Solution to ECO209

8/10/2019 Solution to ECO209

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Prof. Gustavo IndartDepartment of EconomicsUniversity of Toronto

ECO 209YMACROECONOMIC THEORY

Solution to Problem Set 4-5

Question 1

(a) First, we must obtain the aggregate expenditure function:

AE = C + I + G

= (C+ c YD) + (I b i + f Y) + G

where YD = Y TA + TR = Y t Y + TR = (1 t) Y + TR

= C+ c [(1 t) Y + TR ] + I- b i + f Y + G

= A b i + [c (1 t) + f] Y

where A = C+ cTR+ I+ G .

In equilibrium, Y = AE:

Y = A b i + [c (1 t) + f] Y ! [1 c (1 t) f] Y = A b i.

And solving for i, we obtain the equation for the IS curve:

A 1 c (1 t) fi = Y.

b b

(b) The LM curve is found from the money market equilibrium:

L = M / P ! k Y h i = /P.

And solving for i:

i = (k / h) Y (1 / h) ( /P).

(c) To find equilibrium income (Y*) we must equate the IS and LM curves:

A / b (1 / b) [1 c (1 t) f] Y = (k / h) Y (1 / h) ( /P)

(1 / b) [1 c (1 t) f + bk / h] Y = A / b + (1 / h) ( /P)

A + (b / h) ( /P)Y*= .

1 c (1 t) f + bk / h


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And the equilibrium rate of interest (i*) is:

i* = (k / h) Y* (1 / h) ( /P)

(d) The fiscal policy multiplier is:Y* 1

= .G 1 c (1 t) f + bk / h

And the monetary policy multiplier is:

Y* b / h= .

(M/P) 1 c (1 t) f + bk / h

(e) Given i* = (k / h) Y* (1 / h) ( /P),

i*/G = (k / h) (Y*/G)

i*/(M/P) = (k / h) [Y*/( /P)] (1 / h)

Question 2

(a) First, we must obtain the aggregate expenditure function:

AE = C + I + G

= (60 + 0.8YD) + (200 20 i + 0.2 Y) + 300

= 560 + 0.8 YD 20 i + 0.2 Ywhere YD = Y 0.25 Y + 50 = 50 + 0.75 Y

= 560 + 0.8 (50 + 0.75Y) 20 i + 0.2 Y

= 600 + 0.8 Y 20 i. (1)

In equilibrium, Y = AE:

Y = 600 + 0.8 Y 20 i

Y = 3000 100 i. (2)

And solving for i we obtain the equation for the IS curve:

i = 30 0.01 Y. (3)

(b) The LM curve is found from the money market equilibrium:

L = M/P 0.2 Y 10 i = 300 Y = 1500 + 50 i. (4)

And solving for i we obtain the equation for the LM curve:

i = 0.02 Y 30. (5)

(c) To find the equilibrium Y* we must equate the IS and LM curves:


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30 0.01 Y = 0.02 Y 30 0.03 Y = 60 Y* = 60 / 0.03 = 2000.

And the equilibrium i* is found by substituting Y* into either the IS or LM curve:

from the IS curve: i* = 30 0.01 (2000) = 10

from the LM curve: i* = 0.02 (2000) 30 = 10.

(d) AE is now 700 + 0.8Y 20 i, and in equilibrium:

Y = AE

Y = 700 + 0.8Y 20 i Y = 3500 100 i. (6)

And solving for i, the equation for the IS curve is:

i = 35 0.01 Y. (7)

The IS curve has shifted vertically upward by 5 (see equations (3) and (7)) orhorizontally to the right by 500 (see equations (2) and (6)).

(e) The new LM curve is 200 = 0.2 Y 20 i, that is,

Y = 1000 + 50 i (8)

or i = 0.02 Y 20. (9)

The LM curve has shifted vertically upward by 10 (see equations (5) and (9)) andhorizontally to the left by 500 (see equations (4) and (8)).

Question 3

To find the equilibrium values of I, L, and NX we must first find the equilibrium values ofY and i. And to do this we must find the expressions for the IS and LM curves.

To find the expression for the IS curve we must first find the expression for AE:

AE = C + I + G + NX= C + (300 10 i) + 70 + (150 0.2 Y)= C + 520 10 i 0.2 Y

C = YD S = YD [200 + 0.2 YD] = 200 + 0.8 YDYD = Y TA + TR = Y (0.125 Y 40) + 60 = 100 + 0.875 Y

= 200 + 0.8 (100 + 0.875 Y) = 280 + 0.7 Y= (280 + 0.7 Y) + 520 10 i 0.2 Y= 800 10 i + 0.5 Y

In equilibrium Y = AE:

Y = 800 10 i + 0.5 Y0.5Y = 800 10 i 10 i = 800 0.5 Y


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and thus the expression for the IS curve is:

i = 80 0.05 Y

We must now find the expression for the LM curve:

L = M/P100 + 0.25Y 5i = 400 5i = -300 + 0.25Y

and thus the expression for the LM curve is:

i = -60 + 0.05Y

Equilibrium Y and i are determined by the intersection of the IS and LM curves:

80 0.05Y = -60 + 0.05Y Y* = 1400 and i* = 10

Therefore:

I* = 300 10(10) = 200

L* = 0.25(1400) + 100 5(10) = 400

NX* = 150 0.2(1400) = -130

Question 4

(a) AE= 1 / (1 slope of AE curve) = 2 Slope of AE curve = 0.5

AE = 200 5i + 0.5 Y

Y = AEY = 200 5i + 0.5 Y5i = 200 0.5 Yi = 40 0.1 Y

(b) Slope of LM curve = k/h = 0.24/h = 0.2h = 20

400 = 4Y 20i20i = 400 + 4Yi = 20 + 0.2Y

(c) 40 0.1 Y = 20 + 0.2Y0.3Y = 60Y* = 200

i* = 40 0.1(200) = 20

(d) The slope of the AE curve is equal to c(1 t) m, and thus AE= 1 / [1 c(1 t) + m]. Ifm increases from 0.1 to 0.2, then the slope of the AE curve will decrease from 0.5 to 0.4and the value of the multiplier will decrease from 2 to 1.


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Question 5

(a) Points to the left of the IS curve represent points of excess supply of goods.

False.If we consider a point below the IS curve, we observe that at that level ofthe rate of interest the goods market would be in equilibrium at a higher level ofincome. That is, AE > Y and Y will have to increase to restore equilibrium in thegoods market. Therefore, there is a situation of excess demand for goods.

(b) If the central bank keeps the supply of money constant, then the money supplycurve is vertical, which implies a vertical LM-curve.

False.If the central bank keeps the supply of money constant, then the moneysupply curve is vertical as stated. But the LM curve is not vertical. A vertical LMcurve would indicate that the demand for real balances is totally insensitive to

changes in the interest rate (i.e., h = 0 and the liquidity preference curve isvertical). But as long as h > 0, then the LM curve will have a positive slope. Thefact that the money supply is constant determines the position of the LM curve,i.e., the vertical intercept but not the slope of the LM curve.

(c) "A monetary expansion leaves the budget surplus unaffected."

False.Expansionary monetary policy, that is, an increase in money supply, willlower interest rates (the LM-curve will shift to the right). Lower interest rates willlead to an increase in investment spending and the economy will therefore bestimulated. But a higher level of national income increases the governments tax

revenues and therefore the budget surplus will increase.

(d) "If government spending is increased, the quantity demanded of money willincrease."

False.A change in government spending directly affects the expenditure sectorand therefore the IS-curve. But in an IS-LM framework, the money sector is alsoaffected indirectly. An increase in the level of government spending will shift theIS-curve to the right, leading to an increase in income. But the increase in incomewill lead to an increase in money demand, so the interest rate will have toincrease in order to lower the quantity of money demanded and to bring the

money sector back into equilibrium. Overall, no change in the quantity of moneydemanded can occur, since equilibrium in the money sector requires that moneysupply be equal to money demand, and money supply is assumed to be fixed.

(e) Crowding out is complete when money demand is perfectly interest inelastic.

True.Crowding out refers to the fact that an increase in public spending maylead to a decrease in private spending, thus dampening the output expansion. An


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increase in government spending raises interest rates, which leads to a reductionin investment spending. When money demand is perfectly interest inelastic, theLM-curve is vertical at the level of real output that clears the money market. Anincrease in government spending will stimulate income and encourage people to

hold more money balances. The excess demand for money will cause interestrates to rise to the level at which equilibrium in the money market is restored. Ifmoney demand is perfectly interest inelastic, the rise in interest rates will notlower the quantity of money demanded. Instead, income will have to go back toits original level before the money market is back in equilibrium. This means thatinterest rates will have to increase until the level of investment spending hasbeen reduced by the same amount as government spending has been increased.Therefore the level of output demanded is unchanged and crowding out iscomplete.

Solution to ECO209

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