Economics 514
Macroeconomic Analysis
Final Exam
December 18, 2007
1. Consider an economy in which expenditure is a negative function of the real
interest rate:
yt   t  2rt
The central bank sets the real interest rate as an increasing function of the
inflation rate. rt  .1  b   t
The supply of goods (relative to potential output, y ) is an increasing function of
the acceleration in the inflation rate. When the oil price, opt, increases, inflation
will also accelerate at this level of output.
 t   t 1  2  ( yt  y)  opt
a. Solve for the level of inflation and output at time t-1 when inflation is at a
stable level,  t 1   t 2 , and  t 1 =.2, opt-1 = 0. Normalize y = 0.
Assume two possible monetary policies, a strong stance against inflation such that
b = 10 and a weak stance against inflation such that b = .1.
b. Solve for the level of output and inflation under both monetary policies if
there is a demand shock and  t decreases to 0. Solve for the change in output
and inflation (relative to time t-1) under both monetary policies. Under which
monetary policy does inflation change by the most? Under which monetary
policy does output change by the most? Draw a graph explaining your answer.
c. Solve for the change of output and inflation (relative to time t-1), if there is a
supply shock and oil prices increase, such that opt = .2 (assume  t =.2). Under
which monetary policy does inflation change by the most? Under which
monetary policy does output change by the most? Draw a graph explaining
your answer
2. In an economy, workers sign dollar wage contracts to keep their real wages fixed
at an equilibrium level. If prices rise faster than wages, then firms will increase
output above potential output.
yt  y  .5  ( t  gtW )
Demand for goods is governed by the growth of money, t , and velocity,  t :

 t  t  t  yt  y

Velocity growth follows a dynamic process
 t   t 1   t   2 t 2   t 1   t .
Let  =.5, so
 t  .5 t 1   t  .25 t 2  .5 t 1   t
Assume that the shocks to velocity growth are white noise shocks. At any time
before time t, the rational expectation Et-1(  t ) = 0. This means that from the
standpoint of time t, velocity growth is know with certainty. At time t-1, the rational
expectation of velocity growth is Et 1  t   .5 t 1 . If you have to forecast velocity
growth two periods ahead of time the rational expectation is Et 2  t   .25 t 2 .
Normalize y = 0.
The government can set today’s monetary growth in response to yesterday’s velocity
growth so that t  h  t 1 .
a.
Assume that workers with rational expectation sign contracts to set
wages one period in advance to keep expected wage growth constant,
gtW   t E  Et 1 ( t ) Solve for output as a function of  t under two
monetary policies: (i) h = 0 and; (ii) h = -.5.
b.
Assume that workers sign wage contracts two periods in advance.
gtW   t E  Et  2 ( t ) based on their two period ahead forecasts. Solve
for output as a function of  t and  t 1 under two monetary policies: (i)
h = 0 and; (ii) h = -.5. Which monetary policy achieves the shortest
business cycles?
3. The central bank will try to stabilize output near some desired level of output y* =
.5 while simultaneously trying to achieve low inflation. The governments
objective can be expressed as a desire to choose an inflation output combination
2
of the form max   1 2  y  .1  1 2  2  . However, the central bank hires a


y ,
governor who’s long –term contract is fixed in dollar terms. This means, that the
faster is inflation , the less real income the central banker will earn. This changes
the central bankers object as they will now have an extra incentive to have low
inflation.
2
max O    1 2  y  .1  1 2  2  c 


y ,
Workers sign wage contracts, that allow for wage growth to equal their inflation
expectations  t E
a. After workers sign wage contracts, the central bank will choose an inflation
policy to maximize their objective subject to the supply
curve ( t  gtW )  yt  y where y is the level of potential output. To make the


algebra simple, normalize y = 0. Solve for the inflation policy as a function of
wage growth when c = 0 or c =.05.
b. What is the rational expectations equilibrium in terms of output and inflation
when workers have full knowledge of government policy if (i) c = 0 and (ii) c=
.05.
c. How large does c need to be for the rational expectations equilibrium to have zero
inflation?
4. Consider the fiscal policy of a government for a country that lasts two periods, t =
0, 1. The government collects taxes in each period, TAX0 and TAX1; the
government also consumes goods in each period, G0 and G1. The government can
borrow and save at real interest rate, r, but the present value of taxation must
equal the present value of government spending. For simplicity, assume the real
interest rate is r = 0. The government must finance a war today; the war costs G0
= 10 but spends nothing next period G1 = 0. The government can pay for it by
collecting taxes, TAX0, in period 0 and collecting, TAX1, in period 1. The
government considers two plans. Plan A insures a balanced budget TAX0 = 10
and TAX1 = 0. Plan B insures smooth taxes TAX0 = TAX1.
a. If the present value of government spending equals the present value of
taxes collected, what is TAX 0 under plan B. Calculate public savings (e.g.
Public Savings0 ≡ TAX0 –G0) in period 0 under plan B.
A household in this economy has a utility function of U  ln(C0 )  ln(C1 ) . The
income profile of the household will be Y0 = 50 and Y1 = 50 with initial financial
wealth equal 0. Define Household Savings0 ≡ Y0 – TAX0 – C0 and National
Savings0 = Public Savings0 + Household Savings0.
b. Assume that the households borrow or save to maximize utility while
setting the present value of lifetime consumption equal to the present
value of lifetime after-tax income. Calculate public savings, household
savings and national savings in period 0 under both tax plans.
c. Assume that the households face a borrowing constraint so that
households may not borrow. If the household optimum requires
borrowing, but cannot, they consume all after-tax income. Calculate
national savings under each plan. Which plan leads to an outcome closer
to the outcome in which there is no borrowing constraint?
5. Assume that the real interest rate equals the growth rate of output. r = gY = 0. The
output level is fixed at Y = 100. The velocity level is given by V = 2 i . There
are two economies, with different long-run money growth rates, gM = .01 and gM
= .16. Calculate the price level for these two countries when the level of the
money supply is M = 100.