Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

it period (i.e. in period —1 it d 0 equals the level specified

► in period 0 is to keep its ublic observes this inflation nd continues to expect that g to its promise, the policy

.1t takes place.

low is to cheat in period 0. vel attained in period 0 is a under the rule as given in subjected to in period 0 can

a ti ] 2

a2 ± IQ ) 74

(10.29)

• the temptation to cheat if e 10.2 we have plotted this nction are easy to find. If the

T 'ER)

nR

Chapter 10: Macroeconomic Policy, Credibility, and Politics

and T(rR) = 0 if the rule inflation equals the discrete inflation rate irD given in (10.10) (with E = 0 imposed):

The inflation rate under discretion is also the point where temptation is minimized. For higher inflation rates, the T(TR) curve starts to rise again.

But under the second scenario, the policy maker is punished in period 1, because it did not keep its promise in period 0. The public has lost confidence in the policy maker, and expects the discrete solution for period 1. This causes costs in period 1 to be higher than they would have been, since S2D > QR(7TR), and these additional costs must be taken into account in the decision about whether or not to stick to the rule in period 0. From the point of view of the policy maker, the punishment it receives consists of the discounted value of the additional costs it incurs in period 1:

QD 2R(74)

where we have used (10.18) and (10.23). Again, a number of points on the punishment curve can be found easily. First, if the rule inflation 74 = 0, P(0) is equal to:

By comparing (10.33) and (10.30) it is clear that P(0) < T(0). Furthermore, P(nR) = 0 for the discrete inflation rate 7TD:

Finally, for rule inflation rates larger than Ir e, P(74) < 0. The quadratic punishment function P(7R) has been drawn in Figure 10.2.

In period 1 the public expects the policy maker to produce the discretionary inflation rate gm and given this expectation it is also optimal for the policy maker to do so. Hence, in period 1 expected and actual inflation coincide, and confidence in the policy maker is restored (see (10.28)). As a result, the public expects the rule inflation rate to be produced in period 2. And by assumption the policy maker

245

The Foundation of Modern Macroeconomics

does indeed produce the rule inflation because we have investigated the effects of a single act of cheating by the policy maker. No further costs are associated with the cheating that takes place in period 0, and P(7TR) and T(nR) fully summarize the relevant costs and benefits of a single act of cheating in period 0. 3

Clearly, if the temptation of cheating exceeds the punishment, the policy maker will submit to temptation and cheat. The public knows this and does not believe the rule at all in such a case. In technical terms, the rule inflation is then

This immediately explains that the zero inflation rule is not enforceable. The temptation to cheat is simply too large for 7rR = 0 to be enforceable. In terms of Figure 10.2, only rule inflation rates in the interval are enforceable. The optimal enforceable rule inflation rate is of course the lowest possible enforceable inflation rate 7r; (point E). This is because for all rule inflation rates there are no inflation surprises (otherwise a punishment would occur) so that there are only costs associated with inflation and no benefits (through higher than full-employment output). Consequently, the lowest enforceable inflation rate minimizes these costs. Just as in the repeated prisoner's dilemma game analysed inter alia by Axelrod (1984), the enforcement mechanism in the form of loss of reputation ensures that the economy does not get stuck in the worst equilibrium with discretionary monetary policy.

The optimal enforceable rule inflation rate Jr; can be calculated by equating P(nR) and T(7R) given in (10.29) and (10.32), respectively. After some manipulation we obtain:

Hence, the optimal enforceable rule inflation rate is a weighted average of the unenforceable zero-inflation rule and the enforceable but suboptimal discretionary inflation rate 7rD, which equals the term in round brackets (Barro and Gordon, 1983b, p. 113).4

As a final application of this model, consider what happens if the real interest rate r rises. In terms of Figure 10.2, nothing happens to the temptation line T R) but the punishment line rotates in a counter-clockwise fashion around the discretionary point. As a result, the enforceable region shrinks, and the optimal enforceable rule inflation rate rises. This is intuitive. Due to the fact that punishment occurs one period after the offence, higher discounting of the future implies a smaller punishment ceteris paribus. This result is confirmed by the expression in (15).

3 At the beginning of period 2 the policy maker faces exactly the same problem as at the beginning of period 0. Hence, if it pays to cheat in period 0 it also does in period 2. Vice versa, if it does not pay to cheat in period 0 then it also does not pay in period 2. For that reason we only need to check whether cheating pays for one deviation.

4 We assume that the interest rate is not too low (i.e. r > a 2 / fl) so that 0 < < 1 and the optimal enforceable inflation rate is strictly positive. See also Figure 10.2.

10.2 The Voting

In a seminal paper, R central bankers are oftei once again, that the a:

Ulysses' mast). In order of section 1.2 with som we use a median votL: central bank and con& cost function:

where the only differs 4 from person to person. son i were elected to inflation rate and asses In view of (10.10)-(10.1

I

=-/ai [Y* - A -

3

YD = + a2161+

The preferences regar- quency distribution of

Left

Figure 10.3. T aversion p, -3n

Iii

) that 0 < < 1 and the optimal

Figure 10.3. The frequency distribution of the inflation aversion parameter

The Foundation of Modern Macroeconomics

fii are called "left wing" in that they do not worry much about inflation but a great deal about output and employment stabilization. At the other end of the political spectrum, "right-wing" agents with a very high /3i have a strong aversion against inflation and worry very little about output stabilization.

We assume that the agents choose from among themselves the agent who is going to head the central bank. Voting is on a pairwise basis and by majority rule. The agent that is chosen has an inflation aversion parameter #. For this agent there exists no other agent fii such that /3i is preferred by a majority of the people over #. Since there is a single issue (namely the choice of 13) and preferences of the agents are single-peaked in p, the median voter theorem holds (see Mueller, 1989, pp. 65-66). In words this theorem says that the median voter determines the choice of 0. The median voter has an inflation aversion parameter #m that is illustrated in Figure 10.3.

Exactly 50% of the population is more left wing than this voter and 50% is more ap

right wing than the median voter.

8

But the median voter knows exactly what an agent with inflation aversion parameter /3 would choose, since that is given by (10.37)-(10.38) by setting pi = /3. By substituting (10.37)-(10.38) into the median voter's cost function, we obtain:

where we have used EE = 0, .E€2 = a 2 . The median voter minimizes his expected cost level by choice of #. The median voter cannot observe e but knows exactly how agent /3 reacts to supply shocks in general. Hence, the median voter can determine which agent would (if chosen to head the central bank) minimize the expected value of his welfare costs. The first-order condition is given by:

Equation (10.40) implicitly defines the optimal /3 as a function of the parameters of the model and the median voter's inflation aversion parameter Om. It is straightforward to show that the median voter chooses someone more conservative than himself, i.e. # > pm . The proof runs as follows. If we evaluate dS2Al ldp for p = Om , equation (10.40) shows that dS2m10 < 0. Since the second-order condition for

Chap

cost minimization requ. dS- Aildp = 0 for a value ui conduct of monetary pol this manner commits tin,. Furthermore, it is also p with respect to the variani the median voter (#m), an(

I

ap

8a2 — 3/3m _

In words, more uncertain.) 13m) both lead to the appoi Higher output ambition, central banker.

10.3 Dynamic Consi

1

Up to this point the econ. sistency have all been in th

this phi section is to demonstrate as well. We demonstrate th and public goods adaptec two periods, with period 1 representative household hi

C1- 1/Er

-1

— 1 - 1/€1

5 An even easier way to demo,

— /3m),2 _ flm(y* J)2

(a 2 —

from which the result follows in= 6 Indeed, we came across dvna

between wage setting by the ur wage offer of the union is dynes

minimizes his expected cost e E but knows exactly how nedian voter can determine --1;) minimize the expected yen by:

=0

=0. (10.40)

function of the parameters parameter 13M. It is straight- ne more conservative than aluate dS2m 10 for ,8 = 1314, iecond-order condition for

10.3 Dynamic Consistency and Capital Taxation

Up to this point the economic policy applications of the notion of dynamic inconsistency have all been in the area of monetary policy. This is not to say that this is the only area where this phenomenon is encountered.6 Indeed, the purpose of this section is to demonstrate that exactly the same issues are relevant for fiscal policy as well. We demonstrate this with the aid of a simple model of optimal taxation and public goods adapted from Fischer (1980). As in Chapter 6, time is split into two periods, with period 1 representing the present and period 2 the future. The representative household has the following utility function:

from which the result follows immediately.

6 Indeed, we came across dynamic inconsistency in Chapter 8 where we analysed the interaction between wage setting by the union and capital investment by the firm. There we showed that the future wage offer of the union is dynamically inconsistent and thus not credible.

249

The Foundation of Modern Macroeconomics

where Ct is goods consumption in period t (= 1, 2), N2 is labour supply in the future, and G2 is the level of public goods provision in the future. Notice that for simplicity labour supply and public goods provision are zero in the present period. Nothing of substance is affected by these simplifications. At the beginning of period 1, there is an existing capital stock built up in the past, equal to Capital does not depreciate and the constant marginal product of capital is equal to b (see below). The resource constraint in the current period is:

In words, (10.45) says that consumption plus investment in the present period must equal production (and capital income). In the second period, total demand for goods equals C2 + G2, which must equal production F(N2, K2) plus the capital stock (which can be consumed during period 2. Think of capital as "corn"). Assuming a linear production function, the resource constraint in the second period is given by:

where a is the constant marginal product of labour. ?

10.3.1 The first-best optimum

Let us first study the so-called command optimum. Suppose that there is a benevolent social planner who must decide on the optimal allocation by maximizing the utility of the representative household subject to the restrictions (10.45)-(10.46). The Lagrangean for this optimal social plan is:

7 Assuming a linear production function simplifies the exposition substantially. Technically, a linear production function is obtained by imposing an infinite elasticity of substitution between capital and labour, i.e. Do (see Chapter 4). It also means that the demands for labour and capital are infinitely elastic, and that both factors are inessential, in the sense that output can be produced with only one of the two production factors.

which yields the fi r

Equation (10.49) in: = (1 + b)/(1 + p). 131

the optimal values fo

( 1 +

= 1 + ,

1- N2 = (a/0 -'2

G2 = /5 -3.

Finally, by using (10.: of consumption in L.

C2 = (1 + b) 2K1 -

= a+(1+ b1

where we assume th,.. period is non-binding outcome for the rer the state of technolua In practice, the pol goods provision G2 , 1 chosen by the represei centrally planned E: and of itself imply tha economy. Indeed, if tt at its disposal, the fi In the decentralizes rent out to firms at a to these firms, for whi in period 1). The buc

:abour supply in the fir

e.Notice that for simnlic -

-esent period. Noth ginning of period 1, there

Capital does not depre -

b (see below). The rest)._

(10.4

it in the present period total demand for goo,

)1us the capital stock (which "corn"). Assuming a lir - - rid period is given by:

'se that there is a benevolent 'n by maximizing the utilictions (10.45)-(10.46). The

(,2 1-1/E3 )]

4..2`J

1 - 1/63

(10.47)

substantially. Technically, a linear ubstitution between capital and labour and capital are infinitely n be produced with only one of

Chapter 10: Macroeconomic Policy, Credibility, and Politics

which yields the first-order conditions:

Equation (10.49) implies that the marginal utility of income (given by A) is constant: A = (1 + b)/(1 + p). By substituting this value of A into (10.48) and (10.50)-(10.51), the optimal values for C1, N2, and G2 are obtained.

Finally, by using (10.52)-(10.54) in the consolidated resource constraint, the level of consumption in the second period can be calculated:

C2 = (1 + b)2Ki + a - (1 + b)Ci - G2 — a(1 - N2)

= a + (1 + b)2K1 - (1 + P)Ei (1 + - aE2 al - E2 — r3, (10.55)

where we assume that the non-negativity restriction on consumption in the second period is non-binding (i.e. C2 > 0). The command optimum is the best possible outcome for the representative household, given the availability of resources and the state of technology.

In practice, the policy maker may have direct control over the level of public goods provision G2, but it is not likely to have direct control over the variables chosen by the representative household such as and N2 (even in the former centrally planned Eastern bloc countries this proved to be difficult). This does not in and of itself imply that the first-best optimum cannot be attained in a decentralized economy. Indeed, if the government chooses G2 optimally and has lump-sum taxes at its disposal, the first-best plan as given in (10.52)-(10.55) can be decentralized.

In the decentralized economy, households own the capital stock which they rent out to firms at an interest rate r. Households furthermore sell their labour to these firms, for which they receive a real wage W2 (recall that they do not work in period 1). The budget restriction of the representative household in the first

251

The Foundation of Modern Macroeconomics

period is:

C + [K2 - Ki] = (10.56)

where r1 is the interest rate in period 1, so that riKi is the interest income received by the household. This income is spent either on consumption goods or by purchasing additional investment goods. In the second period, the budget restriction is:

C2 = W2N2 + (1 + r2)K2 - Z2, (10.57)

where Z2 is lump-sum taxes and r2 is the real interest rate, both in period 2. The household does not invest in period 2 since the model world ends at the end of that period.

The representative firm produces output by hiring capital and/or labour from the representative household. Profit in period t is equal to:

7rt F(KoNt) - WrNt - rtKr, (10.58)

so that profit-maximizing behaviour implies that rt = FK = b and Wt = FN = a. In period 1 there is no labour supply and only capital is used, and in period 2 both labour and capital are used in production. Hence, for the linear production function we have:

r2 = b, W2= (10.59)

The real interest rate is constant and equal to b and the real wage in the second period is also constant and equal to a. Since both factors of production are paid exactly their respective marginal product, and the production function is constant returns to scale, the representative firm makes no profit.

The government purchases goods in period 2 and pays for these goods by lumpsum taxes levied on the representative household. Hence, the government budget restriction is:

G2 = Z2. (10.60) By substituting (10.59)-(10.60) into (10.56)-(10.57) and consolidating, we obtain:

+C2 + G2 aN2 + (1(10+ b)Ki. .61)

1 + b = 1 + b

The representative household maximizes its utility (10.44) by choice of C1, C2, and

N2, taking G2 and its consolidated budget restriction (10.61) as given. Provided the government sets G2 appropriately (i.e. at the level given in (10.54)) the thus chosen values of C1, C2, and N2 coincide with the first-best optimum values given in (10.52)-(10.53) and (10.55). Hence, the social optimum can be decentralized if the government has access to lump-sum taxes.

252

Ch

10.3.2 The second-IA

In practice the policy n disposal. Instead, it mu income categories. Sum capital income in the se

C1 + [K2 - K1] =

C2 = a(1 - ON2+

where we have already (10.63) we obtain:

which is the counterr utility (10.44) by choki (10.64) as given. The Lai

[Ci + C21 -41 which yields the first-or(

which can be solved for

1+b11 -

=( 1 + P

C2 = a(1 - + ' - (1 +

1 - iv2 = all - tc. a

8 A tax on capital income :-

(10.56)

terest income received by In goods or by purchasing udget restriction is:

(10.57)

te, both in period 2. The rld ends at the end of that

al and/or labour from the

(10.58)

= b and Wt = FN = a. In ied, and in period 2 both near production function

(10.59)

real wage in the second rs of production are paid on function is constant

for these goods by lumpthe government budget

- r)nsolidating, we obtain:-

(10.61)

by choice of C1, C2, and ►o.61) as given. Provided en in (10.54)) the thus hst optimum values given can be decentralized if

Chapter 10: Macroeconomic Policy, Credibility, and Politics

10.3.2 The second-best problem

In practice the policy maker does not have (non-distorting) lump-sum taxes at its disposal. Instead, it must finance its spending by means of taxes on the different income categories. Suppose that tL is the tax on labour income and tK is the tax on capital income in the second period. 8 The household's budget restrictions become:

where we have already imposed the expressions in (10.59). By consolidating (10.62)- (10.63) we obtain:

which is the counterpart to (10.61). The representative household maximizes its utility (10.44) by choice of C1, C2, and N2, taking G2 and its budget restriction

(10.64) as given. The Lagrangean for this problem is:

which yields the first-order conditions:

which can be solved for C1, C2, and N2:

A tax on capital income in the first period is abstracted from as it would amount to a lump-sum tax.

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The Foundation of Modern Macroeconomics

Finally, by substituting these optimal solutions back into the utility function, the is obtained:

where IF is full income of the representative household, which is defined as:

IF a(1 - + [1 + b(1 - + b)Ki. (10.73)

Full income represents the maximum amount of income the household could have in period 2, i.e. by not consuming anything in period 1 and by supplying the maximum amount of labour in period 2.

The government budget restriction in the absence of lump-sum taxes is:

G2 = tKbK2 toN2. (10.74)

Government spending on public goods must be financed by the revenue from the capital and labour taxes. The policy maker maximizes indirect utility of the representative household (given in (10.72)) subject to the government budget restriction (10.74). The Lagrangean for the policy maker's problem is:

where we have substituted the expression for gross saving by the household, K2 =--= (1 + b)K1 — C1, and ,u is the Lagrange multiplier associated with the government budget restriction (10.74). The first-order conditions for the policy maker's problem are the constraint (10.74) and:

In the appendix it is shown that the first-order conditions can be rewritten in the following, more intuitive, form:

254

Ch,

where EL is the uncon uncompensated inters marginal cost of public ft

"costs" to raise a exactly one guilder to ra taxes distort real deci , raise one guilder of publ

Equation (10.79) is the lic goods (see Atkinson benefits of public good,. marginal cost of final. distorting taxes, 7/ = 1, a. consumption. With dist Equations (10.80)-(10. -

(10.80)-(10.81) we obtain

I

Equations (10.82)-(10.83 and labour (named aft.. taxes raise a given amour order to facilitate the it is perfectly inelastic (i.e.

exactly like a (non-distort the MCPF is unity, so t:.. the entire

the savings fun wage elastic. In that case lightly. In the general cas be set at some positive

10.3.3 Dynamic incor

The problem with the c - is dynamically incon,,,,, will tax both labour inc out that once the set :1( policy maker to

model. At the beginn.