GM0701 Advanced Microeconomics
Problem Set 1: Utility, Demand and Uncertainty
Lecturer: Charles Nadeau
Teaching Assistant: Lina Andersson
Review Class: September 30, 9.00-12.00; Room: Volvo Salen
1.
A consumer has preferences represented by the utility function
U(x) = ∑𝑛𝑖=1 ln(xi )
where xi denotes the quantity consumed of good i and n > 3.
2.
(a)
Assuming that the consumer has fixed income y and can buy good i at price pi , find the
ordinary and compensated demand elasticities for good 1 with respect to pj , j = 1, … , n.
(b)
Suppose that the consumer is legally obligated to purchase an amount An of good n (e.g.
heating oil) where pnAn < y. If there are no constraints on the choices of the other n-1
goods, then find the ordinary and compensated demand elasticities for good 1 with
respect to pj , j = 1, … , n. Compare your answers to part (a) above.
(c)
Now suppose that the consumer is legally obligated to purchase an amount Ak of good k
where k = n-r, … , n and 0 < r < n-2 and ∑pkAk < y. Use the arguments in parts (a) and (b)
above to explain what will happen to the elasticity of good 1 with respect to pj as r
increases (i.e. as the number of constraints increases) and comment on the results.
Suppose a representative consumer in the republic of San Serrife has the utility function
U(x) = x1α1 x2α2
where x1 denotes the quantity of milk consumed, x2 denotes the quantity of wine consumed
and αi denotes a strictly positive parameter where (α1 + α2) = 1.
(a)
Calculate the cost function and ordinary demand functions for the typical consumer.
Interpret the parameters α1 and α2 .
(b)
The republic of San Serrife is about to join the European Union. As a consequence the
price of milk will rise to eight times the pre-entry value, but the price of wine will fall by
50%. Use compensating variation to evaluate the impact on consumer welfare of these
price changes.
(c)
San Serrife economists have estimated consumer demand within the republic and have
concluded that it is closely approximated by the demand system derived in part (a)
above. They further estimate that the people of San Serrife spend more than three
times as much on wine as on milk. They conclude that entry into the European Union is
in the interests of San Serrife. Are they correct? Explain. (hint: the spending estimate
by the economists implies that α2 > 3α1.
3.
Suppose a typical consumer has a two-period utility function of the form
U(x) = αln(x1) + (1 – α)ln(x2)
where xi denotes the amount of consumption in period i and α denotes a strictly positive
parameter with α < 1. The resources of the consumer consist of just inherited assets A in
period 1, which is partly spent on consumption in period 1 and the remainder invested in an
asset (e.g. bond) paying a rate of interest r. Assume that p1 = p2 = 1.
4.
(a)
Interpret the parameter α and explain why it might change over time.
(b)
Calculate the ordinary demand functions in both periods (x1*, x2*) (hint: graph the
budget constraint first in order to help derive the budget constraint equation). Explain
how consumption in both periods varies with α according to your results.
(c)
Using graphs, explain how consumption in both periods varies with A and r. In
particular, will a change in A alter the proportion spent on consumption in each period?
Will a change in r alter consumption spending in period 1? Explain.
A risk-averse person with a utility function for wealth of the form U(x) = ln(x) has initial wealth
y0 and faces a risk of loss L < y0 with probability π. An insurance company offers to fully insure
the loss at a premium κ > πL. It is possible to take out partial insurance on a pro-rata basis, so
that an amount tL of the loss can be covered at a cost of tκ where 0 < t < 1.
(a)
If L = 10 and κ = 2 then calculate t* (i.e. the optimal value of t for this person).
(b)
Show how the optimal value of t changes if y0 increases, holding all other parameters
constant.
(c)
Show how the optimal value of t changes if π increases, holding all other parameters
constant.
(d)
Show why this person will never choose full insurance (hint: full insurance means t* = 1).
Additional Problems
In addition to the problems assigned above, also work the following five (5) mini-problems found in
Chapter 4 of the Cowell textbook (note: these mini-problems will not be discussed during the review
class on September 30): 11, 13, 16, 17, 23.