Exercise session 1
Course: Mathematical methods in Economics
Lecturer: Dmytro Vikhrov
Date: February 19, 2013.
Problem 1
Describe the individual preference relation. Explain the representativity theorem, according
to which, if individual’s preferences are complete, transitive, continuous, and strictly monotonic,
there exists a continuous utility function that represents them.
Problem 2
Suppose there is a continuous utility function, U(x) = f(x).
A. Explain the concept of total utility and marginal utility.
B. For U1(x) = 2x − x2 and U2(x) = x2 − 2x draw graphs of the total and marginal utilities.
C. Find x∗ that maximizes U1(x) and U2(x). Interpret your findings.
Problem 3 (application of the utility maximization framework)
Suppose the utility is linear in wage, U = w, and there are N individuals. Each of them
decides whether to study. Without education she gets wL and if she studies, she gets wH.
A. In the simplest case suppose it costs c to study one year. Draw individual’s decision tree.
B. Derive conditions under which individuals decide to study.
C. Suppose now the choice is to study 0 year or 4. The life expectancy is M. Derive conditions
for when the individual decides to study.
Problem 4
In the above problem answer the following questions:
A. How many individuals will choose to study?
B. How many individuals will have wage wL?
C. How do you interpret c?
Problem 5
Solve Problem 3C when individuals are heterogeneous with respect to costs, ci ∼ U[0, ¯c].
Problem 6 Suppose, an individual derives utility from two goods, x1 and x2, U(x1, x2) =
f(x1, x2).
1. Explain the assumptions imposed on U(x1, x2).
2. Derive the marginal rate of substitution of x1 for x1. Draw the indifference curves.