university of toronto scarborough
UNIVERSITY OF TORONTO SCARBOROUGH DEPARTMENT OF MANAGEMENT MGEC06 – TOPICS IN MACROECONOMCS THEORY
Sample Test-1 (Solutions) Ata Mazaheri Instructions:
This is a closed book test.
You have 2 Hours. Good Luck! Last Name: First Name:
ID
FOR MARKERS ONLY: Q1
Q2
Q3
Q4
Total
40
20
20
20
100
Marks Earned Maximum Marks Possible
Page 1 of 13
Answer all following questions: Question-1 [40 Points] Answer the following questions: a) (5 Points) Use a DAD-DAS curve diagram to illustrate the short-run and long-run effect of an unexpected positive demand shock that lasts three years on output, price level, unemployment rate, and inflation rate. Solution:
π C DAS B A D
Y Y
DAD
A positive demand shock that lasts for three periods will shift DAD to the right. DAD stays there as log as the shock last – three periods- and then returns to its original position. In the first three periods, while the positive demand shock persists the DAS adjusts up reflecting the inflationary gap. But when the demand shock dissipates the DAS starts adjusting downwards reflecting the deflationary gap. Eventually the economy returns to its original long run equilibrium.
Page 2 of 13
b) [12 Points)] Assume that an economy's production function is Y = 2000L1/2. Suppose the labor supply curve is given by Ls = 15,625(W/P). Assume that the price level is expected to be 8. i) (5 Points) What are the equilibrium real and nominal wages for the expected price level of 8? If the actual price level turns out to be 16, what will be the actual real wage? ii) (7 Points) How much labour would be demanded if the actual price level turns out to be 16? How much output would be produced? Does aggregate supply increase as the price level increases? Briefly Explain. Solution: i) Equilibrium real wage is (W/P for which labor demand is equal to labor supply, Ld = Ls, that is: (W/P) = 1000L-1/2 => Ld = 1000,000(P/W)2 Ld = Ls = 1000,000(P/W)2 = 15,625(W/P). => W/P = 4 => Nominal wage = 32. Note: This will be the negotiated nominal wage between the firm and workers and will remain fixed in the short-run. Actual real wage = 32/16 = 2. ii) Price Level 16, => Real wage = 2 => labor demand = 1000,000 (P/W)2 = 250,000. => Output produced = 2000L1/2 = 1000,000: Price level = 8, => real wage = 4, demand for labour = 1000,000(P/W)2 = 62,500 => Output Produced = 2000 L1/2 = 500,000. Thus output supply increases as price level increases.
1000,000
500,000 4 2
62500
250000
16 8
Page 3 of 13
c) [13 Points] Assume that an economy is initially at the natural rate of unemployment. i) (5 Points) Use an AS-AD curve diagram to illustrate the short-run and long-run effect of an unexpected negative supply shock on output, price level, unemployment rate, and inflation rate. ii) (8 Points) Use the expectation-augmented Phillips curve [ π = π e − β (u − u n ) + ν ] where ν is the negative supply shock. Show the impact of the contractionary monetary policy on the prices and unemployment (a) assuming prices/wages are sticky in the short term such that π e is fixed (b) assuming rational expectation. Solution: i) In the AD-AS framework, the negative supply shock shifts the AS curve leftward. At new equilibrium, the price level rises and the output falls. So when there is a negative supply shock, the Phillips Curve shifts up and the inflation rate rises (1). The fall in output results in an increase in unemployment rate. So there is a downward move along the Phillips Curve as unemployment rate increases (2). Therefore the effects of the negative supply shock are the increases in π and u:
Page 4 of 13
ii) A decrease in money growth rate (contractionary monetary policy lowers the inflation rate of the economy. If prices are sticky, π e remains unchanged, since the Phillips Curve equation is π = π e − β (u − u n ) + ν , the Phillips curve will not shift. There will be a move along the Phillips curve as shown below.
A lower inlfation and higher unemployment. If the government adopts an expansionary monetary policy. There will be a move along the curve to the left which will result in higher inflation and full employment. According to Lucas rational extectation model, Public adjust their inflation expectation according to the reduction (increase) in money growth as long as the policy is credible. Public anticipate that the decrease (increase) in money growth rate will lower (increase) the inflation rate. So π e decreases (increases) and the Phillips Curve shifts downward (upward). The unemployment rate is not changed.
Page 5 of 13
d) [10 Points] Suppose Canadian economy follows the Phillips curve:
π = π −1 − (u − u n ) and that the natural rate of unemployment is given by the average of the past two years’ unemployment: u n = 0.5(u −1 + u −2 ) . Suppose the natural rate of unemployment (un) has been historically at 8%. The Bank of Canada follows a policy to reduce inflation by 4% permanantly. i) (6 Points) Show the impact of this policy on the natural rate of unemployment over time. Solution:
u0 = 8%, u1 = 12% 0.12 + 0.8 = 0.10 2 u + u1 0.10 + 0.12 u3 = 2 = = 0.11 2 2 u + u 2 0.11 + 0.10 u4 = 3 = = 0.105 2 2 u + u3 0.105 + 0.11 u5 = 4 = = 0.1025 2 2 u2 =
ii) (4 Points) What is the sacrifice ratio for this disinflation? Briefly explain your answer. Solution: Because unemployment is always higher than it started, output is always lower than it would have been. => the sacrifice ratio is infinite. This is called hysteresis: we find that there is a long-run tradeoff between inflation and unemployment: to reduce inflation, unemployment must rise permanently.
Page 6 of 13
Problem-2: [20 Points] Consider an economy in its long-run equilibrium with an inflation
rate, π, of 8% per year and a natural unemployment rate, un, of 4%. The expectationsaugmented Phillips curve is: π = πe – 2(u – un),
Assume that Okun’s law holds so that a 1% increases in the unemployment rate maintained for one year reduces output by 2% of full-employment output. Now consider a four-year disinflation according to the following table:
Year π
T+1 6%
T+2 4%
T+3 4%
T+4 4%
In this economy, agents use adaptive expectations to generate expected inflation rate. To be precise, agents use the average of the inflation rate from the previous two years to form the expected inflation. a) [10 Points] What is the unemployment rate in each of the four years? Illustrate your solution using a Phillips curve graph. b) [5 Points] By what percentage does output fall short of full-employment each year? What is the sacrifice ratio of this disinflation process. c) [5 Points] Use your graph of part (a) and clearly explain what would have happened if the expectation was rational rather than adaptive. What would have been the sacrifice ratio of this disinflation process?
Solution: Year πe u
T+1 8% 6% = 8%– 2(u – un) (u – un) = 1% => u = 5%
T+2 7% 4% = 7%– 2(u – un) (u – un) = 1.5% => u = 5.5%
T+3 5% 4% = 5%– 2(u – un) (u – un) = 0.5% => u = 4.5%
T+4 4% 4% = 4%– 2(u – un) (u – un) = 0% => u = 4%
8%
4%
un
Page 7 of 13
b) Using Okun ratio of 2
2*Cyclical unemployment 2*1% = 2%, 2*1.5% = 3%, 2*.5% = 1%, 2*0%=0 Sacrifice Ratio = (2%+3%+1%)/4=1.5 c) Under rational expectation the inflation would have adjusted instantly and therefore the sacrifice ration would have been zero.
8%
4%
un
Page 8 of 13
Problem-3 [20 Points] Suppose the loss function of the central bank is given by: L(u, π) = u1/2 + (γπ)2 The Phillips curve of this economy is: u = un − α(π − πe) a) (5 Points) What is the actual inflation and the expected inflation if the Bank adopts a credible rule? b) (10 Points) What is the actual inflation under policy discretion? What is the expected inflation and why? Briefly explain your results. c) (5 Points) Holding all else constant, what happens to actual inflation when γ increases and why? What does this suggest as to who should be appointed as the central banker? Solution: a) If rule is chosen
π =πe => u = u n The optimal rule would be: π = 0 b) The first step is to solve for the Bank choice of inflation, for any given inflationary expectations. Substituting the Phillips curve into the loss function to get:
L(u, π ) = (u n − α (π − π e ) )
1/ 2
+ (γπ )
2
Differentiate with respect to inflation π, and set this first-order condition equal to zero: −1 / 2 dL(u, π ) α = − ((u n − α (π − π e ) ) + 2γ 2π = 0 dπ 2
Of course, rational agents understand that the Bank will choose this level of inflation. => π e = π , therefore the above equation simplifies to:
−
α
(u ) 2
π=
n −1 / 2
+ 2γ 2π = 0
α 4(u
)
n 1/ 2
γ2
Since the inflation is expected the unemployment will stay at the natural rate. The outcome of the discretionary policy is higher inflation and no change in the output. The phenomenon is called Page 9 of 13
Time-inconsistency. If the policy makers use their discretion, rational consumers will expect the higher inflation and the policy maker will have no choice but to accommodate. This results into higher inflation which is costy to the policy maker. When the natural rate of unemployment rises, the inflation rate declines, which appears counter intuitive but is by by-product of the loss function presented. c) Higher γ means more dislike for inflation relative to unemployment and therefore lower inflation and therefore lower loss. This suggests appointing a conservative central banker (higher γ).
Page 10 of 13
Problem – 4 (20 Points): Suppose the DAD and DAS equations are as follows and that the economy has been in equilibrium:
Yt = Yt − A(π t − π t* ) + Bε t
π t = π t −1 + φ (Yt − Yt ) + ν t where:
A=
αθπ 1 > 0, B = >0 1 + αθY 1 + αθY
Furthermore, suppose:
Y = 1000, π t* = 4, α = 2.0, ρ = 2.0, φ = 0.2,θπ = 0.5,θ Y = 0.5 a) (7 Points) Briefly explain how the monetary policy works in the DAD framework as compared with the traditional AD framework.
b) (13 Points) In time t the economy faces a supply equal to 1 that lasts for two periods. What would be the impact on the output and inflation. To answer this question make sure to show your solution clearly then fill up the following table and at the end use a DAD/DAS framework to graph your solution.
Period
Supply shock
Demand Shock
t-2 t-1 t t+1 t+2 t+3 t+4
0 0 1 1 0 0 0
0 0 0 0 0 0 0
Inflation in Preceding period
Output
Inflation
Page 11 of 13
Solution:
a) In the DAD framework the monetary policy is endogenized through the monetary rule. For instance, a decrease in the inflation rate forces the Bank of Canada to reduce the overnight rate and as a result the real interest rate will fall and the investment rises leading to a rise in the output. A surprise change in the parameters of the monetary rule such as a reduction in the target inflation rate will shift the DAD leftward. b) Period
Supply shock
Demand Shock
t-2 t-1 t t+1 t+2 t+3 t+4
0 0 1 1 0 0 0
0 0 0 0 0 0 0
Inflation in Preceding period 4 4 4 4.909055 5.735468 5.577662 5.434201
Output
Inflation
1000 1000 999.5453 999.1321 999.211 999.2827 999.3479
4 4 4.909055 5.735468 5.577662 5.434201 5.303783
1 1 Yt = 1000 − (π t − 4) + ε t 2 2 π t = π t −1 + 0.2(Yt − 1000) + ν t 1 Yt = 1000 − (π t −1 + 0.2(Yt − 1000) + ν t − 4) 2 ν 1 0 .1 Yt = 1102 − π t −1 − Y− t 2 2 2 νt 2 .1 1 Y = 1102 − π t −1 − 2 2 2 1 1 Yt = 1001.818 − π t −1 − νt 2 .2 2 .2 4 1 Yt = 1001.818 − − = 999.546 2 .2 2 .2 => π t = π t −1 + 0.2(Yt − 1000) + ν t = 4 + 0.2(999.546 − 1000) + 1 = 4.909 1 1 Yt +1 = 1001.818 − ( 4.909) − = 999.132 2 .2 2 .2 => π t +1 = 4.909 + 0.2(999.132 − 1000) + 1 = 5.735 1 Yt + 2 = 1001.818 − (5.735) = 999.211 2 .2 => π t +2 = 5.735 + 0.2(999.211 − 1000) = 5.578
Page 12 of 13
π DAS
π =4 DAD
Y
Y = 1000
Page 13 of 13
Sample Test-1 (Solutions) Ata Mazaheri Instructions:
This is a closed book test.
You have 2 Hours. Good Luck! Last Name: First Name:
ID
FOR MARKERS ONLY: Q1
Q2
Q3
Q4
Total
40
20
20
20
100
Marks Earned Maximum Marks Possible
Page 1 of 13
Answer all following questions: Question-1 [40 Points] Answer the following questions: a) (5 Points) Use a DAD-DAS curve diagram to illustrate the short-run and long-run effect of an unexpected positive demand shock that lasts three years on output, price level, unemployment rate, and inflation rate. Solution:
π C DAS B A D
Y Y
DAD
A positive demand shock that lasts for three periods will shift DAD to the right. DAD stays there as log as the shock last – three periods- and then returns to its original position. In the first three periods, while the positive demand shock persists the DAS adjusts up reflecting the inflationary gap. But when the demand shock dissipates the DAS starts adjusting downwards reflecting the deflationary gap. Eventually the economy returns to its original long run equilibrium.
Page 2 of 13
b) [12 Points)] Assume that an economy's production function is Y = 2000L1/2. Suppose the labor supply curve is given by Ls = 15,625(W/P). Assume that the price level is expected to be 8. i) (5 Points) What are the equilibrium real and nominal wages for the expected price level of 8? If the actual price level turns out to be 16, what will be the actual real wage? ii) (7 Points) How much labour would be demanded if the actual price level turns out to be 16? How much output would be produced? Does aggregate supply increase as the price level increases? Briefly Explain. Solution: i) Equilibrium real wage is (W/P for which labor demand is equal to labor supply, Ld = Ls, that is: (W/P) = 1000L-1/2 => Ld = 1000,000(P/W)2 Ld = Ls = 1000,000(P/W)2 = 15,625(W/P). => W/P = 4 => Nominal wage = 32. Note: This will be the negotiated nominal wage between the firm and workers and will remain fixed in the short-run. Actual real wage = 32/16 = 2. ii) Price Level 16, => Real wage = 2 => labor demand = 1000,000 (P/W)2 = 250,000. => Output produced = 2000L1/2 = 1000,000: Price level = 8, => real wage = 4, demand for labour = 1000,000(P/W)2 = 62,500 => Output Produced = 2000 L1/2 = 500,000. Thus output supply increases as price level increases.
1000,000
500,000 4 2
62500
250000
16 8
Page 3 of 13
c) [13 Points] Assume that an economy is initially at the natural rate of unemployment. i) (5 Points) Use an AS-AD curve diagram to illustrate the short-run and long-run effect of an unexpected negative supply shock on output, price level, unemployment rate, and inflation rate. ii) (8 Points) Use the expectation-augmented Phillips curve [ π = π e − β (u − u n ) + ν ] where ν is the negative supply shock. Show the impact of the contractionary monetary policy on the prices and unemployment (a) assuming prices/wages are sticky in the short term such that π e is fixed (b) assuming rational expectation. Solution: i) In the AD-AS framework, the negative supply shock shifts the AS curve leftward. At new equilibrium, the price level rises and the output falls. So when there is a negative supply shock, the Phillips Curve shifts up and the inflation rate rises (1). The fall in output results in an increase in unemployment rate. So there is a downward move along the Phillips Curve as unemployment rate increases (2). Therefore the effects of the negative supply shock are the increases in π and u:
Page 4 of 13
ii) A decrease in money growth rate (contractionary monetary policy lowers the inflation rate of the economy. If prices are sticky, π e remains unchanged, since the Phillips Curve equation is π = π e − β (u − u n ) + ν , the Phillips curve will not shift. There will be a move along the Phillips curve as shown below.
A lower inlfation and higher unemployment. If the government adopts an expansionary monetary policy. There will be a move along the curve to the left which will result in higher inflation and full employment. According to Lucas rational extectation model, Public adjust their inflation expectation according to the reduction (increase) in money growth as long as the policy is credible. Public anticipate that the decrease (increase) in money growth rate will lower (increase) the inflation rate. So π e decreases (increases) and the Phillips Curve shifts downward (upward). The unemployment rate is not changed.
Page 5 of 13
d) [10 Points] Suppose Canadian economy follows the Phillips curve:
π = π −1 − (u − u n ) and that the natural rate of unemployment is given by the average of the past two years’ unemployment: u n = 0.5(u −1 + u −2 ) . Suppose the natural rate of unemployment (un) has been historically at 8%. The Bank of Canada follows a policy to reduce inflation by 4% permanantly. i) (6 Points) Show the impact of this policy on the natural rate of unemployment over time. Solution:
u0 = 8%, u1 = 12% 0.12 + 0.8 = 0.10 2 u + u1 0.10 + 0.12 u3 = 2 = = 0.11 2 2 u + u 2 0.11 + 0.10 u4 = 3 = = 0.105 2 2 u + u3 0.105 + 0.11 u5 = 4 = = 0.1025 2 2 u2 =
ii) (4 Points) What is the sacrifice ratio for this disinflation? Briefly explain your answer. Solution: Because unemployment is always higher than it started, output is always lower than it would have been. => the sacrifice ratio is infinite. This is called hysteresis: we find that there is a long-run tradeoff between inflation and unemployment: to reduce inflation, unemployment must rise permanently.
Page 6 of 13
Problem-2: [20 Points] Consider an economy in its long-run equilibrium with an inflation
rate, π, of 8% per year and a natural unemployment rate, un, of 4%. The expectationsaugmented Phillips curve is: π = πe – 2(u – un),
Assume that Okun’s law holds so that a 1% increases in the unemployment rate maintained for one year reduces output by 2% of full-employment output. Now consider a four-year disinflation according to the following table:
Year π
T+1 6%
T+2 4%
T+3 4%
T+4 4%
In this economy, agents use adaptive expectations to generate expected inflation rate. To be precise, agents use the average of the inflation rate from the previous two years to form the expected inflation. a) [10 Points] What is the unemployment rate in each of the four years? Illustrate your solution using a Phillips curve graph. b) [5 Points] By what percentage does output fall short of full-employment each year? What is the sacrifice ratio of this disinflation process. c) [5 Points] Use your graph of part (a) and clearly explain what would have happened if the expectation was rational rather than adaptive. What would have been the sacrifice ratio of this disinflation process?
Solution: Year πe u
T+1 8% 6% = 8%– 2(u – un) (u – un) = 1% => u = 5%
T+2 7% 4% = 7%– 2(u – un) (u – un) = 1.5% => u = 5.5%
T+3 5% 4% = 5%– 2(u – un) (u – un) = 0.5% => u = 4.5%
T+4 4% 4% = 4%– 2(u – un) (u – un) = 0% => u = 4%
8%
4%
un
Page 7 of 13
b) Using Okun ratio of 2
2*Cyclical unemployment 2*1% = 2%, 2*1.5% = 3%, 2*.5% = 1%, 2*0%=0 Sacrifice Ratio = (2%+3%+1%)/4=1.5 c) Under rational expectation the inflation would have adjusted instantly and therefore the sacrifice ration would have been zero.
8%
4%
un
Page 8 of 13
Problem-3 [20 Points] Suppose the loss function of the central bank is given by: L(u, π) = u1/2 + (γπ)2 The Phillips curve of this economy is: u = un − α(π − πe) a) (5 Points) What is the actual inflation and the expected inflation if the Bank adopts a credible rule? b) (10 Points) What is the actual inflation under policy discretion? What is the expected inflation and why? Briefly explain your results. c) (5 Points) Holding all else constant, what happens to actual inflation when γ increases and why? What does this suggest as to who should be appointed as the central banker? Solution: a) If rule is chosen
π =πe => u = u n The optimal rule would be: π = 0 b) The first step is to solve for the Bank choice of inflation, for any given inflationary expectations. Substituting the Phillips curve into the loss function to get:
L(u, π ) = (u n − α (π − π e ) )
1/ 2
+ (γπ )
2
Differentiate with respect to inflation π, and set this first-order condition equal to zero: −1 / 2 dL(u, π ) α = − ((u n − α (π − π e ) ) + 2γ 2π = 0 dπ 2
Of course, rational agents understand that the Bank will choose this level of inflation. => π e = π , therefore the above equation simplifies to:
−
α
(u ) 2
π=
n −1 / 2
+ 2γ 2π = 0
α 4(u
)
n 1/ 2
γ2
Since the inflation is expected the unemployment will stay at the natural rate. The outcome of the discretionary policy is higher inflation and no change in the output. The phenomenon is called Page 9 of 13
Time-inconsistency. If the policy makers use their discretion, rational consumers will expect the higher inflation and the policy maker will have no choice but to accommodate. This results into higher inflation which is costy to the policy maker. When the natural rate of unemployment rises, the inflation rate declines, which appears counter intuitive but is by by-product of the loss function presented. c) Higher γ means more dislike for inflation relative to unemployment and therefore lower inflation and therefore lower loss. This suggests appointing a conservative central banker (higher γ).
Page 10 of 13
Problem – 4 (20 Points): Suppose the DAD and DAS equations are as follows and that the economy has been in equilibrium:
Yt = Yt − A(π t − π t* ) + Bε t
π t = π t −1 + φ (Yt − Yt ) + ν t where:
A=
αθπ 1 > 0, B = >0 1 + αθY 1 + αθY
Furthermore, suppose:
Y = 1000, π t* = 4, α = 2.0, ρ = 2.0, φ = 0.2,θπ = 0.5,θ Y = 0.5 a) (7 Points) Briefly explain how the monetary policy works in the DAD framework as compared with the traditional AD framework.
b) (13 Points) In time t the economy faces a supply equal to 1 that lasts for two periods. What would be the impact on the output and inflation. To answer this question make sure to show your solution clearly then fill up the following table and at the end use a DAD/DAS framework to graph your solution.
Period
Supply shock
Demand Shock
t-2 t-1 t t+1 t+2 t+3 t+4
0 0 1 1 0 0 0
0 0 0 0 0 0 0
Inflation in Preceding period
Output
Inflation
Page 11 of 13
Solution:
a) In the DAD framework the monetary policy is endogenized through the monetary rule. For instance, a decrease in the inflation rate forces the Bank of Canada to reduce the overnight rate and as a result the real interest rate will fall and the investment rises leading to a rise in the output. A surprise change in the parameters of the monetary rule such as a reduction in the target inflation rate will shift the DAD leftward. b) Period
Supply shock
Demand Shock
t-2 t-1 t t+1 t+2 t+3 t+4
0 0 1 1 0 0 0
0 0 0 0 0 0 0
Inflation in Preceding period 4 4 4 4.909055 5.735468 5.577662 5.434201
Output
Inflation
1000 1000 999.5453 999.1321 999.211 999.2827 999.3479
4 4 4.909055 5.735468 5.577662 5.434201 5.303783
1 1 Yt = 1000 − (π t − 4) + ε t 2 2 π t = π t −1 + 0.2(Yt − 1000) + ν t 1 Yt = 1000 − (π t −1 + 0.2(Yt − 1000) + ν t − 4) 2 ν 1 0 .1 Yt = 1102 − π t −1 − Y− t 2 2 2 νt 2 .1 1 Y = 1102 − π t −1 − 2 2 2 1 1 Yt = 1001.818 − π t −1 − νt 2 .2 2 .2 4 1 Yt = 1001.818 − − = 999.546 2 .2 2 .2 => π t = π t −1 + 0.2(Yt − 1000) + ν t = 4 + 0.2(999.546 − 1000) + 1 = 4.909 1 1 Yt +1 = 1001.818 − ( 4.909) − = 999.132 2 .2 2 .2 => π t +1 = 4.909 + 0.2(999.132 − 1000) + 1 = 5.735 1 Yt + 2 = 1001.818 − (5.735) = 999.211 2 .2 => π t +2 = 5.735 + 0.2(999.211 − 1000) = 5.578
Page 12 of 13
π DAS
π =4 DAD
Y
Y = 1000
Page 13 of 13
