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Macroeconomic Theory and Analysis

Problem Set 6Suggested Solutions

1 Problem 1: Ricardian Equivalence

1.1 Government Budget Constraint

The goverment budget constraints in the two periods are

g1 = b (1)

g2+ (1 +r)b = sr2 (2)Notice that government is forced to issue debt since there are no revenuesfrom taxation in the first period. The intertemporal budget constraint is:

g1+ g21 +r

= sr2

1 +r (3)

Equation 3 tells us that the present value of government expenditure mustbe equal to the present value of government revenues from taxation.

1.2 Households problem

The households problem is

max{c1,c2,s}

u(c1, c2) = ln c1+ln c2 (4)

subject to:

c1+s = y1 (5)

c2 = (1 + (12)r)s+y2 (6)

Combining equations 5 and 6 we get the intertemporal budget constraint:

c1+

c2

(1 + (12)r) = y1+

y2

(1 + (12)r) (7)

You can easily write down the lagrangean for this problem and get thefollowing Euler equation:

c2= (1 + (12)r)c1 (8)

The Euler equation depends on taxation.

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1.3 Ricardian Equivalence

In this economy there is no room for fiscal policy in the sense that thegovernment cannot choose between debt and taxation to finance its expen-diture. This is due to the fact that at time 1 the government can only usedebt. The Ricardian Equivalence does not hold in this economy since thetaxation is not lump-sum.

2 Problem 2: Ricardian Neutrality

2.1 Intertemporal budget constraints

The households intertemporal budget constraint is:

c1+ c21 +r

=w1(11)(1l1) +w2(12)(1l2)

1 +r (9)

The governments budget constraint is:

g1+ g21 +r

=w11(1l1) +w22(1l2)

1 +r (10)

We can combine the above equations to get:

c1+

c2

1 +r = w1(1l1) +

w2(1l2)

1 +r w11(1l1)

w22(1l2)

1 +r

= w1(1l1) +w2(1l2)

1 +r g1

g21 +r

2.2 Households problem

The households problem is:

max{c1,c2,l1,l2}u(c1, c2, l1, l2) = ln c1+ ln(l1) +ln c2+ ln(l2) (11)

subject to:

c1+ c21 +r

=w1(11)(1l1) +w2(12)(1l2)

1 +r (12)

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The first order conditions are given by:

c1 : 1/c1 = (13)

c2 : /c2 = /(1 +r) (14)

l1: /l1 = w1(11) (15)

l2: /l2 =w2(12)

1 +r (16)

The Euler equation can be obtained by combining equations (13) and (14):

c2 = (1 +r)c1 (17)

So, the Euler equation does not depend on taxes.

The first order conditions for labor depend on taxes. Combining equations(15) and (16) we get:

MRSl2,l1 =l1l2

= w2(12)

w1(11)(1 +r) (18)

Moreover, by combining equations (13) and (15) we get the following:

MRSl1,c1 =w1(11) (19)

2.3 Competiteve Equilibrium

In order to define a CE we must first solve the problem of the firm. Inthis economy, for each time t (t = 1, 2) the firm chooses ht to maximiseAht wtht. In a CE the firm earns zero profits and the conditionA = wtholds for each t= 1, 2.A CE is an allocation {c1, c2, l1, l2, h1, h2, b ,a}, a price system {w1, w2, r}and a government policy {1, 2} such that, given the price system and thegovernment expenditures {g1, g2}:

households problem is solved

firms problem is solved

government budget constraint is balanced

bonds market clears a = b

labor market clears (1lt) =ht and wt = A for each t= 1, 2

consumption goods market clears yt= ct+gt for each t= 1, 2

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2.4 Ricardian Equivalence

In this case Ricardian Equivalence does not hold since government is notusing lump sum taxation. The Euler equation does not depend on taxes,but this is not sufficient for the Ricardian equivalence to hold. You can usea more sophisticated argument to show that the equivalence does not holdby playing with the intertemporal budget constraint.From the households FOC we can recover that at the optimum:

l1 = c1(11)w1

l2 = c2

(12)w2

The intertemporal budget constraint for the household becomes:

c1+ c21 +r

= w1

1

c1(11)w1

+

w2

1 c2(12)w2

1 +r g1

g21 +r

As we can see, the DPV of household resources is not independent fromthe government fiscal p olicy. So even if a change in fiscal policy will notaffect the agent Euler equation it will affect the budget constraint.

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