Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

i=2

-)rmation of both sides of

and:

must now be calculated. Using (3.29), we

The Foundation of Modern Macroeconomics

output. Using standard (but tedious) techniques, the asymptotic variance of the output path described by (3.49) can be written as:

So, to the extent that fluctuations in output are a good proxy for loss of economic welfare, the policy maker could attempt to minimize the asymptotic variance of output by choosing its reaction coefficients An and A21 appropriately. It turns out that the optimal values for these parameters are equal to:

Intuitively, the policy parameters should be set at values that neutralize the effects of the shocks that occur in period t — 1. In view of (3.49), the coefficients given in (3.51) do exactly that. Of course, not all output fluctuations can be eliminated by the policy maker. This is because the first and the last terms on the right-hand side of (3.49) cannot be affected by the policy maker. The first, because the policy maker has no better information about the innovations in the present period than the public possesses. The last, because ut_2 was known when the oldest contracts were signed in period t — 2, and is hence incorporated in the oldest contract.

In a recent paper, Chadha (1989) has extended Fischer's (1977) analysis to the multi-period overlapping contract setting using the insights of Calvo (1982) that are discussed in detail below in Chapter 13. In his model, he is able to analyse contracts of any particular duration (not just one-period and two-period contracts as in Fischer's model). He is furthermore able to express the asymptotic variance in output as a function of the contract length. This diagram is given in Figure 3.7. The

Figure 3.7. The optimal contract length

a

tic ch. :on is V 7%* •

.,Ch es;,,rn p_ 492). Hence, intuit re are a

Lag discussion. For el

7e r- ,iv be n

.-:y policy may n. anticipated monetr,

. _ ....y n, ,t so Ltry growth rate der— the natural It

aitermez4.0

mptotic vgilance of outpu

ti c (mild) tit...,

the path for ow

Yt = Ayt _1 +

yt is outp and ft is a white no

w, J a

any realizations of o and tI st(x

t41,..4.1L3te the

as follows. First, 1%,..

—=

It

asymptotic variance of the

proxy for loss of economic the asymptotic variance of I appropriately. It turns out o:

(3.51)

s that neutralize the effects ►.-19), the coefficients given ations can be eliminated ist terms on the right-hand e first, because the policy in the present period than when the oldest contracts

In the oldest contract. her's (1977) analysis to the

_ is of Calvo (1982) that odel, he is able to analyse d and two-period contracts the asymptotic variance in n is given in Figure 3.7. The

to -- id length

Chapter 3: Rational Expectations and Economic Policy

conclusion is very surprising indeed: there is an optimal contract length of n* > 0, which Chadha estimates to be around 3.73 quarters for the US economy (1989, p. 492). Hence, intuitively, contracts act as "shock absorbers" of the economy.

There are a number of other reasons why PIP fails-see Buiter (1980) for an interesting discussion. For example, private agents may not have rational expectations, or there may be nominal price stickiness. Furthermore, even though anticipated monetary policy may not be able to cause deviations of output from its natural level, anticipated monetary policy may affect the natural rate itself. A theoretic (albeit empirically not so relevant) example is the Mundell-Tobin effect: a higher monetary growth rate depresses the real interest rate, and this boosts capital accumulation and the natural level of output.

Intermezzo

Asymptotic variance Rational expectations models often use the asymptotic variance of output as a welfare measure. Intuitively, the asymptotic variance measures the degree of fluctuations over time in output. An economy with violent (mild) fluctuations has a high (low) asymptotic variance. Suppose that the path for output is described by the following equation:

where yt is output, xt is some (vector of) deterministic exogenous variable(s), and Et is a white noise stochastic error term with mean zero and variance aE2 . How would a Martian judge the degree of fluctuations in output, not knowing any realizations of output and the error term, but in full knowledge of equation

(a) and the stochastic process of the error terms. The answer is that he would calculate the asymptotic variance:

where the notation Et_,, formalizes the idea of no information about the actual realizations mentioned above. It is as if the Martian makes his calculations at the beginning of time.

The asymptotic variance of output implied by the process in (a) is calculated as follows. First, we write Et-_yt = AEt-coYt-i xt and work out the square:

77

The Foundation of Modern Macroeconomics

The second term on the right-hand side is the variance of the error term (0 -(2 ), and the third term is zero because the error term is independent of lagged output. The term on the left-hand side is the asymptotic variance of y t , and the first term on the right-hand side is A2 times the asymptotic variance of yt_i. Because the process in (a) is stationary (IX! < 1), these two asymptotic variances are identical. Using all this information, the final expression for the asymptotic variance becomes:

Intuitively, the asymptotic variance of output is a multiple of the variance of the error term due to the persistence effect via lagged output. If A is close to unity, there is a lot of persistence and the variance multiplier is very large.

3.4 Punchlines

To most economists, one of the unsatisfactory aspects of the adaptive expectations hypothesis (AEH) is that it implies that agents make systematic mistakes along the entire adjustment path from the initial to the ultimate equilibrium. In the early 1960s, John Muth argued that such an outcome is difficult to square with the predominant notion adopted throughout economics, namely that agents use scarce resources (like information) wisely. He formulated the rational expectations hypothesis (REH) which, in essence, requires the subjective expectation of households regarding a particular variable to be equal to the objective expectation for that variable conditional upon the information set available to the agent.

Muth's idea was introduced into the macroeconomic literature in the early 1970s by a number of prominent new classical economists. They argued that under the REH, monetary policy is ineffective (at influencing aggregate output and employment) because agents cannot be systematically fooled into supplying too much or too little labour. This is the so-called policy irrelevance proposition (PIP) which caused a big stir in the ranks of professional macroeconomists in the mid-1970s. Another implication of the REH is that, according to the Lucas critique, the then predominant macroeconometric models are useless for the task of evaluating the effects of different macroeconomic policies.

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Chapter 3: Rational Expectations and Economic Policy

As was quickly pointed out by proponents of the new Keynesian school, the REH does not necessarily imply the validity of the PIP. Stanley Fischer pointed out that if nominal wage contracts are set for more than one period in advance (and are not indexed) then even under rational expectations monetary policy can (and indeed should) be used to stabilize the economy. Hence, the validity of PIP hinges not so much of the REH but rather on the type of model that is used. If REH is introduced in a classical model then the implications are classical whereas a Keynesian model with REH yields Keynesian implications.

It is almost universally agreed that the PIP cannot be taken seriously, except perhaps as an extreme position taken to promote a discussion. Furthermore, due to the fact that Fischer and others demonstrated that the REH does not necessarily imply PIP, acceptance of the REH as a modelling device is also almost universal. The Lucas critique is valid, but its empirical short-run relevance is seriously doubted by both theoretical econometricians (Favero and Hendry, 1992) and applied policy modellers. A reason for this luke-warm reception may be the absence of a credible theory of how agents learn new policy rules.

Further Reading

The classic articles setting out rational expectations are Lucas (1972, 1973), Sargent (1973), Sargent and Wallace (1975, 1976), and Barro (1976). Papers stressing the stickiness of wages or prices include Fischer (1977), Phelps and Taylor (1977), Barro (1977), Gray (1976, 1978), and Taylor (1979, 1980). For good surveys of this literature, see McCallum (1980), Maddock and Carter (1982), and Attfield, Demery, and Duck (1985). Several key articles on the rational expectations approach are collected in Lucas and Sargent (1981), Miller (1994), and Hoover (1992). The interested reader should also consult the collections of essays by Lucas (1981) and Sargent (1993). See Frydman and Phelps (1983) for a collection of essays on learning under rational expectations.

As was acknowledged by Lucas himself, an early statement of the Lucas critique is found in Marschak (1953). For an early application of the rational expectations hypothesis to finance, see Samuelson (1965). McCallum (1983b) presents a model of the liquidity trap and finds the rational expectations solution. The pre-REH literature on optimal stabilization policy is well surveyed by Turnovsky (1977, chs. 13-14). See also the classic analysis by Poole (1970) on the optimal choice of policy instruments within the stochastic IS-LM model. For an early analysis of economic policy under rational expectations, see Fischer (1980b).

79

Anticipation Effects and

Economic Policy

The purpose of this chapter is to discuss the following issues:

1.To complete our discussion of the dynamic theory of investment by firms that was commenced in Chapter 2,

2.Use the investment theory to determine how the government can use tax incentives (such as an investment subsidy) to stimulate capital accumulation. This is an example of fiscal policy where the government changes a relative price in order to prompt a substitution response,

3.Embed the investment theory in an IS-LM framework. How do anticipation effects influence the outcome of traditional budgetary policies?

4.1Dynamic Investment Theory

In Chapter 2 we sketched a theory of investment by firms that is based on forwardlooking behaviour and adjustment costs of investment. For reasons of intuitive clarity, the model was developed in discrete time. It turns out, however, that working in continuous time is much more convenient from a mathematical point of view. The first task that must be performed therefore is to redevelop and generalize the model in continuous time.

4.1.1 The basic model

Assume that the real profit of the representative firm is given by what is left of revenue after the production factor labour and investment outlays have been paid:

*where z(t) is real proui :ion. 41 t ) is the 7

11111ti.. P-

C 'St function, with a __e same a ,

mum the simpiiticau,

A 1/4:) =1(t)—an,.. •

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2

that is based on forwardFor reasons of intuitive out, however, that work- a mathematical point of develop and generalize

given by what is left of t outlays have been paid:

p

(4.1)

Chapter 4: Anticipation Effects and Economic Policy

where n- (t) is real profit in period t, F(., .) is the constant returns to scale production r. . nction, w(t) is the real wage rate W(t)/P(t)), pi (t) is the relative price of investment goods (EE PI (t) /P(t)), Mt) is the investment subsidy, and (I)(.) is the adjustment cost function, with 0/ > 0 and DH > 0. By assuming that the good produced by the firm is the same as the investment good (the so-called single good assumption), we obtain the simplification pi (t) = 1. In some cases it is convenient to assume that the adjustment cost function is quadratic:

(1)(I (t)) = I (t) b [I (O]2 , b > 0. (4.2) The capital accumulation identity is given by:

K(t) = /(t) - BK(t), 8 > 0. (4.3)

The firm must choose a path for its output such that the present value of its profits is maximized. Since real profits are defined in (4.1), the appropriate discount rate is the real rate of interest on alternative financial assets. This real interest rate is denoted by r and is assumed to be constant over time throughout this chapter. Under these assumptions, the net present value of the stream of profits now and in the future is given by:

V(0) f (t)e7r-rt dt

= f [F(N(t), K(t)) - w(t)N(t) - [1 - (0] (1)(1(t))] e- rt dt (4.4)

To the extent that shares of this company are traded in the stock exchange, and share prices are based on fundamentals and not on the speculative whims and fancies of irrational money sharks, its value on the stock market should equal V(0) in real terms, or P(0) V(0) in nominal terms.

The firm maximizes (4.4) under the restriction (4.3). With the aid of the Maximum Principle the solution to this problem can be found quite easily.' The current-value Hamiltonian can be written as:

7-1(t) e-rt [F (N (0, K(t)) - w(t)N(t) - - Mt)] 'S. (' (0)

+ q(t) [I (t) - 6K(t)1]. (4.5)

Formally, q(t) plays the role of the Lagrange multiplier for the capital accumulation restriction. The economic interpretation of q(t) is straightforward. It can be shown that q(0) represents the shadow price of installed capital K(0). In words, q(0) measures by how much the value of the firm would rise (dV(0)) if the initial capital

1 Note that the method sketched here is a generalization of the Lagrange multiplier method used in Chapter 2. An explanation of the Maximum Principle based mainly on pure economic intuition can be found in Dorfman (1969). Other excellent sources are Dixit (1990) and Intriligator (1971). See also the Mathematical Appendix.

81

The Foundation of Modern Macroeconomics

stock were increased slightly (dK(0)), i.e. q(0)::-=. dV (0)/ dK(0) (see the Intermezzo on Tobin's q below).

The firm can freely choose employment and the rate of investment at each instant, so that the following first-order conditions (for t E [0, opo]) should be intuitive:

The interpretation of (4.6) is the usual one: the firm must choose the amount of labour such that the marginal product of labour equals the real wage rate. Note that (4.7) implies a very simple investment function:

where Iq = 1/[(1 - si )(DH ] > 0 and Is (I)/ /[(1 - si )1 11] > 0. In words, higher values for q and s1 both imply a higher rate of investment. Indeed, for the quadratic adjustment cost function (4.2), the investment function has a very simple form:

The parallel with the expression derived in Chapter 2 (i.e. equation (2.36)) should be noted. Note that we have not used the symbol q for nothing: The investment theory developed here is formally known as Tobin's q-theory, after its inventor James Tobin (1969).

The first expression in equation (4.8) allows a very simple interpretation of the optimality condition for investment. It instructs the firm to equate the marginal cost of investment (equal to (1 - si)(I)/) to the shadow price of capital, which is the marginal benefit of investment. In other words, by spending money today on investment you add value to your company. This added value is measured by the shadow price.

Equations (4.6)-(4.7) are in essence static conditions of the form "marginal cost equals marginal benefit". The truly intertemporal part of the problem is solved by choosing an optimal path for the shadow price of capital. The first-order condition for this choice is:

This condition can be written in several ways, two of which are:

q(t) = (r + 8)q(t) - FK(N(t), K(t)),

a&

1 0: + F40..

Of)

.Tuation (4.12) alk n

Illta. shadow price (to ma

pies tne rate of phy

- - be cow- • J

11,—, 1,,e

optimal path for ‘; dL

P is shown in an

- 10r- • al

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panicular firm (see

termezzo

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J r

`.d by in, ud

ki) .1=Z t kA

.77; kt)

relative price of -

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ad,

eLion kir a

0) (see the Intermezzo on

nvestment at each instant, ,c1) should be intuitive:

4.6)

(4.7)

mist choose the amount of 'tie real wage rate. Note that

(4.8)

to I > 0. In words, higher * Indeed, for the quadratic n has a very simple form:

q(t)

— s/(t) (4.9)

I

i.e. equation (2.36)) should nothing: The investment !gory, after its inventor James

interpretation of the n to equate the marginal price of capital, which is spending money today on d value is measured by the

the form "marginal cost 1 the problem is solved by ii. The first-order condition

(4.10)

'lich are:

(4.11)

Chapter 4: Anticipation Effects and Economic Policy

and:

Equation (4.12) allows for a very intuitive interpretation. The shadow return on the possession and use of physical capital is the sum of the shadow capital gain (4(t)) and the marginal product of capital [FK(N(t), K(t))], expressed in terms of the shadow price (to make it a rate of return). This shadow rate of return must equal the market rate of return on other financial assets (that are perfect substitutes for shares) plus the rate of physical deterioration of the capital stock. The depreciation costs must be counted as a cost item because capital evaporates over time, regardless of whether the firm uses the capital for production or not. Hence, in determining the optimal path for q(t) the firm is guided by the implicit arbitrage equation (4.12).

We have developed Tobin's marginal q-theory of investment in this section. It is shown in an intermezzo to this chapter that, provided some more specific assumptions are made about the adjustment cost function, Tobin's average q-theory coincides with his marginal q-theory. Average q for the firm is defined as 4(0) V(0)/K(0). In words, 7/ represents the value that the stock market ascribes to each unit of installed capital of the firm (at replacement cost, see the Intermezzo).

And this is exactly where the great beauty of the theory lies. In principle one can look up the stock market value of a firm from the financial pages in the newspapers, and divide this by the replacement value of its capital stock (slightly more work), and calculate the firm's q. The value of q that is obtained in this manner reflects all information that is (according to the stock market participants) of relevance to the particular firm (see Hayashi (1982) for further remarks).

ntermezzo

Tobin'sq-theory of investment. In this intermezzo we show that Tobin's average and marginal q coincide under certain conditions. The proof is adapted from Hayashi (1982). Suppose that the profit function in equation (4.1) is adjusted by including the existing capital stock in the adjustment cost function:

n- (t) F (N (0, K(t)) — w(t)N(t) — pl (t) — sr (t)] [1 (0, Mt)] ,

where 7(t) is real profit, w(t) is the real wage rate W (t)/P(t)], (t) is the relative price of investment goods Pi(t)/P(t)}, and sj (t) is the investment subsidy. The adjustment cost function is homogeneous of degree one in I (t)

and K(t), with <13./ > 0, (DK < 0, 4 > 0, (I)/K < 0, and (I)KK > 0. Hence, adjustment costs are decreasing in the capital stock. Large firms experience less

disruption for a given level of investment than small firms.

83

The Foundation of Modern Macroeconomics

The firm is assumed to maximize the present value of profits, using the (timevarying) real interest rate r(t) as the discount factor. Equation (4.4) is altered to:

V(0) f {F (N (t), K(t)) w(t)N(t)

- pi"(t)(1 - s1(t))43 [ (t), K(t)i]e-R(t) dt,

where V(0) is the real stockmarket value of the firm, and R(t) is a discounting factor that depends on the entire path of short interest rates up to t:

The counterpart to (4.5) is:

x(t) e-R(t) [F (N (t), K(t)) - w(t)N(t) - pl (t)(1 s M) l) [I (t), K(t)]] -R(t) x(t) (t) - 6K(01

where A.(t) is the Lagrange multiplier. The first-order conditions for this problem are:

where we have already deleted the (non-zero) exponential term e- R(t) from the expressions. These expressions generalize (4.6), (4.7), and (4.10) to the case of a linear homogeneous adjustment cost function and a time-varying rate of interest.

In order to establish the relationship between the Lagrange multiplier (X(0)), the capital stock (K(0)), and the real stockmarket value of the firm (V(0)), we first derive the definition:

The term in square brackets on the right-hand side of (d) can be expanded by substituting the capital accumulation identity, and equation (c). Ignoring time

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