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# A2.2 Optimal Size of Government Spending

Thus, the private and public sectors can be integrated by substitution of the government budget constraint (3.A5) into the private budget constraint (3.A3) to obtain the effective lifetime budget constraint,

where 0* is defined by

and e (= GE/G) is the government purchases/government spending ratio. The last term arises because a higher level of government spending imposes a negative (positive) wealth effect on the representative individual as long as 0* < (>) 0. If 0* > 0, a higher level of government spending has a desirable wealth effect and the initial level of government spending may be evaluated as too little. However, if 0* < 0, a higher level of government spending has an undesirable wealth effect and the initial level of government spending may be evaluated as too much. In this sense, 0* = 0 is associated with the optimal size of government spending. Substituting 0* = 0 into (3.A6b), we have 0 = e. Thus, the size of government spending may be evaluated as too little (too much) if 0 > e (0 < e).

# A2.3 Optimizing Behavior

The maximization of the representative individual’s objective function (3.A1), subject to the effective intertemporal budget constraint (3.A6a), yields as the first-order necessary condition

together with the intertemporal budget constraint (3.A6a, 3.A6b).

Here, &.Ut+i=d.U(c*t+jj /dC*t+j, and X is a Lagrange multiplier attached to

(3.A6a, 3.A6b) in the consumer’s maximization problem. Consideration of the choice of consumption in the adjacent periods (t, t+1) then leads to the Euler equation,

In order for the representative individual to choose an optimal interior time path for consumption, it must be that he or she cannot improve her or his welfare standing by reducing consumption in one period t and by increasing consumption during another period, say t + 1. The cost of reducing consumption during period t would be a reduction in utility, ДU‘. The benefit of this action would be a gain in utility during period t+1, (1 + г)Ди,+1, which would be subjectively discounted to

(1 + r)/(1 + 8)Ди,+1. A2.4 Indirect Test

We consider an indirect test to evaluate the size of government. For simplicity, government expenditures are initially set at the stationary level, Gt+j = G (j = 0, 1, 2, ...). Let {C+ (j = 0, 1, 2, ...)} and

| C t+j (j = 0, 1, 2, ...)| denote the initially optimal private consumption program and the effective consumption program respectively. From Eq. (3.A8), |c t+j| satisfies

Suppose the government increases the level of government spending in period t+1 only, ДGt+1 > 0. At the beginning of period t, this policy change may be unknown to the representative individual.

(i) Unanticipated Case.

If the representative individual does not anticipate ДGt+1 in period t, how will he or she determine Ct+1 ? Because he or she does not know ДGt+1 until period t + 1, Ct = Ct. In period t + 1, he or she knows ДGt+1. Ct+1 = Ct+1 — еДGt+1 is optimal if and only if 0* = 0 (0 = e) at Gt+1 = G. It follows that ДС*+1 = ДСт + вДGt+1 = 0 at ДСг+1 = —eДGt+1. Hence, the private consumption program

{C<+1 — eДGt+1, Ct+1, Ct+2, Ct+3, ...} satisfies the Euler Eq. (3.A9) at the beginning of period t + 1.

If 0* > 0 (0 > e), ДС{1 > 0 at ДС^ = —eДGt+1. Hence under the private consumption program {Ct+1 — еД^+1, Ct+1, Ct+2, Ct+3, ...}, only C**+1 is

increased compared with the initial equilibrium. Hence, Сг+1 > Ct+1 and C*+1+j (j = 1,2, ..•). Thus, we have

Consequently, in order to restore the Euler equation, the optimal private consumption program in the unanticipated case {Ct+1, Ct+2, Ct^_3,...} satisfies

However, if 0* < 0, we have (ii) Anticipated Case.

We now consider the situation where AGt+1 is known to the representative individual in period t. It is possible for him or her to change Ct. If 0* = 0 (0 = e), as in the unanticipated case, the private consumption program

{Ct+1 — eAGt+1, Ct+1, Ct+2, Ct+3, ...}. is optimal and satisfies the Euler equation at the beginning of period t. It is not necessary to change Ct. If 0* > 0 (0 > e), under the private consumption program {Ct, C'U+1, Ct+2,...} we have

* *

and Ct+1+j > Ct+1+j. Thus, considering Eq. (3.A9), we obtain

In order to restore the Euler equation at the beginning of period t, the optimal private consumption program in the anticipated case {Cta, C^+1; Cf+2,...}. satisfies

However, if 0* < 0 (0 < e), we have

If 0* > 0, the marginal benefit of government expenditure is greater than the marginal cost. G is too little. However, if 0* < 0, the marginal benefit of government expenditure is greater than the marginal cost. G is too much.

Equations (3.A15) and (3.A16) imply that if G is initially too little, unanticipated government expenditures in period t + 1 have a relatively more expansionary effect on private consumption in period t + 1 than anticipated government expenditures, and vice versa. An intuitive explanation is as follows: If G is initially too little, an increase in Gt+1 raises G**+1 and U(C*t+1). If the representative individual anticipates this change in period t, it is desirable for him or her to transfer private consumption from period t + 1 to period t. If he or she does not anticipate government expenditures, it is impossible to transfer from period t + 1 to period t. Thus, unanticipated government action raises private consumption at this time more than anticipated government action.

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