Econ 321- Macroeconomic Problems

6. Labor supply with Taxes (20 Points)
Consider a representative household with the following utility function:
U (C; L) = aC

b2
L ; for a > 0 and b > 0
2

Namely, his utility is increasing in consumption C but decreasing in labor L: His budget constraint (in
real terms) is the following:
w
C= L T
(1)
P
where T are taxes paid to the government. These taxes are levied both on consumption (think of it as
a VAT tax) and on labor income. Namely:
T=
where

W

and

C

W

w
L+
P

CC

(2)

are, respectively, the tax rates on labor income and on consumption.

Answer to the followings.
(a) Taking into account the budget constraint (1) and taxes (2), compute the derivative of U with
respect to L and set it equal to zero (this is the household’ optimality condition)
s
(b) Using the expression found in a., derive an expression for the supply of labor as a function of
the tax rate on labor income W and tax rate on consumption C :

w
P;

(c) Plot the expression you have found in b. (this is the labor supply curve) in a graph with the real
wage on the vertical axis and labor on the horizontal axis. Show what happens when:
1.
2.

W
C

increases
increases

7. Nominal and In‡
ation-Indexed Bonds (15 Points) Some of the bonds issued by the US Treasury
are in‡
ation-indexed. Answer to the followings.
(a) Explain the di¤erence between an in‡
ation-indexed bond and a regular (non-in‡
ation-indexed)
bond.
(b) From a household’ point of view, which of the two is riskier? Explain
s
(c) Explain why one way to measure expected in‡
ation is to look at the di¤erence between the return
on an in‡
ation-indexed bond and the return on a regular bond.
8. Consumption and Savings (15 Points)
Consider the 2-period model. Assume that the household has the following utility:
U (C1 ; C2 ) = ln C1 +

ln C2

(3)

where is the discount rate: To simplify the notation, let Y1 be (real) labor income in period 1, and Y2
be (real) labor income in period 2. The budget constrains in period 1 and period 2 are, respectively:
C1 + S1
C2

= Y1
= Y2 + (1 + i1 ) S1

T2

where T2 are taxes paid in period 2, and equal to:
T2 =

S

(1 + i1 ) S1

In other words, in period 2 the household has to pay a tax on his savings. Think of
assets.
Derive the optimal consumption ratio
what happens when S increases?

C2
C1

S

as a tax on

showing how it depends on the tax rate on savings. Namely

2

9. Seignorage (25 Points)
Suppose that real money demand is given by the following expression:
L (Y; i) =

aY
; for a > 0 and b > 0
bi

where Y is output (or income) and i is the nominal interest rate. Let’ de…ne seignorage - that is, the
s
real revenue to the central bank from money printing - as follows:
S=

L (Y; i)

where > 0 is the (net) rate of money growth (chosen by the central bank). Assume that the real
interest rate r is given and that in‡
ation is equal to :
Answer to the followings.
(a) Write the equation for the nominal interest rate i:
(b) Derive an expression for S as a function of :
(c) Compute the derivative of S with respect to :
(d) Plot S in a graph, with S on the vertical axis and
curve for S ? Brie‡ explain why or why not.