Department of Economics, University of Toronto
Halina Kalita von dem Hagen
ECO 208Y: MACROECONOMIC THEORY
PROBLEM SET #1 – SOLUTIONS for STUDENTS
PART I:
1.
Y=C+I+G
Y = 300+0.6*(Y-0.25Y+150)+110-5.5 r+50 = 460 + 0.45 Y + 90 - 5.5 r = 550 - 5.5 r + 0.45 Y
0.55 Y = 550 - 5.5 r
YIS = 1000 - 10 r
rIS = 100 - 0.1 Y
Slope of the IS curve = -0.1
(b) Product-market multiplier: 1 / [1 - 0.6 * (1-0.25)] = 1 / 0.55 = 100/55 = 20/11
Change in Y on account of G = 50 * 20/11 = 1000/11
Change in Y on account of T = - 0.6*50 * 20/11 = -30 * 20/11 = -600/11
Net change in Y=1000/11 - 600/11 = 400/11: IS shifts by this much horizontally, to the right.
OR: Change in Y = change in G * multiplier + change in C * multiplier =
= multiplier * ( G + C) = multiplier * ( G + MPCYD * YD) =
= multiplier * [ G + MPCYD * (- T)] = multiplier * G (1 - MPCYD) =
= 20/11 * 50 * (1 - 0.6) = 20/11 * 20 = 400/11
(c) 10 + 0.1 Y = 200/2
(d) 1/k=1/0.1 = 10
0.1 Y = 100 - 10 = 90 YLM = 900 Vertical LM, slope: infinity
(e) Fiscal = 0, Monetary = 10.
Notes: LM curve is vertical and Y=900. This implies that shifts in the IS have no impact on Y,
i.e. the system’s fiscal multiplier is zero.
This also implies that if you change money supply and the LM curve shifts horizontally, the
resulting change in the equilibrium Y is equal to the horizontal shift in the LM curve, never mind
the change in interest rates r. This means that the system’s monetary multiplier and the moneymarket multiplier are the same = 1/k = 10 as per (d) above.
(f) Y = 900 r = 10
(g) Y = T - G - TR = 0.25 * 900 - 50 - 150 = 225 - 200 = 25 surplus
2.
Product-market equilibrium condition: Y = C + I + G =
= 200 + 0.75 * (Y - 100) + 400 - 10r + 150 =
= 675 + 0.75Y - 10r
YIS = 2700 - 40r
rIS = 67.5 - (1/40) Y
Slope of the IS: -(1/40)
Product-market multiplier = [1 / (1 - MPC)] = [1 / (1 - 0.75)] = 4
b. Money-market equilibrium condition:
L (Y, r) = MS/P
0.2 Y + 100 - 20r = 1000/2 = 500
YLM = 2000 + 100r
rLM = -20 + 0.01 Y
Slope of the LM = 0.01.
Money-market multiplier = 1/k = (1/0.2) = 5
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c. change in YG = change in G * product-market multiplier = 100 * 4 = 400
change in YT = change in CT * product-market multiplier = -75 * 4 = -300
change in YNet = YG + YT = 100
IS curve shifts horizontally by 100, to the right.
d. YIS = YLM
OR solve from:
2700 - 40r = 2000 + 100r
r=5
Y = 2700 - 40*5 = 2500 (= 2000 + 100*5)
rIS = rLM
67.5 - (1/40) Y = -20 + 0.01 Y
Obtain: Y = 2500, r = 5
e. System's fiscal multiplier:
change in Y / change in G allowing for r different from zero - see IS-LM notes
System's monetary multiplier:
change in Y / [ (MS/P)] allowing for r different from zero - see IS-LM notes
f. P must be treated as a parameter.
LM:
0.2Y + 100 - 20r = 1000/P, solve, obtain:
YLM = 5000/P - 500 + 100r
YIS = YLM, so: 2700 - 40r = 5000/P - 500 + 100r
r = 160/7 - (250/7P)
Substitute the r into either the IS or the LM relation, solve for Y, this is your aggregate demand
relation:
Y (AD) = (2700-40*160/7) + 40 * (250/7P)
3.
(a) Money-market multiplier = (change in Y)/(change in M/P) keeping r constant.
In our model, it equals to 1/k. In our case: 1/k = 1/0.1 = 10
The system's monetary multiplier will be smaller than this number because, unlike the moneymarket multiplier, it takes into account linkages between the money market and the product
market reflected in changing interest rates. As money supply changes, nominal income changes,
and the product-market equilibrium requires a change in investment. The required change in
investment is induced by changes in the interest rate. However, as the interest rate changes, it
weakens pressures on the money demand to adjust to the money supply through the income
channel only. Consequently, an impact of the monetary policy on Y will be lower (i.e., an
absolute change in Y will be smaller) when we consider an adjustment in r compared to the
situation in which r does not change.
The two multipliers would be identical if:
the interest-rate sensitivity of investment were infinity (i.e., IS were horizontal) which is
not the case in this model.
the interest-rate sensitivity of money demand were zero or the income-elasticity of money
demand were infinity or zero - neither is the case in our model.
(b) First, you have to derive the LM curve, treating P as a parameter.
0.1Y - 2r = 100/P
Solve, obtain:
YLM = 1000/P + 20r
Equate YIS = YLM, solve for r = 100/3 - 100/(3P)
Substitute this r back in either YIS or YLM, get the aggregate demand equation:
Y = 2000/3 + 1000/(3P)
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PART II:
1. All points on the IS represent, by definition, equilibrium in the product (goods) market. If the
point is above the LM curve, then, for a given level of income, interest rates are too high (and
hence the money demand is too low) to ensure equilibrium in the money market. Alternatively,
you may say that for a given level of interest rates income is too low (and hence the money
demand is too low) for the monetary equilibrium.
It follows that the goods market is in equilibrium, but money supply exceeds money demand.
Assuming negatively sloped IS curve and positively sloped LM curve (i.e., no "extreme" cases),
the restoration of equilibrium will require a fall in interest rates and a rise in income.
2. Let's consider what weakens the monetary-policy effectiveness:
an increase in money supply, expands nominal incomes, which calls for an increase in investment
to restore the product-market equilibrium. In order to induce the required increase in investment,
interest rates must fall, which reduces pressures on money demand to adjust through the income
channel only. It follows that the more sensitive money demand is to interest rates, ceteris paribus,
the less effective monetary policy will be as changes in interest rates (and not in income) will
ensure to a great extent that equilibrium is restored in the money market. This part of the
statement is false.
Let's consider what weakens the fiscal-policy effectiveness:
An increase in government spending expands income, resulting in excess money demand. In
order to restore the equilibrium in money market, interest rate must increase. This, however,
decreases investment, offsetting thus to some extent the initial impact of government spending on
income.
The more sensitive money demand is to changes in interest rates, the smaller increase in the
interest rate will be required to restore the money-market equilibrium, and hence, ceteris paribus,
the smaller crowding-out effect. This part of the statement is also false.
3. Consider the accounting identity:
Y=C+I+G+X-Q
(Q = imports)
Y - C = I + G + (X - Q)
S + T = I + G + (X - Q)
(S - I) = (G - T) + (X - Q)
If X < Q and G = T, it follows that S < I. The statement is true.
4. The decision means that people may be forced, legally, to retire earlier. Usually, one's income
drops upon the retirement. Hence the decision is going to reduce the expected life-time income
stream of individuals. Assuming that agents are forward-looking and rational, they will take into
account this decline in their life-time (or permanent) income in their consumption decisions.
Further assuming that they want to smooth their consumption over the life-time, subject to the
intertemporal budget constraint, the lower income stream will result in a lower consumption
already TODAY. Assuming that all (or most) economic agents behaves this way (i.e., that the
majority is NOT liquidity-constrained), the aggregate consumption will decline.
5. The Bank "targets" r, i.e. keeps it constant by manipulating money supply.
Shocks to the money demand shift the LM curve. This puts a pressure on the interest rates to
change. However, the Bank prevents r from changing by contracting or expanding money
supply. These changes in MS cause the LM to shift in the direction opposite to its initial shift.
Consequently, the Bank's policy makes the LM return to its initial position. With targeting,
shocks to the money demand will have NO impact on Y. The statement is false.
PARTIII: 1C 2A 3A 4A 5B 6D 7D 8B 9D 10B 11B 12B 13D 14A 15C 16B 17A 18D
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