt, (4.16)

cyt + c∗

yt = yt. (4.17)

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 ëbackdoorí solution strategy of solving the problem

confronting a fictitious social planner. The stochastic dynamic barter

economy can conceptually be reformulated in terms of a static compet￾itive general equilibrium modelóthe properties of which are well known.

The reformulation goes like this.

We want to narrow the definition of a ëgoodí so that it is defined

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