## Abstract

An important obstacle encountered in analyzing interest rate targeting is that standard models usually lead to indeterminacy of the price level or the inflation rate. This paper develops a simple framework that avoids such problems, because the bonds whose interest rate is controlled provide liquidity services. This framework is used to examine interest rate policy in a small open economy under predetermined exchange rates. A permanent increase in the interest rate has no real effects, whereas a temporary increase in the interest rate leads to higher consumption and a current account deficit that worsens over time.

**T**he interest rate enjoys a unique position among macroeconomic adjustment policies. Historically, it played a central role during the gold standard era (despite the insistence of economic theory on specie-flow mechanisms).^{1} Currently, it is one of the most watched variables among the Group of Seven industrial countries.^{2} In developing countries interest rates have been manipulated, among other things, to provide cheap credit to the government, to increase saving and investment, and to try to quell raging inflation.^{3}

The high reputation that interest rate policy enjoys among economic practitioners contrasts sharply with the one it has among their more academic peers. Wicksell (1965) thought that pegging the interest rate, for example, might lead to instability of the inflation rate—a conjecture that also emerges from standard adaptive-expectations models—while the theory of rational expectations demonstrates the possibility that, unless there is some additional nominal anchor, an interest rate policy may lead to indeterminacy of the price level or of the rate of inflation (see Sargent and Wallace (1975), McCallum (1981, 1986), Calvo (1983), and Gagnon and Henderson (1988)).^{4} Determinacy can be recovered, though, by simultaneously setting money supply targets.^{5} For the system not to become overdetermined, however, the interest rate and money supply targets cannot be set independently, which makes it difficult to distinguish the effects of interest rate policy from those of money supply policy.^{6}

Given the importance that policymakers sometimes attach to controlling interest rates in order to achieve certain objectives, it might be worthwhile to develop an analytical framework within which interest rate policy can be analyzed. This paper is an attempt in that direction; it develops a model in which interest rate policy is not subject to the above-mentioned indeterminacy or instability problems. More important—and in contrast to the approaches mentioned above—the effects of interest rate policy are much easier to identify because the interest rate target can be changed without modifying the money supply target.

The key feature of the model is the assumption that the interest rate that is controlled by the central bank is, in fact, not a *pure* rate, but rather the interest rate borne by bonds that possess some of the characteristics of domestic money.^{7} It is shown that just a “pinch of liquidity” in the assets whose interest rate is controlled by the central bank is in general enough to restore full monetary determination. This is an interesting result because it exhibits the relative weakness of the rational expectations case against interest rate policy, given that central banks usually deal with government assets that are not substantially different from high-powered money (see, for example, the discussion in Bryant and Wallace (1979)). This is the paper’s first message.

The analysis concentrates on a small open economy under predetermined exchange rates in which price flexibility and full employment prevail. In addition, and to abstract from other well-known effects, we make sufficient assumptions to guarantee inflation super-neutrality and Ricardian equivalence. These assumptions are strong and prevent the emergence of almost all effects from monetary policy (see Calvo (1989b)). The assumptions are responsible in part for the absence of real effects stemming from *permanent* changes in the interest rate controlled by the central bank.^{8} These assumptions, however, help to highlight the role of *temporary* policies—what almost all actual policymaking is about—because temporary policies are shown to have real effects, even under those strong proneutrality conditions. This is the paper’s second message.

A simple model is developed, which provides an understanding of some of the basic issues raised by interest rate policy. The discussion is carried out in terms of a representative consumer subject to a liquidity constraint. To introduce the liquidity component into domestic bonds, we assume that the consumer “produces” liquidity by means of high-powered money and bonds (which we identify with interest-bearing checking accounts). In the present context, therefore, increasing the interest rate on bonds is equivalent to offering a higher return on money (defined as the sum of high-powered money and interest-bearing checking deposits). Hence, it should not come as a big surprise that an interest rate hike could have expansionary effects, which is contrary to the predictions of more ad hoc models. This is the third message of the paper. We feel that this lesson could be very relevant for countries that attempt to influence aggregate demand by increasing the rate of interest. This policy tends to reduce the cost of holding liquid assets and could, therefore, exert no downward pressure on aggregate demand.^{9}

It is shown that a temporary increase in the rate of interest is equivalent to a temporary *decrease* of the rate of devaluation. They are both expansionary because they reduce the opportunity cost of holding money, and consequently tend to stimulate consumption. The resulting increase in money demand induces an initial accumulation of reserves at the central bank. The current account, however, deteriorates from the first moment, and reserves are eventually lost. The country is ultimately poorer and, more surprisingly, the representative individual also feels poorer from the very start. The policy, therefore, has no redeeming social value.

The paper proceeds as follows. Section I develops the basic model and shows that permanent changes in the nominal interest rate have no real effects. Section II studies the effects of a temporary increase in the nominal interest rate. Section III relates the analysis developed here to interest rate policy in more complex models.

### I. The Model

Consider a small open economy that operates under predetermined exchange rates. There is only one (nonstorable) good in the world whose price is given and equal to unity. The representative consumer is endowed with a constant flow of the good, denoted by *y*.

The utility function of the representative consumer is given by

where the instantaneous utility function, *u*(·), is assumed to be increasing, twice-continuously differentiable, and strictly concave; *c*_{t} denotes consumption; and β is the positive and constant subjective discount rate.

The consumer is subject to a “liquidity-in-advance” constraint. The requirement that a liquid asset (money) be used in order to purchase goods is by now a common feature of monetary models.^{10} A more novel feature introduced into the present analysis—which explains the use of the term “liquidity-in-advance”—is that the consumer is posited to use two distinct liquid assets to carry out his or her purchases. In addition to cash (*H*), the consumer makes use of demand deposits (*Z*) that earn interest at a rate *i* (for instance, NOW or Money Market Accounts).^{11} This formulation is intended to capture, in a simple way, the stylized fact that consumers usually resort to both types of assets to carry out their transactions.^{12} Specifically, the liquidity-in-advance constraint is given by

where *h*_{t} and *z*_{t} stand for real balances of cash and demand deposits, respectively; and *l*(·) is a concave, homogenous of degree 1, twice-continuously differentiable function, which can be viewed as a liquidity-services production function.^{13}, ^{14} The liquidity-in-advance constraint (2) requires that consumption not exceed the liquidity services produced by the use of both cash and demand deposits.

The consumer also holds an internationally traded bond, *f*, whose rate of return is given and equal to *r*. The consumer’s lifetime budget constraint is therefore

where *a*_{0} = *f*_{0}+ *h*_{0} + *z*_{0} denotes initial real financial assets; τ_{t} stands for government lump-sum transfers; and *I*, = *r* + π_{t}, (π_{t}, being the rate of inflation) stands for the *pure* nominal interest rate.^{15} *I*_{t}, and *I*_{t} - *i*_{t} (both of which will be assumed to be positive) represent the opportunity cost of holding cash and demand deposits, respectively.

The consumer chooses paths of *c*_{t}, *h*_{t}, and *z*_{t} to maximize the utility function (1), subject to constraints (2) and (3). In addition to constraint (2), holding with equality, and intertemporal budget constraint (3), the other first-order conditions for this problem are

where λ is the (time-invariant) multiplier associated with constraint (3).^{16}, ^{17} Equation (4) has the familiar interpretation that the marginal utility of consumption equals the marginal utility of wealth times the price of consumption. The relevant “price” of consumption in this model—which will be referred to as the *effective* price of consumption—is given by the term in square brackets on the right-hand side of equation (4). The effective price consists of the market price, unity, plus the marginal cost of producing the liquidity services needed to purchase a unit of consumption, *I*_{t}/*l*_{h} (*h*_{t},*z*_{t}. Intuitively, note that one additional unit of liquidity services is needed to purchase an additional unit of consumption. The production of a unit of liquidity services requires 1/*l*_{h} (*h*_{t},*z*_{t} units of *h*, whose cost is *I*_{t}/*l*_{h} (*h*_{t},*z*_{t}).^{18}

Equation (5) indicates that, at an optimum, the marginal rate of substitution between demand deposits, *z*, and cash, *h*, is equated to the ratio of their opportunity costs. Note that because *l*(*h*,*z*) is homogenous of degree 1, *l*_{z}(*h*,*z*) and *l*_{h}(*h*,*z*) are homogenous of degree 0. Therefore, equation (5) implicitly defines the demand for demand deposits relative to cash as a decreasing function of the opportunity cost of demand deposits, *I* - *i*, relative to that of cash, *I*:

where

An increase in / (that is, an increase in (*I* - *i*)/*I*), which raises the opportunity cost of demand deposits proportionately more than that of cash, induces the consumer to reduce the ratio of demand deposits to cash. An increase in *i* (that is, a decrease in (*I* - *i*)/*I*), which *lowers* the opportunity cost of demand deposits, increases the ratio of demand deposits to cash.

Combining the liquidity-in-advance constraint (2)—holding with equality—with equation (6) yields the demand for cash, demand deposits, and money, which is defined as the sum of cash plus demand deposits (see Appendix I):

where *m* ≡ *h* + *z* and a sign under an argument denotes the sign of the corresponding partial derivative. An increase in the nominal interest rate, *i*, reduces the demand for cash (equation (8)) and raises the demand for demand deposits (equation (9)), as one would expect. The effect on the demand for money of a higher nominal interest rate is positive (equation (10)). In other words, the reduction in the demand for cash is more than offset by the rise in the demand for demand deposits. The intuition is as follows. Since the opportunity cost of demand deposits, *I* - *i*, is lower than the opportunity cost of cash, *I*, the marginal productivity of demand deposits is, at an optimum, lower than that of cash (that is, *l*_{z}(*h*_{t},*z*_{t})<*l*_{h}(*h*_{t},*z*_{t}), as indicated by first-order condition (5)). Therefore, for a given level of liquidity services, a rise in the nominal interest rate, *i*, implies that cash must decrease by less than demand deposits increase, because cash is more productive, at the margin, than demand deposits. Thus, the demand for money (that is, the demand for cash *plus* demand deposits) increases as a result of a higher nominal interest rate.

The function ϕ(·), given by equation (6), can be used to express the effective price of consumption (*p*), given by the term in square brackets on the right-hand side of equation (4), as a function of *I* and *i*:

where

Equation (12) indicates that an increase in the pure nominal interest rate, *I*, raises the effective price of consumption, because it increases the opportunity cost of both cash and demand deposits. Less familiar, but critical to the whole analysis, is the way in which a rise in the nominal interest rate, *i*, affects the effective price of consumption. As follows from equation (13), a rise in *i decreases* the effective price of consumption, because it becomes less costly to hold interest-bearing demand deposits, which are used to produce liquidity services.^{19}

The other actor in this economy is the government. To keep the model simple, we abstract from the banking system and assume that the government issues two nominal liabilities: high-powered money, *H*, and interest-bearing demand deposits, *Z*. There is no government consumption, so that any revenues left after paying interest on demand deposits are transferred back to the consumer in a lump-sum fashion. Formally, the present value of government transfers is given by

where *k*_{0} stands for government’s initial holdings of bonds, and a dot over a variable denotes its time derivative. The government collects revenues from the creation of both high-powered money (ḣ_{t} + π_{t},*h*_{t}) and demand deposits (ż_{t} + π_{t}*z*_{t}).

The government is assumed to control the interest rate paid on demand deposits, *i*, by giving up the control over the *composition* of its liabilities, *H* and *Z*.^{20} Thus, the composition of the government’s liabilities is demand determined. In order to use this model to think about the real world, it is useful to keep in mind the following interpretation. The government issues bonds (for instance, treasury bills) yielding a rate *i*, which are entirely acquired by financial institutions. Financial institutions, in turn, issue demand deposits to consumers, which, in a competitive equilibrium with costless banking, will also yield *i*. It is as though the financial institutions “broke up” the government bonds into small pieces and sold them to consumers as NOW or money market accounts. This is the channel through which the nominal interest rate determined by the government affects the nominal yield of a portion of the consumer’s money holdings and, thus, the consumer’s consumption path.

The combination of equations (3) and (14) yields the economy’s lifetime resource constraint (provided, naturally, that the transversality conditions lim(*h*_{t}*e*^{−rt}) =0 and lim(*z*_{t}*e*^{−rt}) = 0, as *t* →∞ hold):

where *b*_{0} = *f*_{0} + *k*_{0} are the economy’s initial bond holdings. Equation (15) simply says that the present value of consumption equals the present value of tradable resources.

To derive the *equilibrium* path of consumption, assume, for computational simplicity, that *u*(*c*) = log(*c*).^{21} Making use of equations (4) and (15), the expression for the *equilibrium* value of the multiplier follows:^{22}

Substituting equation (16) into equation (4) yields the equilibrium consumption path:

This expression is key to the whole analysis. The ratio on the right-hand side (that is, the factor that multiplies *y* + *rb*_{0}) can be viewed as the *equilibrium* marginal propensity to consume (MPC) out of permanent income, *y* + *rb*_{0}. As will become clear below, the numerator of this ratio can be interpreted as the average effective price over the interval [0,∞). Therefore, the equilibrium MPC is the ratio of the average effective price to the current effective price. In equilibrium, all that matters is the average effective price *relative* to the current effective price, because changes in the effective price of consumption have no wealth effects, so that only substitution effects remain. Hence, if the effective price path is constant over time, the equilibrium MPC is unity for all *t* and consumption is constant over time and equal to permanent income; that is, *c*_{t} = *y* + *rb*_{0}) for all *t*.^{23} The constancy over time of the effective price of consumption implies that there are no incentives to engage in intertemporal consumption substitution. In contrast, if the effective price is lower today than it will be in the future, today’s equilibrium MPC is above unity—because the average effective price is higher than the current effective price—and hence today’s consumption is above *y* + *rb*_{0}. Future consumption, therefore, will be below *y* + *rb*_{0}. The fact that the effective price path is not constant over time induces intertemporal consumption substitution.

Finally, the equilibrium current account path can be derived as follows. Private asset accumulation is given by

The government’s flow budget constraint indicates that the excess of revenues over spending results in asset (or reserve) accumulation:

Combining equations (18) and (19) yields

As expected, the accumulation of net foreign assets (that is, the current account balance) is equal to the difference between income and consumption.

To derive the equilibrium path of reserves, it is necessary to specify the domestic credit rule. As usual, it is assumed that domestic credit expansion just compensates the consumer for the real depreciation of money balances. Therefore, the level of transfers adjusts endogenously, so that the time derivative of domestic credit (measured in real terms) equals π(*h*_{t} + *z*_{t}):

If the domestic credit rule (equation (21)) is substituted into the government’s flow budget constraint (19), it follows that k̇_{t} = k̇_{t} (recall that *m* ≡ h + *z*). Thus, all changes in reserves are associated with changes in real money balances.

It can now be shown that a permanent change in the nominal interest rate has no real effects. For this purpose, assume a constant devaluation rate, so that π_{t} is constant at π and, thus, *I* = *r* + π.^{24} It is assumed here—and throughout the paper—that, prior to the disturbance (which takes place at *t* = 0), the economy is in a stationary state. At the initial steady state, *i*_{t} = *i*^{l}. Then it follows from equation (17) that *c* = *y* + *rb*_{0}; namely, consumption equals permanent income. Consider an unanticipated and permanent increase of *i* from *i*^{l} to *i*^{h} at *t* = 0.^{25} It can be readily verified from equation (17) that consumption remains unchanged at the permanent income level, *y* + *rb*_{0}. The intuition follows from the interpretation of equation (17). Recall that a change in *i*, whether temporary or permanent, has no wealth effects in equilibrium, as can be seen from equation (15).^{26} Therefore, consumption will change only as a result of intertemporal price substitution effects. The permanent change in *i*, however, does not affect the equilibrium MPC, which continues to be unity, because the average effective price decreases by the same amount as the current effective price does. The fact that the effective price path remains flat—even if at a lower level because the rise in the nominal interest rate reduces the effective price of consumption—implies that there is no change in consumption.

The rise in the interest rate increases the demand for demand deposits by more than it decreases the demand for cash; that is, the demand for money rises, as follows from equation (10). This implies that there is a gain in reserves as the consumer exchanges bonds for money at the central bank.

Although there is no change in the pure real interest rate, *I* - π, the rise in *i* increases the real interest rate on the liquid bond, *i* - π. Thus, in the present context, there is no necessary connection between the real interest rate on the liquid asset and real economic activity.

### II. Temporary Increase in the Interest Rate

This section focuses on the central experiment of the paper: a temporary increase in the nominal interest rate, *i*.^{27} Suppose that at time 0 (the “present”), the interest rate is temporarily increased from *i*^{l} *to* *i*^{h}. At time *T*, the interest rate is brought back to *i*^{l}. More formally, for some *T*>0,

where *i*^{h} > i^{l}. Initially (that is, before time 0), consumption is at its permanent income level, given by *y* + *rb*_{0}. Substituting equations (22a) and (22b) into equation (17) yields the consumption path for *t* ≥ 0:

where

For notational convenience, *p*^{h} and *p*^{l}—where *p*^{h} > *p*^{l}—stand for the effective price of consumption associated with *i*^{l} and *i*^{h}, respectively. Since a higher nominal interest rate implies a lower effective price of consumption—recall equation (13)—the effective price of consumption decreases at time 0 from *p*^{h} to *p*^{l}, and it increases back to *p*^{h} at time *T*. In order to make clear that, as already suggested, the term 1/[(1/*p*^{l})(1 - *e* ^{−rT}) + (1/*p*^{h})*e*^{−rT}] in equations (23a) and (23b) can be interpreted as the average effective price, equations (23a) and (23b) can be rewritten as

where 0 ϕ(*T*) < 1; ϕ(*T* → 0) → 0; ϕ(*T* → ∞) → 1;and ϕ′(*T*) > 0 (see Appendix II). The average effective price is thus a weighted average of *P*^{$l$} and *P*^{$h$}, with the weight being determined by the length of the period during which each effective price will prevail. Hence, equations (24a) and (24b) illustrate the notion that the equilibrium MPC is the ratio of the average effective price to the current effective price.

The consumption path that results from policy (22a) and (22b) follows from equations (24a) and (24b). When the nominal interest rate is increased at time 0, there is a once-and-for-all reduction in the average price. The fall in the average price, however, is more than offset by the reduction in the current price. Therefore, the equilibrium MPC increases above unity (as can be verified from equation (24a)), which raises consumption. The fact that the current effective price remains unchanged until *t* = *T* implies that consumption stays constant as well. At *t* = *T*, the current price increases back to *P*^{$h$}, which decreases the equilibrium MPC below unity, so that consumption jumps downward at that point. This anticipated discontinuity in the consumption path is feasible under predetermined exchange rates, because with the price level, and hence the exchange rate, given at *t* = *T*, there are no profit opportunities.^{28} The consumer exchanges money for bonds at the central bank to achieve the desired level of real money balances.

Figure 1 illustrates the consumption path that results from the policy described by equations (22a) and (22b). Initially (that is, before time 0), consumption is at its permanent income level, given by *y* + *rb*_{0}. When the policy (equations (22a) and (22b)) is announced at time 0, consumption jumps upward and remains constant up to time *T*. At time *T*, consumption jumps downward and remains constant thereafter at a level below initial permanent income.

The path of the current account—which is illustrated in Figure 2—is given by equation (20), taking into account equations (24a) and (24b). Due to the increase in consumption at *t* = 0, the current account jumps into deficit. It then deteriorates steadily between time 0 and *T*, even though the trade deficit (that is, *c*_{t} - *y*) remains constant, because interest payments on net foreign assets decline throughout. At *t* = *T*, the current account jumps into balance and the stock of foreign assets stops declining. In the new steady state, net foreign assets are lower than they were initially.

As far as the stock of reserves is concerned, note that both the rise in the nominal interest rate and the rise in consumption increase money demand (recall equation (10)). Therefore, there is an initial gain in central bank reserves at *t* = 0. Reserves remain constant between time 0 and time *T*, but when time *T* arrives there is a sharp drop in reserves (because both the interest rate and consumption fall), which will more than offset the initial gain. This follows from the fact that the steady-state stock of reserves decreases. The reason is that, in the new steady state, the interest rate is back at its initial level but consumption is lower, which implies, by equation (10), that real money demand is lower. Since changes in real money supply reflect changes in reserves, the lower steady-state real money supply implies that the level of reserves is also lower in the new steady state.

This policy experiment illustrates even more sharply the lack of necessary relationship between the real interest rate on the liquid asset and economic activity. Since the effect of the policy described by equations (22a) and (22b) is independent of the level of the nominal interest rate, *i*, at time 0, the real interest rate on the liquid asset could have gone up or down during [0,*T*), and the real effects would still be the same. It is not, therefore, the *level*, but the *expected change* of the real (or nominal) interest rate on the liquid asset that really matters.

Equations (24a) and (24b), together with the earlier finding that permanent changes in *i* do not affect consumption, may be used to examine the role played by *T*. The parameter *T* measures the time during which the interest rate remains at the higher level; namely, the degree of “temporariness” of the policy. Figure 3 depicts the consumption path for different values of *T*. Clearly, if *T* = 0 (that is, there is no change in the interest rate), *c*_{t} = *y* + *rb*_{0} The same is true if “*T* = ∞” (that is, the increase in *i* is permanent).

To examine the initial jump in consumption (and, hence, the level of consumption for 0 ≤ *t* < *T*) for a positive and finite *T*, consider equation (24a) as a function of *T*. It follows from equation (24a) that *c*(*T*) is discontinuous at *T* = 0, because as *T* → 0, lim[*c*(*T*)] = (*y* + *rb*_{0})(*p*^{h}/*p*^{l}). Furthermore, *c*(*T*) is a decreasing function of *T*, and as *T* → ∞, lim[*c*(*T*)] = *y* + *rb*_{0}. Thus, when the degree of temporariness gets arbitrarily large, consumption before *T* tends to its permanent (as of *t* = 0) income level. Equation (24b), as a function of *T*, yields the level of consumption for *t* = *T* (and hence for *t* ≥ *T*) for those cases in which *T* < ∞. It follows from (24b) that (1) consumption at *T* is a decreasing function of *T*; (2) as *T* → 0, lim[*c*(*T*)] =y + *rb*_{0}; and (3) as *T* → ∞, lim[*c*(*T*)] = (*y* + *rb*_{0})(*p*^{l}/*p*^{h}). Note that the latter limit does not coincide with the value of consumption when “*t* = ∞” (that is, when the rise in the interest rate is permanent); this situation is analogous to the discontinuity of *c*(*T*) at *T* = 0.

**Consumption Path for Different Degrees of Temporariness**

Citation: IMF Staff Papers 1990, 004; 10.5089/9781451930788.024.A002

**Consumption Path for Different Degrees of Temporariness**

Citation: IMF Staff Papers 1990, 004; 10.5089/9781451930788.024.A002

**Consumption Path for Different Degrees of Temporariness**

Citation: IMF Staff Papers 1990, 004; 10.5089/9781451930788.024.A002

Thus, as Figure 3 illustrates, the shorter the period of time during which the rise in the interest rate remains in effect (*T* → 0), the larger the initial increase in consumption (and hence the larger the initial current account deficit) and the higher the level of consumption after *T*. Intuitively, the price path is flat except for a very short period of time, which implies that the effects of intertemporal price speculation alluded to above are exacerbated. In contrast, the longer the high interest rate remains in place (*T* → ∞), the smaller the initial increase in consumption and the lower the level of consumption after *T*. The path of consumption for an intermediate value of *T*, such as *T*′, lies somewhere in between the paths for *T* → 0 and *T* → ∞.

To illustrate the effects of a higher ratio of demand deposits to cash on the response of the effective price of consumption (and thus of consumption) to a rise in the nominal interest rate, consider the case in which *l*(*h*,*z*) exhibits fixed proportions.^{29} Formally, let the liquidity-in-advance constraint be given by

where

The effective price of consumption is then (taking into account the policy described by equations (22a) and (22b))

Figure 4 illustrates the effects of a rise in the interest rate for different values of 1 - *q*. Clearly, if 1 - *q* = 0, there is no channel through which the interest rate can affect the effective price of consumption, so that the consumption path remains flat. A given increase in the nominal interest rate results in a lower effective price the larger is 1 — *q*. Thus, a larger 1 - *q* leads, other things being equal, to higher consumption between 0 and *T* and a correspondingly lower consumption after *T*. This suggests that economies in which interest-bearing liquidity plays an important role will exhibit a larger response to temporary interest rate increases.^{30}

Consider now the welfare implications of a temporary rise in the interest rate. To begin with, observe that such a policy is not Pareto-optimal: a planner interested in maximizing the utility function (equation (1)), subject to the economy’s resource constraint (equation (15)), would choose a constant level of consumption equal to permanent income. Therefore, the optimal policy is to choose a *constant* interest rate—*the level is irrelevant*—that induces the consumer to choose the Pareto-optimal consumption path. It seems intuitively clear that the welfare cost of a given temporary rise in the interest rate increases with the proportion of demand deposits held by the consumer. The reason is that, as argued above and illustrated in Figure 4, a higher initial ratio of demand deposits to cash results in a “less smooth” consumption path and would—one would expect—lead to higher welfare losses. (Simulations of the model suggest that this is indeed the case.) In contrast, welfare does not change monotonically with *T* because of the following. First, recall that consumption remains unchanged when *T* = 0 or “*T* = ∞”; second, the welfare loss converges to zero when *T* → 0 or *T* → ∞; and third, the welfare loss is positive for all *T*, such that 0 < *T* < ∞.^{31} Therefore, there exists a *T* ∈ (0,∞) at which the welfare loss reaches a maximum.

**Consumption Path as a Function of the Proportion of Demand Deposits Held**

Citation: IMF Staff Papers 1990, 004; 10.5089/9781451930788.024.A002

Note: (1-*q*

_{1}) > (1-

*q*

_{2}).

**Consumption Path as a Function of the Proportion of Demand Deposits Held**

Citation: IMF Staff Papers 1990, 004; 10.5089/9781451930788.024.A002

Note: (1-*q*

_{1}) > (1-

*q*

_{2}).

**Consumption Path as a Function of the Proportion of Demand Deposits Held**

Citation: IMF Staff Papers 1990, 004; 10.5089/9781451930788.024.A002

Note: (1-*q*

_{1}) > (1-

*q*

_{2}).

It should be noted that this setup could be used to examine a temporary stabilization program under predetermined exchange rates (that is, a temporary reduction in the devaluation rate), as in Calvo (1986). Suppose that π is reduced during the period from *t* = 0 to *T*. It follows from equations (12), (13), and (17) that a reduction in π (which reduces *I*) is equivalent to an increase in *i*, because both affect consumption through the same (and only) channel; that is, by lowering the effective price of consumption. Therefore, the same effects on consumption and the current account obtain for a temporary decrease in the rate of devaluation.

Finally, note that the authorities could control the interest rate paid on demand deposits by manipulating reserve requirements.^{32} To show that this policy is equivalent to the one studied above, note that, under competitive and costless banking, the zero-profit condition implies that

where 0 < γ < 1 denotes the required reserve ratio. Since *I* is exogenously given, controlling γ is tantamount to controlling *i*. Thus, a temporary reduction in γ—by temporarily reducing the effective price of consumption—has an expansionary effect on consumption.

### III. Conclusions

Policymakers have frequently manipulated nominal interest rates in order to achieve different macroeconomic goals. An analytical obstacle that has hindered studies of the effects of such policies in the context of rational expectations models has been that interest rate targeting usually results in indeterminacy of the price level or the inflation rate. This paper provides an analytical framework in which a meaningful examination of interest rate policy is feasible. The key feature is that the authorities control the interest rate borne by a liquid bond—which we identify as interest-bearing demand deposits—rather than the rate of the nonliquid bond. In this context, raising interest rates lowers the cost of holding money. If the rise in the interest rate is unexpected and permanent, there are no real effects because the effective price of consumption remains constant over time. If the rise in the interest rate is temporary, the effective price of consumption is lower today compared to the future, and the increase thus has an expansionary effect on consumption.

We have isolated one specific—and often disregarded—channel through which higher nominal interest rates may work. This is an important first step, from a conceptual point of view, toward understanding the effects of interest rate policy in more realistic (and thus more complex) models. Even in considerably more complex models, this channel will still be present.

We have dealt with the simplest possible scenario: a small open economy with full employment, flexible prices, and predetermined exchange rates. The simplest extension is to consider the case of flexible exchange rates and analyze the effectiveness of raising interest rates in fighting inflation (see Calvo and Végh (1990b)). In that context, it is still true that only temporary changes in the interest rate have any real effects on the economy. The inflation rate always increases on impact as a result of a temporary rise in the interest rate, and increases exponentially thereafter. Consumption may increase or fall on impact. The reason consumption may fall on impact is that the higher inflation rate—which tends to increase the effective price of consumption—may more than offset the effect of the higher interest rate—which tends to decrease the effective price. The conclusion, therefore, is that raising interest rates seems hardly appropriate in the context of a flexible-prices model as a means of fighting inflation.

The expansionary effects that may result from an increase in the interest rate in the flexible-prices models run counter to conventional wisdom, according to which raising interest rates should be contractionary. This has led us to examine interest rate policy in a sticky-prices model. In Calvo and Végh (1989, 1990c), we analyze interest rate policy in a closed economy, staggered-prices model. The main message is that the conventional (that is, contractionary) effects re-emerge in that context. Inflation is brought down—at the cost of a sharp fall in output—both when the interest rate is raised temporarily and when it is raised permanently. When the rise is temporary, however, the initial fall in inflation is followed by an upsurge in inflation over and above its initial level. The expansionary effect of a higher interest rate isolated in this paper is still present in the sticky-prices model, in the sense that the consumer might want to increase consumption. The key difference, however, is that the increased consumption cannot be effected because the real money supply cannot increase instantaneously, since (1) the economy is closed, and (2) the price level cannot jump.

Since the absence of capital mobility might be crucial to the results obtained in the closed economy, staggered-prices model, the next logical (and final) step in our quest for understanding interest rate policy is to open the sticky-prices model to trade in goods and assets. The conjecture is that-—in the Mundell-Fleming spirit—predetermined exchange rates coupled with perfect capital mobility would dramatically affect the results. To this effect, we plan to use the open economy, staggered-prices model developed in Calvo and Végh (1990a) to study interest rate policy. One would expect that, under predetermined exchange rates, the expansionary effects reappear—even in the presence of sticky prices—because the public may increase its real money holdings through the central bank window. Therefore, a temporary increase in the interest rate may have expansionary effects in both the traded and nontraded goods sector. Under flexible exchange rates, however, one would expect to find that the traded goods sector may expand while the nontraded goods sector contracts. Once again, flexible exchange rates prevent the public from increasing real money balances (in terms of nontraded goods).

### APPENDIX I Derivation of Demand for Cash, Demand Deposits, and Money

From equation (2), holding with equality, and equation (6), it follows that

Clearly, all three functions are increasing in c. Differentiation of equations (28)–(30) yields (denoting (*I* - *i*)/*I* by *R*)

The sign of the numerator of equations (32) and (33) follows from Euler’s theorem and the fact that, by equation (5), *l*_{h} > *l*_{z}.

### APPENDIX II Derivation of Φ(*T)*

To illustrate the idea that the equilibrium MPC can be thought of as the ratio of the average effective price to the current effective price, the function Φ(*T*) was used in equations (24a) and (24b). This function and its properties are now derived. Let the function Φ(*T*), 0 < *T* < ∞, be defined in implicit form by

where the right-hand side is the numerator of the ratio that multiplies *y* + *rb*_{0}, in equations (23a) and (23b). For notational simplicity, denote the denominator on the right-hand side of (34) by Γ(*T*). Using (34), Φ(*T*) can be solved for

Clearly, Φ(*T*) is continuous for *T* ∈ (0,∞) since *p*^{l} ≠ *p*^{h}. Recalling that *p*^{l} ≠ *p*^{h}, it follows from (35) that (1) &(*T*) > 0; (2) Φ′(*T*) > 0; (3) as *T* → 0, lim[Φ(T)] = 0; and (4) as *T* → ∞, lim[Φ(T)] = 1. Hence, 0 < Φ (*T*) < 1.

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^{}*

Guillermo A. Calvo is a Senior Advisor in the Research Department. Carlos A. Végh is an Economist in the Research Department. An earlier version of this paper was presented at the Fourth Annual Economic Meeting of the Central Bank of Uruguay, held in Montevideo, November 6–7, 1989; and at the Twenty-Fourth Annual Meeting of the Argentine Association of Political Economy, held in Rosario, November 8–10, 1989. The authors are grateful to Aquiles Almansi, Julio de Brum, Matthew Canzoneri, Pablo Guidotti, Mohsin Khan, David Papell, Carmen Reinhart, and conference participants in Montevideo and Rosario for helpful comments and discussions.

^{}1

Hume (1752) provided the classical exposition of the price specie-flow mechanism. Governments’ frequent disregard for the “rules of the game” under the gold standard is the focus of Bloomfield (1959). At the time, observers were well aware of the active use of interest rates by central banks. Keynes (1924, p. 19), for instance, argued that one of the main characteristics of the British monetary system under the gold standard was “the use of the bank rate for regulating the balance of immediate foreign indebtedness (and hence the flow, by import and export, of gold).”

^{}2

Batten and others (1990) examined the implementation of monetary policy in the Group of Five industrial countries and concluded that monetary authorities focus on influencing key short-term interest rates.

^{}3

See Fry (1988) for a discussion of interest rate policies in developing countries.

^{}4

McCallum (1986) distinguishes between price level “indeterminacy” (the model does not determine the value of any nominal magnitude) and “non-uniqueness or multiplicity” of price level solutions (there are multiple paths of real money balances).

^{}5

See, among others, McCallum (1981, 1986), Calvo (1982), Canzoneri, Henderson, and Rogoff (1983), Goodfriend (1987), Barro (1989), and Reinhart (1989).

^{}8

The expression “real effects” refers to effects on consumption—which determine welfare. Permanent policies, however, will affect the real money supply.

^{}9

Another independent, and more familiar, reason for the counterproductive nature of an interest rate hike in high-inflation countries is that in many cases the government is one of the main borrowers. Thus, the policy tends to worsen the fiscal situation even further.

^{}10

The use of the cash-in-advance constraint in continuous-time models is discussed in Feenstra (1985).

^{}12

The incorporation of demand deposits into the cash-in-advance constraint can be found in Walsh (1984). Brock (1989) assumes that both assets reduce transaction costs or “shopping” time. Englund and Svensson (1988) distinguish between “cash” and “check” goods, both of which are subject to liquidity constraints. Demands for both cash and demand deposits have been derived in a Baumol-Tobin context by Santomero (1979) and Whitesell (1989). Given the different costs of using cash versus debitable accounts, cash is used for small transactions and debitable accounts are used for large transactions (see Whitesell (1989)).

^{}13

A subscript on a function denotes the partial derivative with respect to the subscripted variable.

^{}14

The assumption *l*_{hz}>0 is equivalent to ruling out perfect substitutability between *h* and *z* (Notice that if *l*_{hz}>0,=0. then, by Euler’s theorem, *l*_{hh} = *l*_{zz} = 0, in which case *l*(*k*, *z*) is a linear function.) As discussed below, the analysis still applies to the case of perfect substitution, but it involves a corner solution in which only demand deposits are used.

^{}15

The rate *I* will be referred to as the *pure* nominal interest rate, because it is the rate of return borne by a pure bond—in the sense that the bond does not yield liquidity services. The rate borne by demand deposits, *i*, will be referred to simply as the nominal interest rate.

^{}16

It has been assumed that β = *r* to ensure the existence of a steady state. This implies that there are no intrinsic dynamics in the model, so that all dynamics will result from the implementation of temporary policies.

^{}17

The constraint (2) holds with equality at an optimum, because it has been assumed that the opportunity cost of both cash and demand deposits is positive.

^{}18

This follows from differentiating equation (2)—holding with equality—and setting dz = 0 to obtain *dh*/*dc* = 1/*l*_{h}(*h*,*z*).

^{}19

Notice that if *l*_{hz} = 0, equation (13) does *not* apply because *h* and *z* would be perfect substitutes, as indicated earlier. In that case, the consumer uses only demand deposits because they have a lower opportunity cost than cash. The effective price of consumption is 1 + *I* - *i* (assuming *l*(*h*,*z*) = *h* + *z*), so that an increase in *i* decreases the effective price of consumption.

^{}20

Alternatively (as shown below), the authorities can be viewed as issuing only high-powered money and determining the interest rate paid on demand deposits by controlling reserve requirements.

^{}23

Note that a constant path of the effective price constitutes the only case in which the equilibrium MPC is unity for all _{t}. If the effective price varies over time, the equilibrium MPC may be unity for *some* *t*, but it cannot be unity for *all* *t*.

^{}25

Throughout the paper, superscripts “*l*” and “*h*” will be used to denote “low” and “high” values, respectively, of any given variable.

^{}26

If cash and demand deposits were modeled as reducing transaction costs, as in Brock (1989), then changes in *i* would affect available resources. In this paper, by adopting the liquidity-in-advance specification, we abstract from such effects in order to concentrate exclusively on intertemporal price speculation effects.

^{}27

Since permanent changes in the nominal interest rate have no real effects, this section’s results also hold for an anticipated decrease in the nominal interest rate.

^{}28

In the context of anticipated devaluations, an anticipated discontinuity is possible only if there is no capital mobility (see Calvo (1989a)).

^{}29

The liquidity services production function needs to be specified for this exercise because third derivatives are involved.

^{}30

In the Cobb-Douglas case (*c* = *h*^{θ}), it can be shown that *pi*(*I, i*) = - (*z*/*h*)^{θ}; that is, the fall in the effective price of consumption as a result of a rise in the interest rate is an increasing function of the ratio of demand deposits to cash. Therefore, a higher initial ratio *z*/*h*—because of a higher initial value of *I*/(*I* - *i*)—implies a larger fall in the effective price of consumption and therefore a larger increase in consumption.