《宏观经济学 Macroeconomics》课外读物：What is Left of the Multiplier Accelerator?

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THE ECONOMICS OF RECESSIONS What is Left of the Multiplier accelerator? By OLIVIER J. BlANcHarD" One of the few undisputed facts in macro- y=C+l1+G→ economics is that output is hump shaped Y=(a+y)y-1-Yy1-2+G weights of the moving average representa tion of the deviation of quarterly output where G is autonomous expenditures. This from an exponential trend has a hump shape. has the required implication for (a+y)>I The first eight weights of the distribution In this case white noise disturbances in G are given in Table l, column 1. Nearly generate a hump shape for output quivalent, output is well characterized by The large macro-econometric models also the following AR(2) generate a hump-shaped response of output, although not to white noise but to serially Y=134H-1 42Y,+ε correlated disturbances in G. This is shown (-521) in Table 1, columns(2)to(4), for the MPS model. Although interest rates and prices Y: logarithm of real quarterly GNP minus are endogenous, the hump shaped response linear trend; sample period: 47-3 to 78-4. of output comes from the IS dynamics o This implies that, following a movement interest rates and price movements only of output from its equilibrium value this dampen the effect of G. These IS dynamics period,we expect a movement of output in turn are explicitly constructed around the further away from equilibrium for three multiplier accelerator mechanism(see Carol more quarters before it returns to equi- Corrado for the MPS model). librium. It also implies that, given only the If we consider column ( 3), it is char- past history of output, we can predict acterized by substantial anticipated move- cession (i. e,, sequences EY+i>EY+i-I tial movement in period I is in response to EY>E or the reverse; this would the unanticipated shock in G and therefore not be the case if the best representation of unanticipated itself, the movements in period output was a first-order autoregressive pro- 2 and following are anticipated. It is the cess for example existence of such anticipated movements in either consumption or investment which has L. The Multiplier Accelerator recently come under attack. It has been argued that, given interest rates and tax rates The traditional explanation of the hump most movements in consumption and in- shape relies on the dynamics of private vestment are due to new information and spending and the combination of the multi- that there cannot be large anticipated move- plier and accelerator mechanisms ments in consumption or investment. If the In its original form( see Paul Samuelson), argument is correct, the multiplier accelera it is given by tor and, with it, the IS dynamics of large macro-econometric models are misleading C=a1-1;I=y(y1-1-1-2) and we should look elsewhere for an explanation of the hump shape. Harvard University. I The rest of the paper reviews the the oreti al arguments. The next sections present Foundation and the sloan first the case for the prosecution and then for the defense

VOL, 7I NO. 2 THE ECONOMICS OF RECESSIONS 151 TABLE 1-RESPONSE OF REAL GNP tion, excluding th Tho enter and leave the consumption pool each period, is also a Quarters (1) (sub, super) martingale. Even if individual discount rates differ both from the interest 1.001.00 16 1.16 rate and across individuals, the proposition that anticipated movements in income do ticipated movements in consumption remains valid. When the new and disappearing consumers are taken into l.17 account, this proposition and the martingale 7 characterization are only approximations Col (1): Estimated response of GNP to a one-tir connection with the“" q theory” of invest tation derived from an AR(12)estimated on ment. This theory assumes internal costs of 1947-3 to 1978-4)Cols.( 2),(3),(4): Simulated re- adjustment for capital and derives a relation onse of GNP(r) and private spending(A)to a between investment and the ratio of the one-time deviation of I in e, with G-9G-1+e(G: market value of the firm to its replacement overnment expenditures)in the MPS model cost. It has the following structure: Assum- ing, for example, risk-neutral shareholders, a Il. An Epidemic of Martingales constant interest rate r, a depreciation rate 8, and quadratic costs of adjustment, value For consumption, the point was made by maximization implies the following behav Robert Hall. Assuming that an individual ior maximizes expected lifetime utility, he will always act so as to equalize current mar gin utility and discot anted expected future (2)K=0+o(q,-D) marginal utility. Formally, if C is consump ion, 8 the discount rate r the interest rate assumed known and constant for our purpo- (3) g,=5/1+r ses, and his information set at time t =01-6)E(MR+/2,) (1)U(c)=1+6E[U(c+)】] where MRi+i is the marginal revenue from a the firm at time [+1. CThe exact characterization depends on pa Under the further assumption that utility ticular assumptions such as whether capital is quadratic (or else as an approximation), installed is instantaneously productive and lIs gIves so forth. a closely related derivation is given by Andrew Abel. )The characterization is C|2|=0 intuitive: the rate of investment is a linear function of a shadow price this shadow price is the present discounted value of The implication is that even if 8 is differ- expected marginal returns to capital come, as they belong to S, do not lead o, Being a present value, q, satisfies the fol- nt from r, anticipated movements in in wing relation which is implied by (3) to anticipated movements in consumption Furthermore, if 8=r, there are no antic- ipated movements in consumption: con- (4)E 4+11-6/9MR)2 =0 sumption follows a martingale. Otherwise, if 8 is different from r, it follows a sub- o This is a familiar relation for asset prices supermartingale usually referred to as an arbitrage or no As a sum of(sub, super) martingales is a excess return relation. A similar relation with (sub, super)martingale, aggregate consump- dividends instead of MR, holds for stock

152 AEA PAPERS AND PROCEEDINGS MAY981 prices for example. (Indeed, under further sts are, the lowerφ. A value ofφ=05 assumptions, such as a CRTS technology, implies that a rate of net investment of 5 competitive factor and product markets percent annually entails a marginal installa- equal to the price of a share, i. e, the title to tion cost of 100 percent of purchase price one unit of capital as valued in the stock per unit installed. This may reasonably be taken as an upper bound on the convexity Equation (4) implies that X +1=qu+1 of adjustment costs. a value of =.5 im ((1+r)/(1-8) (q, -MR, )is a fair game plies that a rate of net investment of 5 with respect to but not that q, itself percent entails a marginal cost of 10 percent follows a martingale. It has been suggested per unit. This may be taken as a lower however that q, follows approximately a bound. Further assume in both cases that martingale. If this is the case, equation(2) the production function is Cobb douglas implies that the rate of investment follows with a share of capital of 25 percent, the also approximately a martingale depreciation rate is 12 percent annually, the To be sure, the above theories have fairly real interest rate is 3 percent, that firms take restrictive-and mutually inconsistent (risk output as given and that the wage always averse consumers but risk-neutral share- equals the marginal product of labor. These for example)assumptions The in heroic assumptions allow us to derive the tuition behind equations (I)to(4)suggest following equation for annual net invest however that more complex specifications, ment, IM such as better treatments of uncertainty, are unlikely to yield drastically different impli- (=. 5)IN cations. They therefore suggest that given interest rates, consumption and investment movements are likely to be mostly unan- 20|503∑(69)E(X+12)-K,-1 ticipated Ill. Anticipated Movements in Investment (φ=05)IN Intuition suggests that the martingale "ap 290∑(82)E(Y+4|31)-K1-1 proximation"is simply wrong for invest- ment: If costs of adjustment are nearly lin- ear, the firm will adjust its capital stock The first term in brackets can be thought ainly to current demand conditions; if of as the desired capital stock. Higher ad movements in demand are partly antic- justment costs(=. 05)imply that more ipated and partly d we would veight is expect both large anticipated and unan- ture. They also imply a slower adjustment to ticipated movements in investment. If, on the desired stock. If we further assume for the other hand, adjustment costs are very example that output is equal to a constant convex, the firm will change its capital plus white noise e,, we get smoothly. We would then expect both small anticipated and unanticipated movements in (p=.)IN, -.80IN-1+1(E, -E-1) investment In neither case would we expect the ratio of anticipated movements to unan- (中=05)IN=96/N-1+01(e,-E1-1) ticipated movements to be necessarily small, as required by the martingale approxima- Higher adjustment costs lead to higher serial correlation, i.e., smaller anticipated move To see this more clearly, we can solve ments but also to smaller unanticipated equations(2)and(3)for two different val- movements. To summarize, investment does ues of the convexity parameter. The coeffi- not follow a martingale. For plausible val ient d in (2)is directly related to this esofφ(φ=5 for example), the traditional parameter: the more convex adjustment accelerator theory-with a modified defini

VOL. 7I NO. 2 THE ECONOMICS OF RECESSIONS on of the desired capital stock-still holds constant. Aggregate consumption however nd there can be substantial anticipated grows at rate y: disregarding the change in movements in investment he consumption pool is clearly not innocu- What is the empirical evidence on o? ous. If aggregate income has also a stochas E uations such as(2) have recently been tic component, then to the extent that the estimated and their results are puzzling: they expected consumption of those who start ield implausibly low values of usually consuming is different from the consump- round. 05. Such values of imply, as tion of those who stop, there will be some we have seen, very large adjustment costs anticipated change in consumption and very small anticipated or unanticipated To see whether this anticipated change movements in investment; this is hard to can be large, consider a world in which one reconcile with the actual movements of in- agent is born each period and lives for N vestment. There are reasons to believe that periods, in which r=8=0 and in which ag these estimates of are biased downwards: gregate income follows Y,=a+pY-1+e the shadow price q is usually approximated each agent receiving 1/N of aggregate in- s by the ratio of market come. We can derive the behavior of as value to replacement cost which is likely to gregate consumption C, and look at the ratio be a mediocre proxy. The market value it- of the anticipated change in C to its total self is also surprisingly volatile during the change by computing sample period This is a puzzle in itself; see Robert Shiller, 1981. )Both reasons would E(E(C+1|92)-C)2 lead to a downwards bias in if investment does not follow a mar E(C+1-C)2 tingale, where did the martingale ap- For p=l, i.e if income itself follows a mar- proximationargument of the previous fine tingale, then AN=0: aggregate consumption tion go wrong? It went wrong in assuming also follows a martingale. If on the other that the fair game property of x, implies hand, p=0, then AN is given by that the present discounted value q, follows, even approximately, a martingale. Present discounted values do not in general follow martingales: the present discounted value of an AR(I) variable for example follows also an AR(I) with the same coefficient of serial As the sum in the numerator converges but correlation. This was emphasized by Shiller the second sum in the denominator d (1979, Appendix A), but the mistake is still verges, AN tends to zero as n gets large quite frequently made This implies that the martingale approxima- tion is correct for large N. If we assume for IV. Anticipated Movements in Consumption example that agents consume for 50 years e get Aso=7 percent and the martingale The story is different for consumption. approximation is quite good. To reject the Under the life cycle hypothesis and the ad- martingale result, we therefore have to reject ditional assumptions made in Section I, some of the assumptions made in Section I individual consumption indeed follows a An obvious candidate is the implicit as- martingale. Is it true however of aggregate sumption that wealth can be negative. What consumption Can we really disregard the happens if wealth cannot be negative for an effects of the change in the consumption individual, if there are liquidity constraints? pool? Suppose, for example, that aggregate Consider an individual for whom 8=r,so ncome is deterministic and grows at rate y. that in the absence of liquidity constraints, Suppose also that r=8 so that the consump- his consumption follows a martingale. How on of each individual is constant. If we will he plan consumption if he expects to be look at the total consumption of the agents liquidity constrained? He will still never an present in two successive periods, it is ticipate to decrease his consumption: if he

154 AEA PAPERS AND PROCEEDINGS MAY98/ did there would be a path of constant con- proximately constant and investment will sumption involving saving now and dissav- decline as the capital stock adjusts. The ing later which would yield higher utility overall response of output will not be hump and be feasible. He may, however, antic shaped but declining over time. ipate to increase his consumption if he an If disturbances are anticipated, however, ticipates his income to increase: he cannot say two quarters in advance, they will lead borrow against future income. Liquidity to an initial jump in consumption, an antic constraints therefore imply that consump- ipatory increase in investment. After they tion may not follow a martingale but do not have occurred, investment will slowly de- imply that anticipated decreases in income cline. Overall, anticipated white noise dis- lead to anticipated movements in consump- urbances car an generate a hump shape of output. Whether the hump shape of output Recent empirical evidence suggests the s actually explained by such anticipated existence of liquidity constraints. Marjorie disturbances is a different question that this Flavin finds that the effect of current in- paper does not address, much less answer come on consumption is too strong to be consistent with the life cycle hypothesis. REFERENCES Robert Hall and Frederic Mishkin, using micro data, conclude that liquidity con- Abel, "Investment and the value of Cap- saints may affect approximately 20 percent ital, "New York: Garland Publishing, Inc of consumers. This suggests that, although liquidity constraints exist, they may not be C. Corrado, "The Steady State and Stability so prevalent so as to lead to large antic Properties of the MPS Model, "unpub- ipated increases in consumption in response lished doctoral thesis univ. pc to increases in income 1976. M. Flavin, "The Adjustment of Consumption V. Conclusion oout Future Income, "J. Polit. Econ., forthcoming. Boldly stated the conclusions of this R. Hall, "Stochastic Implications of the life paper are that anticipated movements in Cycle Permanent Income Hypothesis output-especially anticipated decreases Theory and Evidence, "J. Polit. Econ will not lead to changes in consumption but Dec.1978,86,971-87 may lead to large changes in investment. In and F. Mishkin, "The Sensitivity of his sense, the multiplier is dead and the Consumption to Transitory Income accelerator alive Estimates from Panel Data on House- Given these conclusions, can we for ex holds, "mimeo,, July 1980 ample, generate a hump-shaped output only P. Samuelson,"Interactions Between the from the dynamics of private spending in Multiplier Analysis and the Principle of response to disturbances in autonomous Acceleration, " Rev. Econ Statist, May pending? In response to a disturbance, 939,21,75-78 consumption will react and adjust to a new R. Shiller, "Do Stock Prices Move Too Much constant level; the anticipated movements to be Justified by Subsequent Changes in in private spending must therefore come Dividends?, "Amer. Econ. Rev. forthcom- Ing. If disturbances are unanticipated, their "The Volatility of Long Term Inter- occurrence will lead to an increase in con est Rates and Expectations Models of the Investment Term Structure,”’J. Polit.Econ,Dec. Over time, consumption will remain ap- 1979,87,1190-229

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