the inflation tax in rbc: a revision

8/7/2019 The Inflation Tax in RBC: A revision

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Review Article

by Jorge Rojas

October 19th, 2010

The Inflation Tax in a Real Business Cycle Model. By Thomas F. Cooley and Gary D.Hansen, (The American Economic Review, LXXIX (1989), pg. 733)

1. Introduction

The article written by Cooley and Hansen (C&H) attempts to answer three major ques-tions in modern economics. They attempt to see whether money and its supply ruleimpact on the nature and amplitude of the business cycle. Next, they evaluate the welfarecosts related to alternative constrained periods and a constant growth rate for the money

supply. Finally, they study the way in which anticipated inflation could affect the value ofreal macroeconomic variables in the long-run such as consumption, output, investment,labour and capital stock.

To do this research, the authors build a real business cycle model that takes into accountthe role of money. The way in which money is incorporated into the model is basicallyusing a cash-in-advance constraint (or Clower constraint) that applies only to consump-tion. In other words, this is a condition that establishes that each consumer or firm musthave sufficient cash available before they can purchase goods. In addition, there are twoways to supply money: a constant growth rate and in a erratic but correlated way. Themodel assumes that there is a representative household for the whole economy, and distin-

guishes among cash goods (consumption) and credit goods (leisure). According to the def-initions given by Lucas and Stokey1 (1983), a cash good has to be bought with a formalorder of currency previously stored, while a credit good can be purchased with labourincome contemporaneously accumulated. In addition, there is only one good that is pro-duced and consumed by the agents in the economy.

Overall, the paper is very well-structured and follows the classical tradition in economicmodels. However, there are important elements that are omitted in order to simplify themathematics.

2. The Model

The utility function has the constant-relative-risk-aversion form with CRRA coefficientequal to one. The function only considers consumption and leisure at time t.

U=E0

t=0

t (log ct+A log lt) (1)

1. Lucas, Robert E., Jr. and Stokey, Nancy L., Optimal Fiscal and Monetary Policy in an Economy WithoutCapital,Journal of Monetary Economics, XII (July 1983), 491-514.

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where (0, 1) is the discount factor. Households have an endowment of one unit of timeeach period and supply labour to the firms in the economy. Furthermore, the agents accu-mulate capital and they rent it to the firms. An interesting characteristics of this model isthat the households enter at period t with nominal money balances equal to mt1 fromthe previous period. Notice that consumption is independent of employment, and this

holds in equilibrium according to proof provide by Hansen (1985).There are two money supply rules. The first one follows a constant gross growth rate ofmoney, gt, and is given by equation (2), while the second one follows an autoregressiveprocess in its growth rate depicted by equation (3).

Mt = gt Mt1 (2)

log gt+1 = log gt+ t (3)

where t iid (log(g )(1), 2

and log(g ) is the unconditional mean of the logarithm ofgt, and Mt is the per capita money supply in period t.

The households consumption choice has to satisfy the following constraint:

pt ctmt1+(gt 1)Mt1 (4)

pt is the price level, ct is consumption per household, all at time t.

In the paper written by C&H, this constraint is binding under the argument that thegross growth rate of money always exceeds the discount factor and this is observed inthe actual money supply. However, they recognize that if this assumption is not made,then the solution procedure will be more complicated and there might be corner solutions.Usually, what is looked for in economics are the interior solutions because these are non-

trivial.

Another important assumption is that labour is indivisible. That is, households supply agiven positive number of hours (h) as labour, or they do not supply anything at all. Thisassumption implies that the economy is non-convex, but as Rogerson (1988) did, they con-vexify the set using employment lotteries that the agents trade. The concept of a convexeconomy can be better understood as a consumption set A that is convex. This set A isdefined following the approach in Rogerson, where kt is capital stock per household:

A = {(c , h , k)R3: c 0, hB , 0 k 1}2 (5)

Given that in this economy, labour is a discrete set B with only two choices, namely,supply a certain level or nothing at all. This introduces non-convexity in the above set.

This assumption could have been valid in the 1980, but in my opinion, that reality haschanged over the past decade with the use of Information Technology in many fields whichhas contributed to the advance of part-time jobs.

2. Rogerson, Richard, Indivisible Labor, Lotteries and Equilibrium, Journal of Monetary Economics, XXI (Jan-

uary 1988), 3-16.

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Therefore, in their model the household chooses consumption (ct), investment (xt) and thenominal money holdings (mt). This gives the following budget constraint:

ct+xt+mtptwt ht+ rt kt+

(mt1+(gt 1)Mt1)pt

(6)

where wt is the wage rate, rt is the rental rate of capital and xt is the investment, all attime t.

The law of motion for capital has the form given by equation (7):

kt+1=(1 ) kt+xt, [0, 1] (7)

is the depreciation rate.

The production function is given by:

Yt= exp(zt) Kt Ht

1, (0, 1) (8)

zt is an exogenous shock to technology that follows an AR(1). In addition, they assume

that markets are competitive, which implies thatYt

Kt= rt and

Yt

Ht=wt.

The representative household wants to maximise their value function, which in equilibriumis as follows:

V(zt, gt, mt,Kt, kt) = max {U(c, h) + E[V(zt+1, gt+1, mt+1,Kt+1, kt+1|zt, gt, mt,Kt, kt) (9)

This function is subject to the previous constraints. For more details on this, a suggestedreading is Morimoto (2010) pages 13 to 16, and Lucas (1987) section II. This function sat-isfies the Bellmans equation, a well developed topic in dynamic programming3. Due tothe inefficiency induced by money, the second welfare theorem4 cannot be applied, and thesolution has to be found directly. After solving the equation with somewhat advancedmathematical tools they found the functions using a quadratic approximation in a neigh-bourhood of the steady state for the deterministic problem.

3. Results

The following discussion is related to the results of the paper. First of all, the paper par-tially achieves its goal of answering the questions proposed at the beginning, at least tosome extent according to the model described above.

3. Morimoto, Hiroaki, Stochastic Control and Mathematical Modeling, New York: Cambridge University Press,2010.

4. Hansen, Gary D., Indivisible Labor and the Business Cycle, Journal of Monetary Economics, XVI (November1985), 309-28.

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Cyclical properties

From table 1, taken from C&H, can be seen that when the money supply grows at a con-stant rate, even if it implies a high average inflation rate, the characteristics of the busi-ness cycles are not affected. This behaviour is very close to that observed in models

without money. On the other hand, when money supply is erratic (in a similar way duringthe period of the postwar, WWII) its effect is small but significant on the cyclical proper-ties of the economy. Namely, the effect of the volatility of the money supply impacts onthe standard deviation of consumption and prices, and in their correlation with output. Inother words, consumption and prices become more volatile and less correlated withoutput.

Figure 1. Cooley and Hansen (1989)

Welfare costs of the inflation tax

The paper analyses the welfare costs comparing steady-states for an economy with amoney supply that grows at a constant rate, and in which the cash-in-advance constraintis binding. C&H do not analyse the case with erratic money supply because the Business

Cycle has different properties. The former case is chosen by the authors because the char-acteristics of the business cycles of such an economy are unaffected, and thus they maycompare for different levels of inflation. The way in which welfare is measured is expressedin terms of the ratio of the change in consumption to the steady-state real output C/Y,and the steady-state real consumption C/C. Notice that C is based on a measure ofthe increase in the amount of consumption such that the representative agent will requireto be as well off as under the Pareto optimal allocation. There are two assumptionsrelated to the period of time that the individuals are constrained to hold money, quarterly

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and monthly. From table 2 in the original article may be seen that if the constrained ismonthly, then the welfare costs are importantly diminished.

Figure 2. Cooley and Hansen (1989)

Furthermore, these results are consistent with Fischer (1981) and Lucas (1987), eventhough the assumptions in this case are different. Fischer and Lucas assume that outputis constant and vary the velocity of money, while in this paper the assumption is thatvelocity is constant and output is allowed to change. Since aggregate money is made upby deposits, that may be protected against the inflation tax through competitive interestrates, and there are also currency and reserves that are subject to the effect of the infla-

tion tax. So, the authors claim that the monthly constraint provide a lower bound to thewelfare loss.

Steady-State Implications of Inflation

If the growth rate of the money supply g is greater than the discount factor (theoptimal level for g), then the individuals in this economy will substitute leisure for con-sumption. Recall that leisure is a credit good and consumption is a cash good. Therefore,if there is inflation, the agents will prefer to have less consumption to pay less of this infla-tion tax, while prefering more leisure. As a result of the increase in leisure, there is adecrease in labour, and hence a decrease in output. This leads to a decrease in consump-tion, investment and capital. Notice that the share of output to investment remainsunchanged because investment will provide future consumption that will be subject to thesame inflation tax as the present.

A somewhat surprising and interesting result obtained using this model is that higherinflation rates are associated with lower employment rates. This is completely different tothe result obtained by A.W. Phillips (1958). However, the new result is supported withempirical evidence by the authors themselves and also by Milton Friedman (1977).

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Figure 3. Cooley and Hansen (1989) Figure 4. Milton Friedman (1977)

Figure 5. A.W. Phillips (1958)

4. General critique and Conclusions

The main contribution of the article is that the effects of money are taken into account in

a real business cycle model, through the use of a cash-in-advance constraint and money

supply forms. The results of this model are consistent with previous literature and explain

some striking outcomes such as the positive relationship between inflation rate and unem-

ployment rate, in constrant with the Phillips curve.

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However, there are a number of simplifying assumptions that diminish the power of themodel.

There is only one representative agent5. This does not take into account differentamount of skills or income distribution in the individuals of the economy.

There is production of only one good that is consumed, traded and produced. Inreality, there are many different goods and substituion effects between them thathave an impact on the economy.

Unexpected inflation has no role in the model. This is explicitely recognized by theauthors.

The economy is competitive, but in reality there are many examples in which firmshave an important market power. As a result, the firms do not act as price takers.

The cash-in-advance constraint is always binding in the analysis to simplify themath involved. Nevertheless, this could not be always the case.

Labour is assumed to be indivisible, but there are many cases in which agentsprovide different amounts of labour (part-time jobs).

Consumption is independent of the labour income earns in a period of time. This isclearly not the case in actual economies. According to consumer theory, peopletend to consume more when their income is higher and less when their income islower (budget constraints).

The empirical evidence to support the positive relationship between inflation rateand unemployment rate does not have a detailed explanation in the choice of thecountries. In econometrics, researchers are very careful about data since differentsets of data may lead to different conclusions.

Therefore, a way to improve the model is to relax any of the aforementioned assumptions,and enrich the econometrics deployed.

5. For more details about having different representative agents the following is a good book to read: Salvadori,Neri, Economic Growth and Distribution: On the Nature and Causes of the Wealth of Nations, Chapter 4.

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References

[1] Fischer, Stanley, Towards an Understanding of The Costs of Inflation, Carnegie-Rochester Conference on Public Policy, K. Brunner and A. Meltzer, eds. Autumn, XV(1981), 5-42.

[2] Friedman, Milton, Nobel Lecture: Inflation and Unemployment, Journal of PoliticalEconomy, LXXXV (June 1977), 451-72.

[3] Hansen, Gary D., Indivisible Labor and the Business Cycle, Journal of MonetaryEconomics, XVI (November 1985), 309-28.

[4] Lucas, Robert E., Jr., Models of Business Cycle, New York: Basil Blackwell, 1987.

[5] Lucas, Robert E., Jr. and Stokey, Nancy L., Optimal Fiscal and Monetary Policy inan Economy Without Capital, Journal of Monetary Economics, XII (July 1983), 491-514.

[6] Lucas, Robert E., Jr. and Stokey, Nancy L., Money and Interest in a Cash-in-Advance

Economy, Econometrica, LV (May 1987), 491-514.

[7] Morimoto, Hiroaki, Stochastic Control and Mathematical Modeling, New York: Cam-bridge University Press, 2010.

[8] Phillips, A.W., The Relation between Unemployment and the Rate of Change ofMoney Wage Rates in the United Kingdom, 1861-1957, Economica, XXV (November1958), 283-99.

[9] Rogerson, Richard, Indivisible Labor, Lotteries and Equilibrium, Journal of MonetaryEconomics, XXI (January 1988), 3-16.

[10] Salvadori, Neri, Economic Growth and Distribution: On the Nature and Causes of theWealth of Nations, Chapter 4 (Cheltenham, UK: Edward Elgar Publishing, Inc., 2006).

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the inflation tax in rbc: a revision

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