lOMoAR cPSD| 45211451
CHAP 7 Modeling Monetary Economies
4th Edition
MONEY AND BANKING
, lOMoAR cPSD| 45211451
Chapter 7
Capital
1 Roadmap
So far, people in our model economy have had only one
way to acquire consumption at a later time – by holding
fiat money. In the real world, however, there are many
other assets. In this chapter, we concentrate on one particular
alternative asset, capital. Capital is different from fiat money in
that when people acquire capital this period, the capital
produces goods next period and thus affects an economy’s
output. In contrast, fiat money allows a person to
purchase some of the existing output next period but does
not increase the quantity of output. We will see how the
presence of an alternative asset affects people’s willingness to
hold fiat money. At a basic level, we see that different stores
of value compete against each other. In this chapter, we start
with the simplifying assumption that different storesof value are
perfect substitutes. It is easy to see how changes in
monetary policy could affect the stock of capital and thus the level
of output.
Before takingall the challenges on at once, we begin by
looking at the simplest model in which capital is the only
store of value.
2 Capital
Consider the following production technology: If units of the
consumption good are converted into capital goods at time ,
at time you will receive
consumption goods, where is some positive constant.
This implies that the gross real return on capital is . In
this book, we assume that date- capital goods disintegrate in
, lOMoAR cPSD| 45211451
date . In other words, capital goods mature, producing
consumption goods but are useless themselves. This is the
same as saying that the capital depreciates at a 100
percent rate in the production period and therefore has no
1
salvage value.
As in previous models, people in the single-country
economy are endowed with units of the consumption good
when young and zero units when old. Population grows at the gross rate
. Each member of the initial old begins with a stock of
capital that produces goods in the first period.
Let us first analyze an equilibrium without fiat money. The
capital technology enables the young to use some of
today’s consumption good to produce the consumption good
at a later date. When young, people can convert part of
their endowmentinto capital and consume the rest. This implies that
in the first period of life, the budget constraint facing a
person born in period is
(7.1)
When old, the person will consume the goods produced by
capital . The secondperiod constraint is then
(7.2)
We can combine Equations 7.1 and 7.2 into a lifetime
budget constraint. Equation 7.2 tells use that .
Substituting this into Equation 7.1, we obtain the
lifetime budget constraint
(7.3)
, lOMoAR cPSD| 45211451
We can see that determines the slope of the budget line.
If, for example, , then the vertical intercept of the
budget line will lie farther from the origin than the horizontal
intercept. How much capital will a person desire? As
before, the answer is derived by superimposing the person’s
indifference map on the budget set. This is done in Figure
7.1.
Figure 7.1. The person’s choice of capital. This figure
depicts the budget line when capital pays the gross return
. Each person maximizes utility by choosing the consumption
pattern . Each person’s capital holdings will be
.
This simple model of capital assumes that the
output from each unit of capital is simply assumed to be
some number unaffected by any economic forces. Although the
assumption of a fixed rate of return of capital makes
the model very easy for us to use, we sometimes need a
CHAP 7 Modeling Monetary Economies
4th Edition
MONEY AND BANKING
, lOMoAR cPSD| 45211451
Chapter 7
Capital
1 Roadmap
So far, people in our model economy have had only one
way to acquire consumption at a later time – by holding
fiat money. In the real world, however, there are many
other assets. In this chapter, we concentrate on one particular
alternative asset, capital. Capital is different from fiat money in
that when people acquire capital this period, the capital
produces goods next period and thus affects an economy’s
output. In contrast, fiat money allows a person to
purchase some of the existing output next period but does
not increase the quantity of output. We will see how the
presence of an alternative asset affects people’s willingness to
hold fiat money. At a basic level, we see that different stores
of value compete against each other. In this chapter, we start
with the simplifying assumption that different storesof value are
perfect substitutes. It is easy to see how changes in
monetary policy could affect the stock of capital and thus the level
of output.
Before takingall the challenges on at once, we begin by
looking at the simplest model in which capital is the only
store of value.
2 Capital
Consider the following production technology: If units of the
consumption good are converted into capital goods at time ,
at time you will receive
consumption goods, where is some positive constant.
This implies that the gross real return on capital is . In
this book, we assume that date- capital goods disintegrate in
, lOMoAR cPSD| 45211451
date . In other words, capital goods mature, producing
consumption goods but are useless themselves. This is the
same as saying that the capital depreciates at a 100
percent rate in the production period and therefore has no
1
salvage value.
As in previous models, people in the single-country
economy are endowed with units of the consumption good
when young and zero units when old. Population grows at the gross rate
. Each member of the initial old begins with a stock of
capital that produces goods in the first period.
Let us first analyze an equilibrium without fiat money. The
capital technology enables the young to use some of
today’s consumption good to produce the consumption good
at a later date. When young, people can convert part of
their endowmentinto capital and consume the rest. This implies that
in the first period of life, the budget constraint facing a
person born in period is
(7.1)
When old, the person will consume the goods produced by
capital . The secondperiod constraint is then
(7.2)
We can combine Equations 7.1 and 7.2 into a lifetime
budget constraint. Equation 7.2 tells use that .
Substituting this into Equation 7.1, we obtain the
lifetime budget constraint
(7.3)
, lOMoAR cPSD| 45211451
We can see that determines the slope of the budget line.
If, for example, , then the vertical intercept of the
budget line will lie farther from the origin than the horizontal
intercept. How much capital will a person desire? As
before, the answer is derived by superimposing the person’s
indifference map on the budget set. This is done in Figure
7.1.
Figure 7.1. The person’s choice of capital. This figure
depicts the budget line when capital pays the gross return
. Each person maximizes utility by choosing the consumption
pattern . Each person’s capital holdings will be
.
This simple model of capital assumes that the
output from each unit of capital is simply assumed to be
some number unaffected by any economic forces. Although the
assumption of a fixed rate of return of capital makes
the model very easy for us to use, we sometimes need a
