Diseconomies of Money and Optimal Inflation

Transcription

1 Diseconomies of Money and Optimal Inflation Edward J. Balistreri Colorado School of Mines April 2006 Abstract This paper develops a growth model that is affected by the rate of inflation. The problem of matching savings to tangible capital investments indicates exposure to inflation (in what is otherwise a fully-competitive intertemporal exchange equilibrium). Inflation acts as a tax on unmatched savings and, therefore, might be used as a policy instrument that better aligns private incentives to search for productive matches. Simulations indicate that a small positive rate of inflation is the optimal stationary policy. Further, the optimal rate of inflation is negatively correlated with transitory productivity shocks. The model offers a unique illustration of monetary policy in a micro-consistent real general equilibrium. (JEL E00, E13) Engineering Hall 311, Division of Economics and Business, Colorado School of Mines, Golden, CO ; ebalistr@mines.edu.

2 Diseconomies of Money and Optimal Inflation 1 Introduction It is difficult to deny the real effects of monetary policy. It is equally difficult to relinquish the transparency and logic of a real Arrow-Debreu general equilibrium. How can money matter in such a world? In recent theoretic work, the answers are as logical as they are ingenious. In particular, the work of Kiyotaki and Wright (1989) and Kocherlakota (1998) have helped enormously in our understanding of the operation of money. In practice, however, these theories are a backdrop to the money-in-the-utility-function and Clower-constraint models that dominate general-equilibrium applications. 1 This paper builds on the new micro theories of money to offer a direct reconciliation of real monetary-policy effects and our notion of a real neo-classical growth model. To reconcile the fundamental tools that we apply to exchange equilibria and the study of monetary phenomena, I formulate a model that is homogeneous degree zero in real presentvalue prices but is affected by the rate of inflation. I accomplish this by first noting that money is an inferior claim that results from a failure to match real savings to physical capital. Inflation is an instrument that taxes these idle claims. Like with other matching problems, there are coordination failures and search externalities associated with investment, and these indicate exposure to inflation despite the fact that perceptions are correct and real prices are perfectly flexible. The matching distortions also indicate a non-zero optimal rate of inflation, which is qualitatively at odds with the negative optimal rate of inflation suggested by the Friedman rule. 2 1 Wallace (1998), introduces the Journal of Economic Theory s symposium on models of money and the study of monetary policy (which is lead by Kocherlakota s paper). Wallace clearly notes the disconnect between contemporary monetary theory and monetary-policy models that use short-cuts to derive money demand. In a more recent paper Wallace (2001) refines his critique of models that assume money demand. He suggests that a more fruitful path for monetary economics is to embrace the micro-consistent theories in which money operates as an essential mechanism that helps overcome well specified frictions. 2 Friedman (1969) proposes a rate of price deflation that optimally drives the nominal interest rate to 1

3 I make no attempt to formulate money demand. From the perspective (taken here) that money is unmatched income, formulating demand for money makes as much sense as formulating demand for unemployment. Money (held during the elapse of time) is unemployed capital. Money is a claim that transmits wealth through time at zero (or near zero) matching costs. Investment in physical capital, in contrast, transmits wealth through time (is productive along the way) but faces a nontrivial matching process. If physical-capital matching is not perfect monetary holdings result, as the default. In fact, in the real formulation presented here and under the assumption that the commodity value of money is negligible, inflation is the only mechanism by which monetary policy affects the economy. 3 Money itself is not tracked as a commodity in the exchange equilibrium. Households perceive inflation as a tax on unmatched real savings. All commodities are real and all prices are measured in terms of a real numeraire good. 4 The primary literature that I draw on to frame the model is the theoretic work that explores the nature of money. Kiyotaki and Wright (1989) illustrate that money (a medium of exchange) will emerge naturally in a noncooperative exchange equilibrium that involves matching problems. They go on to show that fiat money (a commodity with no intrinsic value) might well serve as the medium of exchange. A key lesson from the work of Kiyotaki and Wright (1989) is that fiat money is useful as a store of value when exchange matching in real commodities is not perfect the double coincidence of wants problem. Kocherlakota (1998) explains, in a more general context, that money takes the place of memory (when zero. Friedman argues that households should not be penalized for having to hold money (which is costless for society to produce) so deflation should match the real interest rate. Articles by Chari et al. (1996) and Correia and Teles (1999) are more recent examples that support the Friedman rule in contemporary general-equilibrium models with transactions technologies (cash-in-advance constraints) or money in the utility function. The Friedman rule is not valid in the theory presented here because money is not forced on agents through an artificial constraint. 3 Assuming that the commodity value of money is negligible is convenient but not essential. One might extend the model to include commodity money. The key feature of money in this formulation is its ability to transmit wealth through time without matching costs (not its nature as a tangible commodity). 4 One could always use the real prices and the rate of inflation to back out prices in money units, but this is neither necessary nor desirable in the model presented. 2

4 Figure 1: The flow of physical output and real claims memory is not perfect). Money operates in these models as a tally, indicating something real is owed, for previous transactions the value of which was not instantaneously exhausted. Operationalizing the ideas of Kiyotaki and Wright (1989) and Kocherlakota (1998) in a growth model requires specific assumptions about how ideal contracting breaks down. I assume, specifically, that the matching and storage problems manifest themselves as a matching wedge between savings and real investment. Consider the illustration of physical and financial flows in Figure 1. Production at time t generates physical output and an equal number of claims to that output (Y t ). A household may exercise these claims immediately through current consumption, but saved claims (S t ) face a matching problem. With a matching rate of µ some of the savings convert into current investment (I t ), which becomes productive capital in the following period (K t+1 ), but some portion (1 µ) fails to match. On the physical side some capital produced at t fails to become employed at t + 1, and on the financial side some savings must be held in idle real claims during the elapse of time, M t 3

5 (real money balances). The value of idle claims (held as money balances) is eroded by inflation. Under ideal circumstances the household would prefer not to hold money, but stochastic rationing prevents µ from reaching one. Stochastic rationing prevents the household from finding the full set of physical-capital ownership contracts (which would form a perfect hedge against inflation). Matched savings are hedged because their value is linked to the physical asset (not money). In contrast, the value of unmatched claims will fall if a monetary authority dilutes the pool of claims (by printing its own claims). Essentially, the quantity of unmatched physical capital has not changed, but the number of idle claims that chase that fixed quantity has increased (by any new claims made by the government). The value of unmatched savings is, thus, eroded at the inflation rate. Inflation is a direct tax on unmatched real claims. We assume that the revenues from this tax return to the household, lump sum. The revenues must be recycled to maintain a balance between the physical and financial flows. 5 Although money is inferior to a (matched) physical-capital ownership contract, it does transmit (at least some portion of) the memory of the real claim on idle physical capital through time. A lack of ideal contracting indicates that agents are exposed to the inflation tax. My intention in this paper is to demonstrate a neo-classical growth model that roughly fits the physical and financial flows outlined in Figure 1. The model becomes very interesting if I allow the household to affect its matching rate through a costly search activity, and if there are aggregate coordination failures (matching externalities) that also affect the matching rate. I propose that optimal social policy might include a small rate of inflation that induces search. This paper proceeds as follows. In Section 2 a relatively transparent two-period general 5 For example, in Figure 1, if output is 10 units which supports 6 units of current consumption, 3 units of current investment, and 1 unit remains idle during the elapse of time then this is consistent with a matching rate of 60 percent and a real inflation tax equal to one unit (π = 50% acting on a base of (1 µ)s t = 2). In this case the household s income in t is 11 claims, although one claim is lost to the inflation tax. 4

6 equilibrium is set up to illustrate the basic operation of the matching externalities and the inflation tax. The major lesson to take away from the two-period model is that inflation may be used as an instrument that counteracts the matching distortions. In Section 3 the infinite-horizon model is formulated including a detailing of the steady-state calibration of the simulation model. In Section 4 various simulations are explored that show a small-positive optimal rate of inflation, and the possibility that transitory inflation reactions might be used to counteract transitory productivity shocks. Section 5 concludes. 2 Two-period Allocation in General Equilibrium To build the intuition behind the fundamental matching distortion, and the operation of inflation, we first consider a simple two-period allocation problem. An individual household (i) seeks to maximize utility (U i ), which is a well behaved function of consumption in periods one and two (C1 i and C2 i ). The household is constrained by its real endowment in period one (E i ) and the technology by which the current endowment might be transmitted into period 2. When the household chooses to transmit some of this wealth into period 2 via savings (S i ) it faces a matching problem. The portion of savings matched to physical capital will earn a return (r), but the portion that fails to match earns no return and is eroded by a tax, which we will call inflation (π). The rate of return is assumed exogenous in the two-period model, and the rate of inflation is an exogenous policy instrument. The inflation tax generates real lumpsum revenues (R i ). 6 The household has the ability to affect its matching rate (µ i ) by devoting more or less current wealth to a search activity (Z i ). The household problem is to maximize 6 The real value of the revenue generated by the inflation tax might be thought of as the purchasing power of the money injection. These are claims on real resources made by the government, which we assume are directly transferred back to the household. The household, therefore, acts on a real income of E i + R i. 5

7 U i = U i (C1 i, C2 i ) (1) subject to its real budget constraint E i + R i = C1 i + S i + Z i (2) and the transmission constraint C2 i = S i [(1 + r)µ i (Z i ) + (1 π)(1 µ i (Z i ))]. (3) From the household s perspective the matching rate, µ i, is only affected by Z i, but in the general equilibrium µ i is affected by scale and congestion in the market for physical capital. We explicitly assume that the scale and congestion effects are external to the small household, and thus not considered in their calculus. The resulting externality indicates that households will not engage in savings and search for real investment opportunities to the socially optimal levels. To be more precise consider the following matching technology faced by an individual household i: µ i = µ i (µs, (1 µ), Z i ). (4) The matching function is assumed to be continuous, nonnegative, increasing in each argument, and concave, with µ i (0, (1 µ), Z i ) = µ i (µs, 0, Z i ) = µ i (µs, (1 µ), 0) = 0. 7 The first argument in the function, µs, is economy-wide matched savings. This indicates a key scale effect, by which deeper investment markets improve the matching opportunities of individual households. The scale effect is tempered, however, by a congestion effect represented in the second argument, (1 µ). At high aggregate matching rates the term (1 µ) approaches zero and congestion begins to dominate individual matching opportunities. As mentioned, the household is assumed to be a small enough player such that µs and (1 µ) are taken as constants in the decentralized programming problem, but in a competitive equilibrium with 7 A Cobb-Douglas form works and is adopted in subsequent simulations. 6

8 many homogeneous households µ = µ i. If µ = µ i the restrictions placed on the matching technology ensure that µ (0, 1). 8 From the first-order conditions we can characterize household behavior. We drop the subscript at this point as all households are assumed to be the same. Consumption in each period is chosen such that the ratio of marginal utilities equals one plus the effective rate of return; U C1 U C2 = [(1 + r)µ + (1 π)(1 µ)]. (5) It is important to note that (5) indicates a wedge between the rate of return on capital r and the pure rate of time preference in a steady state. In general, we will see that installed capital earns a premium that compensates for the losses associated with savings that fail to match. Savings that fail to match earn no interest and are further eroded by inflation. The second important household condition determines search intensity. Search intensity is chosen such that its marginal cost, in terms of forgone current consumption, equals its marginal benefit; µ U C1 = U C2 Z C2(π + r) [(1 + r)µ + (1 π)(1 µ)]. (6) Although useful in terms of characterizing household reactions, equations (5) and (6) only partially inform the operation of the competitive equilibrium. We must endogenize µ as a function of all of its arguments and track the real rents generated when society chooses a non-zero inflation rate. Without a proper link between π and R it should be clear that from the household s perspective the optimal rate of inflation would be. In contrast, we are interested in a general equilibrium where inflation acts as a tax on unmatched balances, the revenues of which are returned to the representative household, lump sum. Subsequently, we substitute R out of the model noting that in equilibrium the real transfer is given by 8 The matching technology outlined is admittedly a reduced form. It is very convenient because it builds on the extensive external-economies literature [Markusen (1990)]. Essentially, if we only include the scale effect term then the technology reverts to the well studied external-economies technology. Following, Balistreri (2002) the second term is added to transform the external impact into a matching rate. 7

9 R = π(1 µ)s/(1 + µr). 9 Simulating the decentralized general equilibrium follows the template forwarded by Rutherford (1995). The nonlinear system is represented as a relatively compact Mixed Complementarity Problem (MCP). The MCP is a natural format for representing and computing problems that cannot be reduced to a social-planner optimal-allocation problem. The method entails specifying three primary types of conditions: 1) optimality, 2) market-clearance, and 3) income-balance; in three primary types of variables: 1) activity levels, 2) prices, and 3) incomes (where incomes are measured in real numeraire units). In addition we include some equations that simply define certain variables that might be substituted out, but are better retained for clarity. Table 1 shows the overall scope of the two-period MCP in terms of variables and associated conditions. We introduce the relative price terms (P 1, and P 2 ) in Table 1, which are associated with market clearance in the decentralized problem. Only relative prices will be determined, however, so P 1 is assigned the duty of numeraire (P 1 = 1) and the associated market clearance condition drops out of the system (by Walras law). For clarity we maintain P 1 explicitly in the equilibrium conditions that follow, but its value must be determined outside the model for a unique equilibrium. Notice also that the conditions associated with C1 and S are definitional, rather than optimality conditions. This is because these follow recursively from the optimal choices of C2 and Z. The first condition in the general equilibrium is for the optimal transmission of wealth into the second period. If we have market prices (P 1, and P 2 ), which reflect the relative value 9 It may not be obvious why the inflation-tax revenue term includes the denominator (1 + µr). In the equilibrium let the decentralized market price of consumption in period one be, P 1, then the net-of-tax price of period-two consumption is P 1 /[(1 + r)µ + (1 µ)] = P 1 /(1 + µr), where as the gross-of-tax price of period-two consumption is P 1 /[(1 + r)µ + (1 π)(1 µ)]. The denominator, (1 + µr), gives the tax revenue term the proper base. 8

10 Table 1: Scope of the two-period decentralized general equilibrium Equations Optimality conditions: Optimal period-2 consumption Optimal search intensity Utility maximization Associated Variable C2 Z U Market-clearance conditions: Period-1 market clearance P 1 = 1 (numeraire) Period-2 market clearance P 2 Income-balance condition: Representative Agent Income A Auxiliary definitions: Matching function µ Period-1 Consumption C1 Savings S that consumers place on consumption in each period, then optimal transmission is satisfied when the marginal cost of C2 is equal to the marginal benefit; P 1 (1 + r)µ + (1 π)(1 µ) P 2 = 0. (7) Equation (7) is simply a restatement of (5) with the marginal utilities replaced by market prices. Similarly, we can specify the condition for optimal search effort by equating marginal cost to marginal benefit; ηµ P 1 P 2 Z C2(π + r) (1 + r)µ + (1 π)(1 µ) = 0, (8) where η is the constant elasticity of matching with respect to search effort. Equation (8) is a restatement of (6) in terms of the market prices (and with an explicit representation of 9

11 µ/ Z = ηµ/z). The third optimality condition is associated with overall utility maximization given market prices. Utility is maximized when the expenditure function equals the portion of income devoted to consumption. Assuming a Cobb-Douglas utility function with a period-1 value share of θ and rate of time preference ρ we have U[P 1 ] θ [(1 + ρ)p 2 ] (1 θ) = A ZP 1. (9) The right-hand term is income (A) less search effort (ZP 1 ), in numeraire units. Although ultimately removed from the system by Walras Law, the market-clearance condition for the commodity that trades at P 1 is given by: E = C1 + S ( 1 ) π(1 µ) + Z. (10) (1 + µr) Supply of the period-one commodity is given by the exogenous endowment E and demand is the sum of three terms that represent its equilibrium disposition. The first term is consumption in period one. The second term is real saving net of the real claims that are returned, lump sum, to the agent through the inflation tax. The final term is the quantity of the initial endowment allocated to the search activity. The supply of the commodity that trades at P 2 is given by the optimal level of transmission of real wealth into period two (C2), and demand is derived by applying Shephard s lemma to the expenditure function. The market clearance condition associated with P 2 is, thus, ( [P1 ] θ [(1 + ρ)p 2 ] (1 θ) ) C2 = (1 θ)u. (11) P 2 The next equilibrium condition represents income balance with a careful tracking of the implicit lump-sum revenues associated with the inflation tax. Let E represent the exogenous 10

12 quantity endowed in period one, then income balance is given by A = P 1 E + π(1 µ)p 1S. (12) (1 + µr) The right-hand side of (12) equals receipts including the inflation-tax revenues in numeraire units. The value of total receipts must equal expenditures on consumption and search effort (A). Next we define the matching rate as a function of all of its arguments. We assume that aggregate matching takes on a Cobb-Douglas form such that we have constant elasticities to search, scale, and congestion; µ = φ(z) η (µs) β (1 µ) 1 β η, (13) where η and β are positive fractions, and η + β < 1. The parameter φ is simply a scaling parameter. This somewhat arbitrary reduced-form of the matching function is useful in that it is relatively easy to control via the two elasticity parameters. 10 The remaining conditions simply report the levels of period-one consumption and savings as functions of the other variables. Period-one consumption is given by its (compensated) demand function C1 = θu[p 1 ] (θ 1) [(1 + ρ)p 2 ] (1 θ), (14) and the level of saving is given directly by period-2 consumption and the transmission technology S = C2 (1 + r)µ + (1 π)(1 µ). (15) Equations (7) (15) form a complete general equilibrium in relative prices that can be 10 Balistreri (2002) uses a similar form, of constant external elasticities, to represent matching externalities in a general equilibrium formulation of equilibrium unemployment. 11

13 calibrated and computed. There are eight equations in eight unknowns [when we fix P 1 = 1 and remove the redundant equation (10)]. The nonlinear MCP is represented in GAMS syntax and is solved using the PATH algorithm. 11 Relative to a standard exchange equilibrium, we have introduced two key distortions in the economy. First, the level of matching will not be efficient because the scale and congestion effects are external to household decisions. Second, inflation acts as a tax on the portion of savings that fails to match. Depending on the parameterization and policy choice these distortions may reinforce or counteract one another. In general, the optimal policy reaction to the matching externalities is a non-zero rate of inflation. To illustrate the operation of the distortions, we calibrate the model to a benchmark equilibrium and then run two different simulation exercises. First, we isolate the inflation tax effect by fixing the matching rate and the search intensity at an exogenous level. The optimal rate of inflation is shown to be zero. Second, we simulate the full model with endogenous search effort and matching. In this case a small positive rate of inflation is optimal. Calibration of the model involves specifying values for the initial inflation rate, the initial matching rate, the interest rate, the elasticities (η and β), and the benchmark consumption levels. The remaining parameters (ρ, θ, φ, and E), and the initial search intensity are found by inverting the equilibrium conditions at the benchmark equilibrium. The GAMS code, which includes the full details of the calibration and simulation model, is available upon request. Somewhat arbitrarily the following parameterization is applied to a benchmark equilibrium in which C1 = C2 = 1: 11 All simulation code used in this paper is available upon request. Rutherford (1995) illustrates how to formulate a general set of MCP problems using GAMS software. For GAMS documentation see Brook et al. (2005), and for documentation on PATH, the specific nonlinear solver used, see Ferris and Munson (2005). 12

14 π = 0.00 (benchmark) µ = 0.70 (benchmark) r = 0.07 η = 0.15 β = In the first simulation exercise we examine the welfare effects of different inflation rates when matching and search are fixed at their benchmark values. This eliminates equations (8) and (13) from the equilibrium system and the remaining equations are computed as if µ and Z are parameters. Figure 2 shows that, when matching is fixed, welfare is maximized at a zero inflation rate. Inflation behaves like any other tax in an undistorted economy. In this case, households simply accept the fact that some portion of savings will not be matched, and inflation erodes this inferior claim. Negative inflation rates subsidize savings (and period-two consumption), but this is costly because the subsidy must be financed by the household. Quantitatively, when matching is fixed the effects of inflation are very modest, given that inflation only taxes that portion of saving that is not matched and because the implied revenues are distributed back to households. In contrast, when matching and search effort are endogenous we see (in Figure 2) that adding a small amount of inflation can enhance welfare. This is because inflation stimulates private search effort, partially offsetting the distortion inherent in the matching equilibrium. This is a straightforward application of the familiar theory of the second best, where by the addition of a distortion (inflation) in the presence of an existing distortion (the matching externalities) may improve welfare. Of course, with alternative parameterizations, or with the addition of other distortions, the optimal rate of inflation could be very different (even negative). Choosing a calibration that results in a small positive optimal inflation rate is appealing from the perspective of demonstrating, and potentially rationalizing, contemporary monetary policy. To gain more insight into the comparative statics of the two-period general equilibrium, 13

15 Figure 2: Welfare under fixed and endogenous physical-capital matching (two-period model) Fixed Matching Endogenous Matching Welfare Index zero Inflation Rate (%) Table 2 presents the numeric solution for the benchmark case (π = 0) and for the case where we set π = 3%. Notice that increasing the inflation rate has the effect of increasing consumption in both periods. The relative price of consumption in period two falls with the increase in matching efficiency; µ increases from 70% to 80%. This represents a substantial increase in the portion of savings that earns real interest. Income rises because, although the endowment quantity (E) and its price (P 1 ) are fixed, lump-sum revenue is generated by the inflation tax. The two-period general equilibrium examined is useful in building intuition about optimal inflation and the matching problem. This framework cannot, however, inform a study of business-cycles and other real economic features that are presumably of interest to policy makers. To look at these issues we must develop a true intertemporal model. 14

16 Table 2: Numeric Solutions to the Two-period Model π = 0 π = 3% Variable (benchmark) C C Z S U P P A µ Infinite-horizon General Equilibrium 3.1 Overview of the Intertemporal Model The infinite-horizon general equilibrium is setup essentially as an exogenous-growth Ramsey model, with the addition of the matching and inflation distortions. Exogenous growth is achieved by assuming that labor grows at a fixed rate (g) and by assuming that the arguments in the matching technology are per capita measures. 12 The representative household is assumed to maximize welfare over the infinite-horizon, where we assume a fixed rate of time preference and a constant elasticity of intertemporal substitution. The household is constrained by limited resources (the labor endowment in each time period and the endowment of first-period capital) and the technologies for producing output and future capital. Specifically, we assume that output in a given year, Y t, is generated via a constant-returns 12 Defining the arguments in the matching technology as per capita measures ensures that there are no intertemporal scale effects. So, although the market for capital gets larger over time the per capita matching opportunities remain the same in a steady-state equilibrium. Along a steady-state growth equilibrium the matching rate is stationary. 15

17 function of labor inputs, L t, and the capital stock, K t ; Y t = f(l t, K t ), wheref > 0 and f < 0. (16) Output at time t can be used in three possible activities; consumption (C t ), savings (S t ), or search effort (Z t ): Y t = C t + S t + Z t. (17) Capital evolves based on the depreciation rate, δ, and augmentations due to matched savings from the previous year. Capital is also augmented by unmatched savings with an additional lag. Assuming that unmatched savings converts directly into physical capital with an additional lag simplifies the general equilibrium by avoiding the need to track an additional intertemporal asset, which represents unmatched claims. Thus, capital evolves as follows: K t+1 = (1 δ)k t + µ t S t + (1 µ t 1 )S t 1, (18) where we would consider the final two terms as real investment at time t. Simulating the intertemporal equilibrium requires a finite approximation of the infinite horizon model. Furthermore, we are interested in the decentralized competitive equilibrium as opposed to a social planner s maximization. Utilizing the methods developed by Rutherford (2004) and Lau et al. (2002) the equilibrium is formulated as a complementarity problem. The problem is terminated (at t = T ) using a state-variable targeting method [suggested by Lau et al. (2002)] to determine the terminal capital stock. Following Harrison and Rutherford (1999) the infinite-horizon welfare level is approximated using information from the computed within-horizon equilibrium. Infinite-horizon welfare is calculated as the constant-elasticity-of-substitution composite of within-horizon welfare and post-horizon welfare, where post-horizon welfare is approximated by assuming that final period consumption 16

18 (C T ) is on a steady-state path. Changes in welfare will be accurate if the counterfactual in question does not greatly change the value shares attributed to within-horizon versus posthorizon welfare, and to the degree that the equilibrium is sufficiently close to a steady-state trajectory at t = T. 3.2 Algebraic Model Formulation The intertemporal simulation model is specified in optimal activity levels, present-value prices, and present-value wealth. Table 3 presents the overall scope of the complementarity problem including the auxiliary conditions that define the matching rate, the terminal constraint, and the infinite-horizon welfare approximation. The first equilibrium condition ensures intertemporal welfare maximization. Consider that the representative household s objective is to maximize an activity that generates a (constant-intertemporal-elasticity-of-substitution) composite of consumption in each period. 13 Let P W represent the price of the intertemporal composite commodity and P C t represent the price of consumption in year t, then welfare is maximized at the point where the marginal cost of the composite equals the price index; [ ] ( ) 1/(1 σ) λ t (1 + ρ) t 1 σ P C t P W = 0. (19) t The first term is the unit intertemporal expenditure function, where the parameter ρ is the rate of time preference and λ t is the steady-state value share of C t in within-horizon welfare (which depends on ρ and the steady-state growth rate, g). 14 The intertemporal elasticity of 13 The technology for producing the intertemporal composite is household utility as a function of consumption in each period. As we did in the two-period problem, we treat the transformation of consumption commodities into welfare just like any other activity. We take advantage of duality theory, in that the unit expenditure function is analogous to a unit cost function, where the unit cost function is the (minimized) unit cost of an activity that produces utility. 14 See Rutherford (2004) for a conversion of the standard additively-separable constant-elasticity-ofintertemporal-substitution utility function into the linearly-homogeneous form. Both forms generate identical demand systems, but the linearly-homogeneous form is preferable because changes in the utility in- 17

19 Table 3: Scope of the intertemporal decentralized general equilibrium Equations Optimality conditions for activities: Optimal intertemporal (within-horizon) welfare Optimal consumption at time t Optimal output at time t Optimal savings at time t Optimal search intensity at time t Capital-stock evolution at time t (dual) Market-clearance conditions: (all prices are present-value indexes) Market clearance for the intertemporal composite Market clearance for output Market clearance for consumption Market clearance for capital rental Market clearance for physical capital Market clearance for labor Income-balance condition: Present-value income Associated Variable W C t Y t S t Z t K t P W P Y t P C t RK t P K t P L t A Auxiliary conditions: Matching function µ t Terminal condition K T +1 Infinite-horizon welfare W 18

20 substitution is σ. The second condition is associated with the optimal level of consumption in each period. The consumption activity satisfies marginal cost equals marginal benefit. The marginal cost is the market price of output, P Y t, and the marginal benefit is the price of consumption units: P Y t P C t = 0. (20) Alternatively we could eliminate this condition by using P Y t = P C t to substitute P C t out of the equilibrium system. The consumption level could then be computed off line without a loss of information. The system is not overburdened with dimensions, however, (even with a 250 year horizon) so the consumption activity is explicitly maintained in the general equilibrium to simplify reporting. The optimal level of output in a given year will satisfy a zero-profit condition, where the profit function is given by an assumed Cobb-Douglas technology; P L 1 α t RK α t P Y t = 0, (21) where α is capital s value share in production, P L t is the wage, and RK t is the rental price of capital. The next optimality condition is for the savings activity. In a given time period a household will save up to the point that the marginal cost equals the marginal benefit. The marginal cost of saving in t is the present-value price of output in that period. The marginal benefit of saving is the present-value price of the future capital that savings purchase. Letting P K t indicate the price of a unit of capital at time t, the optimality condition for the dex W directly indicate equivalent variation and we can formulate the model in the standard CES dual cost/expenditure functions. The linearly-homogeneous form depends on the steady-state value share parameters. ( Based on the growth and discount rates these are simply computed as λ t = [(1 + g)/(1 + T ρ)] t / s=0 ). [(1 + g)/(1 + ρ)]s 19

21 saving activity is given by P Y t [µ t P K t+1 + (1 µ t )(1 π t )P K t+2 ] = 0. (22) The optimal savings condition reflects the assumption that unmatched savings is converted into physical capital but with an additional lag and that unmatched savings is eroded by the inflation tax. The optimal level of search is also given by balancing its marginal cost with its marginal benefit. The marginal cost of a unit of search effort is the price of output units. To derive the marginal benefits of search, first, consider that equation (22) might logically be interpreted as a zero-profit condition for the savings activity. Equation (22) is, in fact, the opposite of the unit-profit function for S t, and if we multiply it by S t it is the profit function. Let us denote this profit function as Π St. Now using the chain rule, the marginal benefit of search effort equals Π S t µ t µ t Z t. The optimality condition for Z t is given by P Y t ηµ t Z t [P K t+1 (1 π t )P K t+2 ]S t = 0. (23) The final optimality condition in the decentralized equilibrium is a dual representation of the capital evolution technology. Consider an activity that converts the current capital stock into units of future capital. Zero profits in this activity require P K t RK t (1 δ)p K t+1 = 0. (24) In words, the present-value price of a unit of capital at time t equals the present-value gross 20

22 rental return on that unit today plus the present-value of the undepreciated portion of the unit, which carries over to t + 1. The next set of conditions establish market clearance for each commodity. Associated with each of these conditions is a present-value price. These conditions follow directly from the demand and supply functions, and endowments. The first condition is non standard, however, so deserves some explanation. Utility maximization is given by condition (19). The expenditure function [the first term in (19)] embeds intertemporal optimization (the optimal mix of consumption across time is the cost minimizing method of generating the intertemporal composite). Subsequently, the intertemporal composite commodity, which trades at a price P W, is exhausted in final demand. Supply of the composite is the (activity) level of within-horizon welfare. The quantity demanded is given by present-value income, less search expenditures, scaled by the intertemporal price index: (A T t=0 P Y t Z t )/P W. Thus, market clearance in the intertemporal composite commodity (which determines P W ) is given by w 0 W A T t=0 P Y t Z t P W = 0. (25) The scalar w 0 is added to allow for a convenient normalization; in the simulations the benchmark values of W and P W are normalized to one. The remaining market clearance conditions are relatively transparent equations that follow directly from the supply and demand functions implied by the optimality conditions, and endowments. Market clearance for output (which determines P Y t ): Y t C t S t Z t = 0. (26) Market clearance for the consumption good (which determines P C t ): ( ) σ C t c 0 (1 + g) t P W W = 0, (27) (1 + ρ) t P C t 21

23 where c 0 represents the benchmark level of consumption at t = 0. Market clearance for capital in production (which determines RK t ): K t αy t P Y t RK t = 0. (28) Market clearance for labor (which determines P L t ): L t (1 α)y t P Y t P L t = 0. (29) Market clearance for the capital stock (which determines P K t ): K t+1 = (1 δ)k t + µ t S t + (1 µ t 1 )S t 1. (30) The market clearance condition for the capital stock is modified slightly around the first and terminal periods. For example, at t = 0 we have K 1 = (1 δ) K 0 + µ 0 S 0 + (1 µ 0 )s 0 /(1 + g), where K 0, µ 0, and s 0 indicate constants determined in the initial steady-state calibration. Note that the assumption that unmatched savings convert directly into capital with an additional lag has important implications for investment at t = 0. Real investment at t = 0 equals µ 0 S 0 + (1 µ 0 )s 0 /(1 + g). The first term is endogenous but the second term is exogenously determined by the steadystate calibration of µ 0 and s 0, and therefore cannot be adjusted in a counterfactual equilibrium. The income balance condition equates receipts with expenditures. From equation (25), 22

24 present-value expenditures are given by A, so income balance is simply given by equating A to the present-value budget constraint (including the inflation-tax revenue term): T T A = P L t Lt +P K 0 K0 +P K 1 (1 µ 0 )s 0 /(1+g) P K T K T +1 + π t (1 µ t )P K t+2 S t. (31) t=0 t=0 The next equilibrium condition defines the matching technology as a function of the per capita variables. µ t = µ 0 ( ) η ( Z t ) β ( ) 1 η β µ t S t 1 µt. (32) (1 + g) t z 0 µ 0 (1 + g) t s 0 1 µ 0 Using the state-variable targeting constraint suggested by Lau et al. (2002) demand for the terminal capital stock adjusts to satisfy an equalization of the savings and output growth rates at the point of termination; S T /S T 1 = Y T /Y T 1. (33) The final equilibrium condition defines the infinite-horizon welfare index; W = (1 ψ)w σ 1 σ ( + ψ C T c 0 (1 + g) T ) σ 1 σ σ σ 1, (34) where ψ is the value share of post-terminal consumption along the benchmark steady-state. Again, this approximation will be accurate if the within-horizon transitional dynamics do not significantly change the post-terminal value share, and if C T is close to its steady-state trajectory. Conditions (19) through (34) are a complete intertemporal general equilibrium, which can be calibrated and solved using modern computational software. To calibrate the system we assume a benchmark steady-state in which all activities (indexed by t) grow at the 23

25 exogenous labor growth rate and all present-value prices (indexed by t) decay at the rate of time preference. The model maintains the property that it is homogeneous degree zero in real present-value prices, so arriving at a unique equilibrium requires the choice of a numeraire commodity. The system is determined by adding the restriction P W = 1 (so the numeraire is the intertemporal composite commodity), although any other arbitrary designation of a numeraire is equally valid. In the central cases the model is computed for a deterministic 250 year horizon. A relatively long horizon is chosen to prevent end effects while searching for the optimal inflation trajectory. A considerably shorter horizon might be used for simulations in which the policy instrument is exogenously fixed. Even with a 250 year horizon the system is computationally tractable given relatively small changes in the inflation rate Steady-state Calibration The simulations in the remainder of this paper are based on the following illustrative calibration to a steady-state equilibrium. The initial set of assumed parameters are given by ρ = 0.05 Rate of time preference g = 0.02 Growth rate δ = 0.07 Depreciation rate µ 0 = 0.70 Initial matching rate π 0 = 0.02 Initial inflation rate σ = 0.50 Intertemporal elasticity of substitution η = 0.10 Elasticity of search in matching β = 0.75 Elasticity of scale in matching y 0 = 100 Benchmark steady-state output level (at t = 0) i 0 = 36 Benchmark steady-state real investment (at t = 0) 15 With a 250 year horizon the nonlinear system includes over 2700 equations in 2700 unknowns, which are solved simultaneously using the PATH algorithm [Ferris and Munson (2005)]. Given minor changes in the inflation rate, solutions are arrived at within a few seconds on a Pentium computer running Windows. All programs are available from the author. 24

26 The remaining parameters are found by using the steady-state conditions to derive benchmark trajectories for each of the variables. To simplify the process we assume that the output price in the initial time period is one (P Y 0 = 1). Under this normalization the output level equals the value (in numeraire units) of output in the initial period of the benchmark. Given the initial matching rate and the level of first period benchmark investment we can find the level of first period savings s 0 = i 0 µ 0 + (1 µ 0 )/(1 + g), (35) and, given that investment covers growth and depreciation of capital, the initial stock is given by K 0 = i 0 g + δ. (36) Let the net interest rate, in the calibrated benchmark, be r 0. The interest rate does not equal the rate of time preference because it operates only on matched savings. The calibrated net return on physical capital can be calculated as r 0 = (1 + ρ) 2 1. (37) µ 0 (1 + ρ) + (1 µ 0 )(1 π 0 ) This rather complicated expression is found by noting that the price of a unit of first period capital (which in the calibrated benchmark earns r 0 ) is given by pk 0 = (1+r 0 ), and that this price evolves according to the rate of time preference, such that pk t = (1 + r 0 )/(1 + ρ) t along the benchmark. Substituting these prices (pk 1 and pk 2 ), the initial output price (P Y 0 = 1), and the assumed matching and inflation rates into (22) at t = 0 and inverting to find r 0 gives us (37). 25

27 The benchmark gross return on capital, rk 0, also reflects the wedge between r 0 and ρ; rk 0 = (ρ + δ)(1 + r 0) ; (38) 1 + ρ Given the gross return we calculate capital rental payments at t = 0 to be rk 0 K0, and from this we can calculate capital s value share in production; α = rk 0 K 0 y 0. (39) Normalizing the initial wage to one, labor supply is given by L t = (1 α)y 0 (1 + g) t. (40) The initial level of the search activity is given by solving equation (23) for Z t=0 = z 0 in the first period of the benchmark steady state; z 0 = η µ 0 s r ρ ( 1 1 π ρ ). (41) The final calibration parameter is the initial benchmark level of consumption, which is given by c 0 = y 0 s 0 z 0. (42) 4 Optimal Inflation 4.1 Optimal Stationary Inflation and Transitional Dynamics In the first simulation of the intertemporal model we find the relationship between the level of welfare and the stationary rate of inflation. Figure 3 plots the welfare index for scenarios 26

28 Figure 3: Welfare at different stationary inflation rates Infinite-horizon Welfare Welfare Index zero Inflation Rate (%) where the inflation rate is permanently changed (relative to the benchmark steady-state level of inflation). The benchmark calibration is very close to the optimal. Identical to the results from the two-period model a small positive rate of inflation is preferable. Also consistent with the two-period model, when matching and search are fixed exogenously, the optimal rate of inflation is zero (not shown in the figure). 16 The results presented in Figure 3 include transitional adjustments associated with moving away from the benchmark steady-state to the new equilibrium (except, of course, in the case that π t = 2%). The policy shift to a new constant inflation rate is unanticipated and implemented from t = 0 into the future. To illustrate the transitional dynamics we look at the effect on the trajectories of permanently increasing the inflation rate from π t = π 0 = 2% to π t = 4%. Figure 4 shows the impacts on output, consumption and capital over the first 16 The curve from the simulation, in which matching and search are fixed, is not shown because the curve is nearly a horizontal line relative to the curve that is plotted. Numerically, under fixed matching and search, welfare is maximized at at a zero inflation rate. 27

29 100 years, and Figure 5 shows the impacts on savings, investment, and matching over the first 100 years. 17 Figure 4 indicates the temporary stimulus generated by increasing inflation. The inflation tax discourages saving and temporarily stimulates the economy. Relative to the benchmark, consumption is higher in the first 18 years of the horizon, but in the long run settles at a level that is about 0.2% below the benchmark. Output also increases in the short run due to the increase in real investment, shown in Figure 5. It might seem odd that real investment increases in the initial period given that savings falls, but this is due to the initial increase in matching efficiency. Inflation encourages search, which results in an increase in the matching rate (from µ 0 = 70.0% to µ 0 = 72.6%). The impact on savings of the higher inflation tax is a decrease from the calibrated level, s 0 = , to S 0 = in the scenario case. So, on net, matched saving (µ 0 S 0 ) increases by over 3%. Now given that the portion of real investment determined by unmatched savings in t = 1 is a constant, real investment increases by over 2%. The effect of increased matching efficiency is largely neutralized, however, in t = 1 and beyond as the unmatched portion of investment operates on the new lower level of savings. In the presence of transitional dynamics we cannot find the optimal rate of inflation by simply imposing different rates on the calibrated benchmark. The welfare results as presented in Figure 3 are inherently dependent on the original assumed calibration parameters. If we find an optimal stationary π t that is different than π 0 then we have not found the optimal steady-state equilibrium, but rather the optimal stationary policy reaction to the calibrated benchmark. To find the optimal steady-state inflation rate we find the optimal stationary rate of inflation and then recalibrate the benchmark to this new rate. That is, we find the stationary πt. Set this equal to π 0 and recalibrate the benchmark, then iterate until at the solution 17 Beyond 100 years the trajectories of the changes in the variables relative to the original steady-state are virtually flat. 28

30 Figure 4: Dynamic impacts on Output, Consumption, and Capital of a permanent increase in the inflation rate Reference Output (%Change) Consumption (%Change) Capital (%Change) 0 Percent (%) Year Figure 5: Dynamic impacts on Matching, Savings, and Investment of a permanent increase in the inflation rate 5 4 Reference Matching (Change) Savings (%Change) Real Investment (%Change) 3 Percent (%) Year 29

31 π t = π 0. To find π t the system is setup as a Non-Linear Programming problem (NLP) where W is maximized subject to all of the general equilibrium conditions. 18 Using this procedure we find π t = 1.99%, which is very close to the original calibration point. All subsequent analysis is based on a steady-state calibration where π 0 is adjusted to its optimal value, Optimal Non-stationary Inflation Although it is interesting to explore optimal stationary policies, productivity shocks and intertemporal misallocation indicate the possibility that non-stationary policies might be welfare enhancing. In fact, calibrating the model to the optimal steady state implies that the monetary authority has convinced the public that it will not change policy. Having believed the monetary authority the agents are assumed to have made decisions at t < 0 that depend on π t. Given that these decisions cannot be reversed the optimal path is likely to be different. This is the well known dynamic policy inconsistency problem found in many macroeconomic models. To find the optimal intertemporal policy trajectory the problem is set up as an NLP with the restriction that π t = π t+1 lifted. In the NLP, W is maximized subject to the intertemporal general equilibrium. Additional computational problems arise, however, when stationarity is lifted. The algorithm is likely to manipulate the termination conditions in a way that increases W, but that is generally inconsistent with the infinite-horizon problem that we are trying to approximate. We deal with this problem in two ways. First, we impose stationarity on the policy instrument at some point prior to termination, and second we terminate the problem in the distant future (250 years out). Figure 6 explores the sensitivity of the optimal policy trajectory to the point at which 18 More generally the problem is referred to as a Mathematical Program with Equilibrium Constraints (MPEC), but ruling out corner solutions this reverts to the special case of an NLP. GAMS software is utilized in conjunction with the CONOPT solver to find the optimal inflation rate. 30