ECON202 – HOMEWORK 2
Q1. An economy has the production function
Y = 3K(α)N(1-α) , such that, K=K0 , N=N0 , α∈(0,1) , K0>0 , N0>0
a. Graph the relationship between output and capital, holding labour constant at its current
value. What is the MPK? Does the marginal productivity of capital diminish?
b. Graph the relationship between output and labour, holding capital constant at its current
value. Find the MPN for a 10% increase in labour. Compare this result with the MPN for a 5%
increase in labour. Does the marginal productivity of labour diminish?
c. Show that the production function Y has a constant return to scale.
d. Find the labour demand of this economy (meaning write N as function of the wage).
Tip: use the usual MB=MC condition but leave the wage as an unknown variable.
Q2. In one economy there are two workers: Amy and Bill. They are living 1 period. They both
consume all their income immediately and they have the following utility functions
UA = 2C0.5+4L0.5 and UB = 2C0.5+2L0.5
L represents hours of leisure and C is the amount of consumption. Suppose that they both
work maximum 50 hours in a period. The market wage is 10 and prices are fixed at 1.
a. Find the budget constraint for the two workers
b. Find the optimal amount of consumption, leisure and hours worked for Amy and Bill
c. Find the individual supply function for A and B (meaning N as a function of the wage).
Tip: instead of putting a number for the wage, both in the budget constraint and in the
condition for marginal costs and benefits, let it be an unknown variable w. Then, find L as a
function of that w for both A and B. Remember that w>0, so, if necessary, don’t be afraid to
divide your equation by w! Finally, remember that hours worked are the maximum hours
minus the hours of leisure.
d. Suppose that government introduce a labour tax of 20% on labour income. Find the new
optimal choice for Amy and Bill.
e. Suppose now that the government gives back the total amount of taxes collected, equally
shared between the two workers. The workers spend immediately everything in consumption
(they have already made their work decision; they cannot increase leisure). Compare workers
in point b and now. Are they better off? Worse off? Same?
Tip: to understand if an agent is better off you need to compare their previous level of utility
(substituting C and L in the utility function) with the current one (in which C is increased by
the government transfer).
Q3. In one economy there are two consumers, Rebecca and Sarah. They are living 2 periods: 1 the
present and 2 the future. Their utility functions are the following
UR = log(c1)+log(c2) and US = log(c1)+2log(c2)
Rebecca and Sarah work at the same firm, Rebecca has a salary equal to 150 per period, while
Sarah has 100 per period. The real interest rate in the economy is 10%.
a. Compute the optimal allocation of consumption for both consumers
b. Suppose that the government wants to increase taxation on the rich, so it announces that it
is going to tax 5% of accumulated savings in period 2. Compute the new optimal allocations.
Tip: this taxation changes the budget constraint, but also the marginal cost of consumption.
1 dollar saved in s going to give less than (1+r) dollars in period 2, specifically 0.95(1+r)
c. Suppose that at the end of period 2, the government gives back the money equally between
the 2 workers and they spend it all immediately. Who between R and S is better off after this
policy, with respect to the status quo? Has the government obtained its aim of “taxing the
rich”? Explain, stating the difference between the wealth in period 2 and overall present
value of life income.
Tip: To understand who is better off, compare the level of the utility function before and
after the policy implementation.
Q4. Name the type of unemployment occurring in each of the causes below and explain your
answer.
a. Fatih recently get fired from the furniture manufacturing company that he works for, because the
company recently suffers from the recession in the economy during the Covid-19 outbreak.
b. Melih recently lost his job as the sports shoes manufacturing firm he worked for moved their plant
overseas. The other factories in the nation have been also following this trend due to lower
production costs.
c. Barçın recently left her job as a salesperson because she found that cooking interests her more.
Now, she is looking for a job in a bakery shop.
Q5. Suppose that in an economy the aggregate labour supply is the following
Nagg = 10+2w
While the labour demand is the following
Nagg = 1000/w
a. Find the full-employment labour market equilibrium (wage and number of hours worked)
b. Show in a graph the two curves and the equilibrium. Suppose now that the government
imposes a minimum wage of 30. How does the equilibrium changes? How many people
become unemployed?
c. Suppose that the government gives up the idea of the minimum wage, but instead decides
to pay a higher unemployment benefit to all workers that are not working. Show graphically
how the labour supply will respond to this policy, determining in which direction the labour
market equilibrium (w and N) moves. Does helping people outside the labour force improve
workers’ conditions?
Tip for Part c : You are not required to make any calculation in this point. You need to use
the standard assumptions about workers’ decision about N and drawing the expected
changes in the curve.
Q6. Suppose that the natural rate of unemployment in a particular year is 7 percent and the actual
rate of unemployment is 13 percent. Use Okun’s law to determine the size of the GDP gap in
percentage point terms. If the potential GDP is $100 billion in that year, how much output is being
forgone because of cyclical unemployment?
Additional Exercises
You do not need to solve these exercises for the homework, but they could help you
in understanding the concepts of these chapters
EX1
Suppose that a worker has 50 hours to be dedicated to leisure or work. The real wage W/P is 100
and the preferences of the worker are the following
U = log( C ) + 0.5 log (L)
a. Compute the budget constraint of the worker and graphically show it.
b. Compute the optimal allocation of the worker
c. Suppose that the worker’s firm decides to raise his hourly wage to 120. Compute the new
allocation after showing in the graph the shift in the budget constraint.
d. What can you conclude about the substitution and income effects in this specific case?
Explain in detail what these effects are and how they interact to arrive at the solution.
EX2
Macroland is a close economy characterized by the following aggregate labour supply function
N = 10+2w
And the following production function
Y = 6K0.5N0.5
Moreover, the capital accumulated by firms is equal to K=100. The price of capital is fixed at 1 and
its depreciation rate is 5%.
a. Find the full-employment labour market equilibrium (wage and hours worked)
b. Find the amount of desired capital in the economy as a function of the real interest rate
c. Compute the amount of investment needed to reach that desired amount (including the
part used to replace the deteriorated capital). How much savings there will be in Macroland?
(Express always everything as function of r)
d. Suppose that the economy arrives at the desired amount of capital when r=10%. Compute
the new equilibrium wage, GDP and investment.
e. Suppose the government decides to tax firms’ profits by 20%. Suppose that firms initially
cannot change capital K, does this tax affect firm’s decision about N?
Tip: Look at the condition marginal benefits=marginal costs, keeping in mind it is a tax on
profit, not on revenues!
f. What will happen to the desired level of capital when the tax is implemented? What about
the real interest rate? Explain with the help of the saving-investment diagram. Assume that
the firm owns already the capital, so that the tax does not modify the cost of capital that
remains the usual (δ+r).
g. Suppose that after the tax the capital has moved in the direction you pointed out in the
previous section. What happened to equilibrium wage of workers? Explain mathematically
or with the help of graphs.