Exercise session #2
Lecturer: Dmytro Vikhrov
Date: February 28, 2012
Problem 1
Suppose Karel’s utility function is U(x1, x2) = x1x2.
1. Fix the utility at U = 12 and draw his indifference curve in (x1, x2) locus. On the
graph explain the notion of the marginal utility and total utility. Draw both
graphs.
2. Suppose that we are at the point (2, 6), and x2 suddenly drops to x2 = 5. By how
much must the consumption of x1 increase to keep Karel’s utility constant at the
level U?
3. Take the total derivative of the utility function and formalize the concept of the
marginal rate of substitution.
Problem 2
In the above Problem 1 prices of goods were absent. Let us extend the setup to include
prices, p1 for good x1 and p2 for good x2.
1. Reproduce the first-order conditions of the utility maximization problem from
Lecture 1. Compute the optimal consumption bundle for (p1, p2) = (4
3
, 1) and
w = 8. Compute the elasticity of demand with respect to prices.
2. Suppose that p1 increases to p1 = 2. Compute the utility loss.
3. By how much should the initial budget increase for Karel to get back on his initial
indifference curve, U = 12? Setup a system of three equations with three
unknowns. Interpret your findings.
4. Generalize the notion of compensating variation (CV).
5. Now suppose that the budget decrease is measured in terms of old prices.
Generalize the notion of equivalent variation (EV).
6. How can the CV/EV framework be applied to the case of the recent tax increase
in the Czech Republic?
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Problem 3
Veronika’s utility function is U(x, y) = min{x, y}. Her monthly salary is 150 units of
currency. The price of both x and y is 1. Her boss is thinking of sending her to work to
another town, where everything is the same, but py = 2. The boss offers no rise in pay
and the transportation costs are zero. Veronika says that for her the move is as bad as a
cut in pay of A units. She also does not mind moving if she gets a rise in pay equivalent
to B units. What are the values of A and B?
Problem 4 (left-over from the previous exercise session)
The studied utility maximization framework is also used to analyze how economic
agents (i.e. individuals) supply labor. Agent’s utility consists of how much labor to
supply and how many other goods, x, to buy: U(x, L) = xγ
γ
+ L, where L is free time,
i.e. leisure, and γ = 0. The price of labor is the wage,w, and the price of the
consumption good is p. The total endowment of time is 1.
1. Define the agent’s constraint and setup the utility maximization problem.
Graphically represent the budget constraint and discuss the location of the
indifference curve for the following cases:
i. The minimum number of hours worked is 8 hours per day.
ii. The maximum number of hours worked is 40 hours per week.
iii. The agent is offered to work overtime at a higher wage, w > w.
2. Derive the optimal allocation of the labor supply and consumption. Are
leisure (or labor) and consumption substitutes or complements? Are they normal
or inferior goods?
3. Define the equilibrium in this model.
4. Suppose that the labor income is taxed at a per-unit rate τ. Re-derive the optimal
labor supply and analyze the effect of the taxation.
5. Now think that, unlike to the previous point, there is a lump-sum tax on labor T.
Re-derive the optimal labor supply and compare it with the labor supply under
per-unit tax.
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