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Topic 5: Capital and Private Debt
Monetary Economics ECOS3010
Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 1 / 32

Introduction
So far, the only asset in our model is Öat money. In the real world, there are many other assets:
capital: goods that are saved for production;
private debt: loans that facilitate borrowing and lending; and more.
In this chapter, we will examine the interaction between money and other assets (capital and private debt).
A model with capital.
A model with private debt.
How can money and other assets coexist? Ináation rate and interest rate.
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A Model with Capital
Consider the following production technology:
at time t, kt units of consumption goods are saved as capital goods at time t;
at time t + 1, xkt units of consumption goods can be produced with kt units of capital; x is a constant;
capital fully depreciates after production.
Back to our basic OLG model, suppose that there is no money. Instead, technology allows the young to save in the form of capital. The old can use capital to produce consumption goods. In addition,
each initial old is endowed with k0 units of capital; population is growing at a constant rate n, Nt = nNt1.
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A Model with Capital
Suppose we focus on stationary allocations. An individual faces the Örst-period budget constraint
c1 + k y ; the second-period budget constraint
c2 xk. the lifetime budget constraint
c1+c2 y, x
combining the two period budget constraints.
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A Model with Capital
For any given x, we can depict the lifetime budget constraint.
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A Model with Capital
How much capital will an individual desire? The individual chooses (c1,c2) such that the indi§erence curve is tangent to the budget constraint. The optimal choice of k is
k = y c 1 .
This simple model assumes that the marginal product of capital is a constant x. A more realistic assumption is that capital exhibits a “diminishing marginal product”. Let f (k) denote a general production function. In general, f 0 (k) > 0. A diminishing marginal product of capital means that
f 00 (k) < 0. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 6 / 32 A Model with Capital An illustration of f (k) with diminishing marginal product. f(k) f’(k) f(k) Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 7 / 32 A Model with Private Debt Consider private debt as IOUs issued by individuals ñ private loans. Let there be two types of individuals: borrowers: endowed with nothing when young and y units of goods when old; lenders: endowed with y units of goods when young and nothing when old. In each generation, half of the people are borrowers and the rest half are lenders. To begin with, suppose that private debt is the only asset in the economy. There is neither money nor capital. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 8 / 32 A Model with Private Debt For a lender, the Örst-period budget constraint is c1,L + l y, where l denotes the amount of loans; the second-period budget constraint is c2,L rl, where r is the gross real interest rate on loans; the lifetime budget constraint is c1,L+c2,L y, r by combining the two period budget constraints. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 9 / 32 A Model with Private Debt We can depict the lenderís lifetime budget constraint. The lender chooses (c1,c2) such that the indi§erence curve is tangent to the budget constraint. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 10 / 32 A Model with Private Debt We assume that preferences are such that as r increases, c1,L decreases so that l increases. Let L denote the aggregate supply of loans. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 11 / 32 A Model with Private Debt For a borrower, the Örst-period budget constraint is c1,B b, where b denotes the amount of debt (loans); the second-period budget constraint is c2,B yrb, the lifetime budget constraint is c1,B+c2,B y, by combining the two period budget constraints. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 12 / 32 A Model with Private Debt We can depict the borrowerís lifetime budget constraint. The borrower chooses (c1,c2) such that the indi§erence curve is tangent to the budget constraint. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 13 / 32 A Model with Private Debt We assume that preferences are such that as r increases, c1,B decreases so that b decreases. Let B denote the aggregate demand for loans. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 14 / 32 Rate of Return Equality Suppose that we introduce capital to our model with private debt. The marginal product of capital is x. The rate of return on loans is r. How should lenders choose between capital and private loans? What would happen if x > r? What would happen if x < r? What would happen if x = r? Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 15 / 32 Rate of Return Equality For people to be willing to hold both capital and loans as assets, we must have or more generally f 0 (k) = r. Suppose that there are many assets available to individuals. Without any uncertainty about returns and any government restrictions (perfect substitutes), the rate of return on these assets must be identical for people to hold all available assets simultaneously. We refer to this as the principle of "rate-of-return equality". Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 16 / 32 Coexistence of Money and Other Assets If we introduce money into our model with capital and private debt, the rate-of-return equality requires that for all assets to be held by individuals, Here n is the growth rate of population and z is the growth rate of nz = r = x . money supply; so n/z is the rate of return on money. If all assets are perfect substitutes, then rate-of-return equality holds for all assets to coexist. In this case, how does money interact with other assets? Interaction between money and capital: the Tobin e§ect. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 17 / 32 Coexistence of Money and Other Assets The Tobin E§ect Consider a standard OLG model with money and capital. Each young is endowed with y units of consumption goods when young and nothing when old. Suppose that capital displays a diminishing marginal product. That is, f 00 (k) < 0. For both money and capital to be valued, we must have f 0 ( k ) = nz . For any given (n,z), we can Önd a desired level of capital stock. What if there is a permanent increase in z? Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 18 / 32 Coexistence of Money and Other Assets The Tobin E§ect Graphically, we show the determination of k. When z increases, n/z decreases and k increases. (In the Ögure, z0 > z.)
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Coexistence of Money and Other Assets The Tobin E§ect
The substitution of capital for money in reaction to an increase in ináation, described by Tobin (1965), is called the “Tobin e§ect”.
Does the Tobin e§ect suggest that an increase in z could help to increase output? In our model, the output in period t is
GDPt =Nty+Nt1f(kt1).
If we live in a world where money and capital are perfect substitutes, an increase in z would raise k and hence output. Should the policy maker use z as a tool to raise output?
Output 6= Welfare.
The Tobin e§ect is not large in the real world.
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
In the real world, Öat money and other assets are not perfect substitutes. In particular, the rate of return on money is generally lower than the rate of return on other assets. What are the e§ects of anticipated ináation on interest rate, capital and output?
This raises an obvious question: why would money still be valued? We postpone this important question for later chapters. For now, consider a simple argument: legal restriction. Each young is required by law to acquire money worth a Öxed number of goods q.
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
Nominal interest rate and real interest rate.
Nominal interest rates: the number of dollars paid in interest for each dollar lent.
Real interest rates: the number of goods paid in interest for each good lent.
Nominal interest rates are the ones cited by Önancial intermediaries and the press.
In times of ináation, nominal interest rates do not reáect the real rate of return.
Let Rt and rt denote the nominal interest rate and the real interest rate. Let pt denote the price of a good.
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
We can express the gross real interest rate rt as r =Rtpt =R pt .
If we rearrange (1), we can obtain Rt 1 =
net nominal interest rate
(rt 1) | {z }
+ pt+1 1 | pt{z }
net real interest rate
+(rt 1)pt+1 1.
net ináation rate
For low values of the real interest rate and the ináation rate, we have approximately:
net nominal interest rate = net real interest rate + net ináation rate.
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
Anticipated ináation and the nominal interest rate: the predicted full adjustment of the nominal interest rate to anticipated ináation is called the “Fisher e§ect” ñnamed after .
Consider a model with money, capital and private debt. Suppose that the marginal product of capital is x. The rate-of-return equality implies that
x=r=R pt . t t pt+1
Note that here money is held by people because of the legal restriction. The rate-of-return equality applies to capital and private debt. As the rate of return on money is pt/pt+1 = n/z, we have
The real interest rate is a constant x. The nominal interest rate rises
with anticipated ináation to keep the real interest rate constant x. Monetary Economics (ECOS3010) Topic 5: Capital and Private Debt 24 / 32
x=Rn orR=xz. tztn

Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
Ináation and nominal interest rate in the U.S.
insert US data
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
Ináation rate and nominal interest rate in Australia.
insert AU data
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
There is a tendency for nominal interest rates and ináation rates to move together in accordance with the Fisher e§ect. However, the gap between the nominal interest rate and the ináation rate is not constant due to changes in the real interest rate.
Will ináation a§ect the real interest rate? If so, how? In our previous example,
x=r =Rn. ttz
The real interest rate is always a constant x by the rate-of-return equality. An increase in ináation will only a§ect the nominal interest rate, but not the real interest rate. The key assumption is a constant marginal product of capital.
What if capital exhibits a diminishing marginal product?
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
Anticipated ináation and the real interest rate: an exception to the Fisher e§ect could occur if money, private debt, and capital are substitutes and capital has a diminishing marginal product. In this case,
an increase in the ináation rate leads to an increase in capital ñthe Tobin e§ect;
an increase in capital leads to a decline in the real interest rate because of diminishing marginal product of capital;
overall, an increase in ináation will still lead to a rise in the nominal interest rate, however, because of the simultaneous decrease in the real interest rate, the nominal interest rate will not rise by the full amount of the rise in ináation.
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
Ináation rate and real interest rate in the U.S.
insert US scatterplot
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Coexistence of Money and Other Assets When Money and Other Assets are not Perfect Substitutes
Ináation rate and real interest rate in Australia.
insert AU scatterplot
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So far, we have assumed that all assets pay a rate of return that is known with complete certainty. What would happen to rate-of-return equality if instead one asset has a random rate of return?
For example: there is some positive probability that a loan will not be repaid by the borrower. If the marginal product of capital is a constant x, then capital and private debt are not perfect substitutes.
If people do not care about risk (are “risk neutral”), then the rate-of-return equality still holds on average. Suppose that an asset pays return r1, r2, … , rn with probabilities π1, π2, … , πn, respectively. The expected rate of return on this asset can be calculated as
E(r) = π1r1 +π2r2 ++πnrn. The rate-of-return equality is modiÖed to
E (r) = x.
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Are people risk neutral? Probability not.
If people are risk averse, people may still accept a risky asset. However, the rate-of-return equality will not hold in this case. In fact, the expected rate of return on this risky asset must exceed that of the risk-free asset, compensating for the risk.
The extra average rate of return that is necessary to entice people to hold a risky asset is called a risk premium,
risk premium = E (rrisky ) rsafe .
The greater the potential loss and the greater the probability of the
loss, the larger the risk premium must be.
In many real economies, why do people choose to hold Öat money when many alternative assets appear to o§er greater rates of return?
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