ecn313

Intermediate Macroeconomics

Quiz 3

Name: __________________________________________________

I. Cyclical Unemployment

Consider the general equilibrium model studied in class. The representative household

solves the problem

max 𝑐,𝑛

𝑈 = 𝑐 − 𝑛2

2

s.t. 𝑃𝑒 𝑐 = 𝑤𝑛

where 𝑃0 𝑒 = 1. The representative firm solves the problem

max 𝑛

𝜋 = 𝑃𝑌 − 𝑤𝑛

where 𝑌 = 𝐴𝑛 and 𝐴 = 2. Finally, the aggregate demand equation is given by

𝑀𝑉 = 𝑃𝑌

where 𝑉 = 0.8 and 𝑀 = 4.

In period t=0, the economy is simultaneously in short-run and long-run equilibrium.

Then, in period t=1, an increase in price expectations occurs and 𝑃1 𝑒 = 8

(a) Find the change in the cyclical unemployment rate from t=0 to t=1.

II. Structural Unemployment

Now, suppose that each agent solves the same problem stated above, but consider two

different scenarios:

I. Efficiency wages. The production function is 𝑌 = (𝑒𝑛) 1

2, where 𝑒 = (𝑤 − 2) 1

3.

II. Labor union. The production function is 𝑌 = 𝑛 1

2, there is a labor union with objective

function 𝑈 = (𝑤 − 2)𝑛, and the social welfare function is 𝑆𝑊 = 𝜋 𝛼 𝑈1−𝛼

(a) Find the value of 𝛼 such that the wage 𝑤 under the two scenarios is the same.

III. Frictional Unemployment

Suppose that the unemployment insurance 𝑏 is fully financed by the wage tax on the

employed worker. This is, 𝑏 = 𝜏𝑒 𝑤. In addition, the value function of being employed is:

𝑉𝑒 (𝑤) = √𝑤(1 − 𝜏𝑒 )

𝜎 ,

where 𝜏𝑒 = 0.5, 𝑝 = 0.1, and 𝜎 = 0.8. In addition, 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦(𝑤 < 𝑤 ∗) =

𝑤∗

1000 .

(a) If the unemployment rate at time 0 is 𝑢0 = 0.98, compute the unemployment rate at

time 1 and time 2. Is unemployment increasing or decreasing over time? Explain

(b) Find the unemployment rate in the long-run.