Download 1 1. Suppose the economy is described by the following equations

Economics 435
Fall 2016
University of Wisconsin-Madison
Menzie D. Chinn
Social Sciences 7418
Problem Set 1 Answers
Due in lecture on Wednesday, September 28. Be sure to put your name on your problem set. Put “boxes”
around your answers to the algebraic questions.
1. Suppose the economy is described by the following equations (so we are looking at a closed
economy):
• Real Sector
Y Z
(1)
Output equals aggregate demand, an equilibrium condition
Z

C

I

G
Definition of aggregate demand
(2)
C

c

c
Y
o
1 D
(3)
Consumption fn, c1 is the marginal propensity to consume
(4)
YD  Y  T  Tr
(5)
T  t1Y
Definition of disposable income
Tax function; t1 is marginal tax rate.
(6)
Tr  TR0
Transfer payments; TR0 is lump sum transfers.
(7)
I  b0  b1Y  b2 i
Investment function
(8)
G  GO0
Government spending on goods and services, exogenous
• Asset Sector
Md
Ms

P
P
(9)
Equilibrium condition
s
(10)
M
M
 0
P
P
Real money supply
d
(11)
M
  0  Y  hi
P
Real money demand
1.1 Solve for the IS curve (Y as a function of i).
Y=Z=C+I+G
Y = c0 + c1YD + b0 + b1Y - b2 i + GO0
Y = a0 + c1 (Y - T + Tr) + b0 + b1Y - b2 i + GO0
functions
Y = a0 + c1 (Y - t1Y + TR0) + b0 + b1Y - b2 i + GO0
substitute in for C, I, G
substitute in for Yd
substitute in for tax, transfers
bring the "Y" terms to left hand side.
Y - b(Y – t1Y-λY) = Y(1-b(1-t-λ)) = c0 + c1 TR0 + b0 + GO0 – b2i
divide both sides by (1- c1 (1-t1)-b1) and let Λ0 ≡ c0 + c1 TR0 + b0 + GO0
 
Λ
let
1
[1  c1 (1  t1 )  b1 ]
1
1.2 Solve for the LM curve (i as a function of Y). What is the channel by which monetary
influences affect the real goods sector in this model?
M0 M s M d


  0  Y  hi
P0
P
P
Solving for the interest rate, i, yields the LM curve:
i
0
 1  M
   0
h  h  P0
 1
   Y
 h
Monetary policy influences (in part) interest rates. Interest rates in turn affect investment, and via the
simple Keynesian multiplier (  ) affects the entire real sector.
1.3 Solve for the equilibrium value of Y.
To solve for the equilibrium value of income, substitute the LM into the IS equation from 1.1:
 


1
1 M0 1 
   0  b2 0 
 Y 
Y  
h h P0 h 
 1  c1 (1  t1 )  b1  
Move the term in parentheses (.) and the (b2/h)Y term to the LHS; factoring out the Y's on the LHS yields:
Dividing both sides by the term in the parentheses yields:
Y0  ˆ[ 0 
b2  0  b2  M 0
  
h
 h  P0

1
]
ˆ 
1  c1 (1  t1 )  b1  b2 / h
 where
1.4 Graph the IS and LM curves on one diagram. Clearly indicate the intercepts and the slopes.
2
Λ0/b2
LM|M0, P0,
Slope
= 1/h
R0
IS|Λ0
µ0/h-(1/h)(M0/P0)
Y
Y0
Label the equilibrium income and interest rate Y0 and i0.
2.1 Assume G increases by ΔGO, and is completely bond financed (assume no portfolio effects
here). Calculate the government spending multiplier.
Take the total differential of your answer to 1.3.
b

b
Y  ˆ A  2 0   2
h
h

 M
 
  P



To find the government spending multiplier, set the changes in real money to zero and the money
constant, and divide both sides by ΔGO:
Y  ˆGO 
Y
1
 ˆ 
GO
1  c1 (1  t1 )  b1  b2 / h
2.2 Suppose instead Tr increases by ΔTR. Calculate the government transfers multiplier.
Take the total differential again:
b

b  M
Y  ˆ   2 0   2 
h
h  P




To find the government transfers multiplier, set the changes in real money to zero and the money
constant, set the change in ΔΛ to equal c1 ΔTR, and divide both sides by ΔTR:
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Y  ˆc1 TR 
c1
Y
 ˆc1 
TR
1  c1 (1  t1 )  b2 / h
2.3 Redraw your answer to 1.4. Then in the same graph, show what happens to the equilibrium
income and interest rate if government spending on goods and services transfers is increased by
ΔTR. Include in your graph the level of income that would be achieved if somehow the interest
rate stayed constant (label this point YA).
̅∆
Λ0/b2
LM|M0, P0,
∆
i1
i0
IS|Λ1
IS|Λ0
µ0/h-(1/h)(M0/P0)
Y0 Y1 YA
Y
2.4 At the new equilibrium, do we know if investment is higher or lower than the level it started
out with? Do we know if it is higher or lower than at YA?
Recall the investment function is given by:
I  b0  b1Y  b2 i
So the change in investment is given by:
I  b0  b1 Y  b2 i
Notice that at the new equilibrium, income is higher (Y1), but the interest rate is higher as well (i1). Hence,
there are offsetting effects on investment, and the end results could be higher or lower, depending on the
magnitudes of the changes in income and interest rates and the parameter values (b1, -b2).
Regarding the second question, if one were at YA, and interest rate i0, then investment would
unambiguously be higher.
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2.5 Suppose the Fed targets the interest rate at i0 (call this itarget). Returning to 2.3, show
graphically what happens if government is increased. What happens to the level of investment?
̅∆
Λ0/b2
LM|M0, P0,
i1
LMEffective
i0
IS|Λ1
IS|Λ0
-(1/h)(M0/P0)
Y0 Y1 YA
Y
When the Fed targets the interest rate, and the target interest rate remains constant, then the LM is now
the Effective LM (flat). An increase in government transfers, increasing autonomous spending, induces an
unambiguous increase in investment, since income is higher, but interest rates are unchanged.
Note that the answer is the same if we are in a liquidity trap.
3. Consider the case where the economy is in a liquidity trap.
3.1 Draw an IS-LM graph corresponding to this case.
5
i
LM| M0, P0, μ0
LM| M1, P0, μ0
IS|Λ0
0
Y0
Y
3.2 Show what happens if business owners become more pessimistic, such that the autonomous
component of investment declines.
i
IS|Λ1
LM| M0, P0, μ0
IS|Λ0
̅∆
0
Y0
Y
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3.3 What happens to tax revenues in 3.2? If the budget were initially in balance, what happens to
the budget deficit?
If tax revenues are as given in the IS-LM handout, then:
(5)
T  t 0  t1Y
Tax function; t 0 is lump sum taxes, t1 is marginal tax rate.
So, as income falls, tax revenues fall (holding lump sum taxes constant):
∆
∆
Where
0
∆
̅∆
The budget balance is given by:
≡
∆
∆
∆
̅∆
0
∆
So the deficit will increase.
3.4 Suppose policymakers wish to improve the budget balance. Would an increase in lump sum
taxes accomplish that aim? What would happen to GDP?
∆
∆
≡
∆
∆
∆
̅∆
̅∆
∆ 1
̅
∆
∆
This looks ambiguous, but in this model, tax revenues will rise.
4. Consider the Aggregate Demand-Aggregate Supply framework. Suppose government
spending is reduced when we are not in a liquidity trap (and do not end up in a liquidity trap),
and the Fed does NOT target the interest rate. You can assume for simplicity expected inflation
is always zero.
4.1 Show what happens in an IS-LM and AD-AS graph in the period the government spending
reduction occurs. See answer to 4.2.
4.2 Show what happens over time to output, the price level, and the interest rate.
I will answer questions 4.1 and 4.2 starting at full employment, to simplify the exposition. The reduction in
government spending shifts in the IS, AD curves (black arrows). Interest rates fall from i1 to i2; income falls
from Y1 to Y2. The price level remains fixed at P1. In period 3, the price level falls, driving up the real
money stock, dropping interest rates to i3, spurring investment, so that income rises to Y3. (gray arrows).
Over time (white arrows), the price level continues to fall, so that M/P continues to rise, driving the interest
rate down to iFinal, and output back to Y*.
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LM | M1, P1
i
LM | M1, P3
LM | M1, PFinal
i1
i2
i3
iFinal
IS | Λ1
IS | Λ2
Y
P
ASLR
P1
PFinal
P3
AD|A1, M1
AD|A2, M1
Y2
Y3
Y
Y1 =Y*
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4.3 Show what the IS-LM and AD-AS graphs look like if initially, the economy is not in a
liquidity trap, but interest rates are very close to zero, and output is below potential GDP/full
employment. Re-answer 4.1, assuming the economy does end up in a liquidity trap.
5.1. Look up on the WSJ or Bloomberg the yield on (as close as possible to) a one year T-bill.
Calculate the price as if the bond were to mature one year from now (specify the date you looked
up the data). Show your calculations.
5.2 Once again, look up the one year and two year yields. Assuming the expectations hypothesis
of the term structure holds, what is the expected interest one year interest rate, one year from
today. Show your calculations.
5.3 Draw the yield curve, for 3 months to 30 years (at 3, 6, 12 months, 2, 5, 10, 30 years)
E435ps1a_f16.doc
30.9.2016 (1_4_f)
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