Expected inflation and the real rate of interest

Journal of Banking and Finance 9 (1985) 491--498. North-Holland

EXPECTED INFLATION AND THE REAL RATE OF INTEREST* A Note Theodore E. DAY Vanderbilt University, Nashville, TN 37203, USA Received December 1983, final version received September 1984

This paper derives an alternative explanation for the Mundell effect in the context of a state preference framework. In contrast to the real cash balance effect discussed by MundeU, the arrival of new information concerning the future course of economic events is shown to simultaneously affect both the real rate of interest and the expected rate of inflation. A negative relation between changes in expected inflation and the real rate of interest is shown to occur in spite of the fact that investors in this model hold no cash balances.

1. Introduction

Irving Fisher is perhaps the most widely quoted source on the theory of interest under inflation. In The Theory of Interest, he asserted that in a world of perfect foresight, the nominal rate of interest should equal the real rate of interest plus the anticipated rate of inflation. Later, Fisher observed that, ex post, the nominal rate of interest did not seem to adjust to completely offset the actual rate of inflation. One attempt to explain Fisher's empirical observation is the real balance effect posited by Mundell (1963). Mundell suggested that during periods when the expected rate of inflation is relatively high, investors choose to hold lower real cash balances. Investors are assumed to adjust their real balances to the desired level by investing in bonds. The increase in demand pushes up the prices of bonds, resulting, ceteris paribus, in a fall in the real rate of interest. Thus, although the nominal rate of interest increases to reflect the expected rate of inflation, a portion of the 'inflation premium' is offset by the decline in the real rate of interest. This real balance effect has come to be known as the 'Mundell effect'. The work of Fisher and Mundell examines the behavior of real and nominal interest rates when the expected rate of inflation is given exogenously. This paper takes a different approach in that the expected rate of *This research was supported by the Vanderbilt University Research Council and the Dean's Fund for Faculty Research at the Owen School. 0378-4266/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

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T.E. Day, Expected inflation and real rate of interest

inflation is derived endogenously. Money is assumed to enter the model through a 'cash-in-advance' constraint on transactions similar to that discussed in Clower (1967) and Stockman (1981). Changes in the equilibrium price level are caused by random variations in the economy's endowment and by the deficit spending of a government agency in order to finance its demands for the endowment. These random shocks to the economy also determine the real rate of interest. The results derived from the model are consistent with the Mundell effect in that the real rate of interest is shown to vary inversely with changes in the expected rate of inflation. However, in this framework, the effect noted by Mundell occurs due to a change in the demand for state contingent claims on future consumption, rather than a change in the demand for real balances. The model differs from previous work in that the expected rate of inflation does not play a causal role in the determination of the real rate of interest. Instead, expected inflation and interest rates are jointly determined by current and expected future economic events. This is in contrast to models in which the expected rate of inflation has a causal effect on the real rate of interest through the tax system [e.g., Darby (1975) and Feldstein (1976)-I or through portfolio adjustment, as in Mundell's model. The model is introduced in sections 2 and 3. The relation between the real rate of interest and the expected rate of inflation is analyzed in section 4 by considering the effects of unanticipated information about the future course of economic events. 2. The economy

Consider a multi-period exchange economy composed of identical individuals. Each period, t, the economy is endowed with a random per capita supply of a single perishable good, Or. Per capita consumption of the good by individuals (the private sector) is denoted by C,. The economy's endowment is also demanded by a government agency in order to make real transfer payments which have no effect on the prices of securities in the private sector of the economy. ~ Assume that the economy's endowment accrues to a single firm, which is owned by the economy's identical investors. Transactions costs prevent the firm from distributing the endowment directly to investors. Instead, investors purchase output with a cash dividend based upon a 100~ payout of the firm's revenues in the preceding period. The assumption that money must be used to purchase output is similar to the 'cash-in-advance' constraint discussed by Clower (1967) and Stockman (1981). ~Government transfers can be thought of as either foreign aid or transfers to investors who do not affect prices in the securities market. An alternative formulation which results in identical implications for asset pricing is that the government agency produces a public good whose consumption has no effect on the marginal utility of private consumption.

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The requirement that money be used to acquire endowment enables the government agency to purchase output using newly printed money (i.e., deficit spending). All government debt is immediately monetized so that the relation between the per capita money supply in periods t and t + 1 is

Mr+ 1 =(1 +g,+x)Mt,

(1)

where gt+ x represents the growth rate of the money supply in period t + 1. The government demand for output is a serially uncorrelated random variable; independent of the aggregate endowment. This assumption implies that government policy is chosen without knowledge of the economy's current endowment. The random nature of policy can be attributed to random shocks to an exogenous policy making process. The price level in this economy is determined by the Quantity Theory,

Mtv = PRO,,

(2)

where Pt is the nominal price per unit of output at time t and v is the velocity of money. For notational convenience, the velocity of money is assumed to be constant and equal to one. 2 The assumption of a constant unitary velocity of money implies that individuals hold no real cash balances between periods. Instead, consumers hold a dividend receivable equal to the date t money supply, Mr The absence of any demand for real balances by consumers implies that any systematic relation between expected inflation and the real rate of interest must operate through a mechanism distinct from the real balance effect suggested by Mundell. Note that the monetary shocks in this model (if) are similar to pure endowment shocks (8) in that they have a direct effect on the quantity of output available to the private sector. The difference lies in the fact that monetary shocks also affect the money supply (Mr). The fact that the seigniorage from the creation of money is not explicitly returned to consumers can be interpreted as an implicit transfor to individuals who do not affect price in the securities market (i.e., an individual owning no shares of the economy's endowment whose current consumption equals a transfer payment). Alternatively, the monetary shocks can be thought of in terms of a long-run relation between prices and the money supply. The creation of gtMt_x dollars at time t enables the government agency to purchase gt/(1 +g,) of the endowment, since (by assumption) the money supply has a velocity of one. This implies that at any time t, the government agency 2Although the seigniorage from the creation of money (i.e., a government deficit) increases (decreases) for a velocity of money less (greater) then one, government deficits can always be thought of in terms of the magnitude necessary to purchase a desired fraction of output.

T.E. Day, Expected inflation and real rate of interest

494

purchases [gt/(1 +gt)]Ot units of current output, which leaves investors [1/(1 +gr)]Ot for current consumption, Ct. The expected rate of inflation will refer to the expectation at t of the price relative of the endowment in periods t and t + 1,

(3) Using (1) and (2), this can be expressed as "+gt+ 1

(4)

The expected rate of inflation is, as might be expected, increasing in the expected growth rate of the money supply and decreasing in the expected level of the endowment.

3. Equilibrium interest rates Although bonds have no explicit role in a model with identical investors, the price of a real bond (and thus the real rate of interest) can be derived from the prices of state contingent claims on real wealth. 3 In the present framework, future states of the world are defined by the potential realizations of ~ and 6. The prices of state contingent claims must satisfy the first-order conditions for the intertemporal consumptive optimum of a representative investor, subject to a budget constraint. Assume that a representative investor chooses current consumption, Ct, and state contingent claims on wealth, W~t+ 1, to maximize a utility function of the form 4

y(w,)=u(c,)+ E

v(w,,+ 1),

(5a)

S

subject to a budget constraint

<=w,,

(Sb)

S

and subject to

Ot Ct < 1 + g----~t'

(5c)

3Since investors are identical, securities will be priced as ff the market for state contingent claims were complete. *The presence of a time preference parameter in the specification of the utility function would not effect the results to be presented.

T.E. Day, Expected inflation and real rate of interest

495

where Wt = r e a l wealth at time of t, ~bst+1= the real price (in terms of current consumption) of real wealth in state s at date t + 1, 7gst+ 1 " ~ the probability of state s at date t + 1, U = utility function for current consumption with partial derivatives Uc > 0 and U~ < O, V = indirect utility function for future wealth with partial derivatives Vw>O and V~w
C st+, =re,t+1 Uc'

(6)

where Vws is used to denote the marginal utility of real wealth in state s at date t + 1. The real rate of interest can be derived by using expression (6) to determine the price of a real bond. Since a real bond is a security which pays off one unit of real wealth (i.e., one unit of consumption) in each state of the world at date t + 1, the current price of a real bond is simply the value of one unit of certain consumption, discounted at the real rate of interest, r. In other words, the price of a real bond is 1/(1+1"). Using (6), the value of this portfolio of state contingent claims can be expressed as

E(Vw) U~ '

l+r

(7)

which implies that

r=--

Uc

E(Vw)

1.

(8)

In section 4, this expression will be used to examine the simultaneous response of expected inflation and the real rate of interest to new information about the future course of economic events.

4. Changing expectations and the real rate of interest New information about the future course of economic events will, in general, result in a change in both the real rate of interest and the expected

T.E. Day, Expected inflation and real rate of interest

496

rate of inflation. This section formalizes the notion of unanticipated information in order to examine the correlation between changes in the real rate of interest and the expected rate of inflation. The resulting relation between interest rates and expected inflation is then compared with the result derived by Mundell. Both the real rate of interest and the expected rate of inflation at date t are determined by investor expectations concerning the realizations of Or+ and g,÷a (i.e., the state of the world at date t + l ) . The arrival of new information about the distribution of either Ot+~ or gt+~ will, in general, alter the real rate of interest. New information will also cause a change in the expected rate of inflation. Thus, a random shock to the economy in the form of information about future economic conditions will result in simultaneous changes in the real rate of interest and the expected rate of inflation. Assume that new information arrives in the form of an additive shift to the distribution of Ot÷~ such that the new distribution of the economy's endowment becomes 5 O;+1 =Ot+l

(9)

q-h.

The nature of the relation between expected inflation and the real rate of interest can be determined by examining their comparative statics response to variations in the shift parameter h. 6 For notational convenience, assume that gt+ ~ is equal to zero and let 0 be normalized so that 0 t is equal to one. The effect of new information about the future endowment on the expected rate of inflation can be determined by differentiating eq. (4) with respect to h, using (9), and evaluating the result at h=0, -

E


(lO)

In other words, given unchanging expectations as to government demand for endowment, an increase (decrease) in the expected endowment will increase (decrease) potential consumption relative to the money supply and decrease (increase) the expected rate of change in the price level. The simultaneous effect on the real rate of interest can be found by differentiating (8) with respect to h, using (9), and evaluating at h =0, t3r V Uc gh -- - E(ww) (E(-V-))2 >0.

(11)

5Although the analysis is somewhat more complicated, multiplicative or 'mean preserving' shifts (e.g., O~÷ 1 = Or+ 1 + h(O,+ 1 - Or+ 1)) have similar comparative statics effects. 6Identical results can be derived for shifts in the distribution of ~.

T.E. Day, Expected inflation and real rate of interest

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Eq. (11) indicates that an increase (decrease) in the expected future endowment will be accompanied by an increase (decrease) in the real rate of interest. The change in the real rate of interest reflects a change in the demand for state contingent claims on future consumption. An increase in expected future consumption (h>0), relative to current consumption, will cause the prices of state contingent claims, and thus the prices of real bonds to fall, resulting in an increase in the real rate of interest. Eq. (10) shows that the increase in the real rate of interest will be accompanied by a decrease in the expected rate of inflation. Conversely, when expected future consumption becomes more scarce, relative to current consumption, the prices of state contingent claims on consumption will increase, resulting in an increase in the prices of real bonds and a fall in the real rate of interest. The expected scarcity of future consumption results in a simultaneous increase in the expected rate of inflation in this case. Thus, the model predicts a negative relation between changes in the expected rate of inflation and the real rate of interest. The source of this relation differs from the real balance effect posited by Mundell. Here, new information causes prices in the market for state contingent claims to adjust until investors are satisfied to hold their endowments. In other words, an increase in the expected rate of inflation (caused by a decrease in the expected endowment) will be accompanied by an increase in the price of a real bond (i.e., future consumption) and a decrease in the real rate of interest. Note that the timing of the change in the price level is an implication of the assumption that the monetary sector of the economy is characterized by a cash in advance constraint with a unitary velocity of money (i.e., investors hold no cash balances between periods). In a monetary economy in which investors hold cash balances (e.g., Mundell), new information concerning the endowment or government demand in future periods might cause investors to adjust their cash balances by purchasing current output. For example, a decrease in next period's expected endowment will cause the expected level of next period's prices to rise, and the expected real value of investors' current cash balances to fall. Consequently, investors may attempt to reduce their cash balances by purchasing current consumption, causing part or all of the increase in the price level to occur in the current period. This reaction would tend to reduce or eliminate any increase in the expected rate of inflation. The timing of the price increase in this model (and thus the increase in the expected rate of inflation) is justified to the extent that claims on future payments denominated in units of currency (e.g., wages and salaries, or dividends) are large relative to the current level of investors' cash balances. Since, in the aggregate, investors cannot borrow against these future payments, the increase in the price level will tend to be delayed until the next period's endowment is available for consumption and the fixed cash pay-

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T.E. DaY, Expected inflation and real rate of interest

ments (the previous period's money supply in the model) have been received. The relative importance of the two effects is an empirical question. 5. Conclusion The results which have been presented in this paper show that the Mundell effect can be derived in a model where individuals' demand for real cash balances is unaffected by the expected rate of inflation (i.e., the velocity of money is, by assumption, invariant to changes in the expected rate of inflation). The relation between changes in expected inflation and the real rate of interest which has been derived is an associative relation, in contrast to the causative relation which Mundell has shown to exist when individuals' holdings of real cash balances are a function of the expected rate of inflation. The model which has been developed has a microeconomic foundation in the sense that the real rate of interest has been derived from the consumptioninvestment decision of a representative investor. This is in contrast to Mundell, who considered a macroeconomic model based upon the Hicksian IS and LM curves. The effects described in these two models capture different aspects of investor behavior in financial markets. Consequently, the two sources of variation in the real rate of interest need not be mutually exclusive. One empirical implication of the model is that when the demand for real cash balances is sensitive to the expected rate of inflation, a change in the expected rate of inflation may be accompanied by a change in the real rate of interest which is greater than that predicted by Mundell. References Clower, Robert A., 1967, A reconsideration of the microfoundations of money, Western Economic Journal 6, 1-9. Darby, M.R., 1975, The financial and tax effects of monetary policy on interest rates, Economic Inquiry 13, June, 266--276. Feldstein, Martin, 1976, Inflation, taxes and the rate of interest: A theoretical analysis, American Economic Review 66, Dec., 889-920. Fisher, Irving, 1930, The theory of interest (Macmillan, New York). Mundell, Robert A., 1963, Inflation and real interest, Journal of Political Economy 71, June, 622-626. Stockman, Alan C., 1981, Anticipated inflation and capital stock in a cash-in-advance economy, Journal of Monetary Economics 8, 387-393.