Economics 405, University of Calgary, Winter 2021
Assignment 3: 12 marks
Due 5:00pm Friday, March 5, 2021
1. Farmers face uncertainty over their earnings because rain is random. Let a farmer’s yield be
Y = 0 in the dry outcome and Y = 1 in the wet outcome. However, not all farmers are the
same:
(
p1 = 2/3 if type = 1
Probability(Y = 1) =
p0 = 1/3 if type = 0
√
All farmer’s are risk averse, with utility U (Y ) = Y as their utility function. Farmers know
their type.
(a) Calculate each farmer’s expected yield, E(Yi ) = pi 1+(1−pi )0, expected utility, E(Ui ) =
pi U (1) + (1 − pi )U (0), and calculate the utility at E(Yi ), U (E(Yi )), for i = 0, 1. Show
these all on a graph with utility on the vertical axis and yield on the horizontal axis.
(b) Suppose that the insurance company cannot identify which farmers are of which type,
but knows that q = 3/4 of all farmers are of type 0 and 1 − q = 1/4 are of type 1.
If the insurance company offers a contract that ensures a farmer of income E(Y ) =
qE(Y0 ) + (1 − q)E(Y1 ), will all farmers wish to buy the insurance? Will the insurance
company break even with this contract?
(c) Suppse the insurance company offers a contract that ensures that it breaks even. What
insurance contract will be available and who will buy it? What proportion of farmers
are ”uninsured”?
(d) Relative to the equilibrium in part (c), who gains and who loses if the government
requires that everyone buy actuarially fair insurance to ensure income E(Y ) as was
contemplated in part (b). Explain. Could the winners compensate the losers? Explain.
(e) If the proposal in part (e) were put to a simple majority rule vote by farmers, each of
whom votes according to their economic gain relative to the world with no insurance,
would it pass?
2. In the “Lighthouse in Economics,” Coase examined private provision of lighthouses in England.
(a) Samuelson argued that even if a private firm was able to somehow charge users for the
use of a lighthouse, that doing so would not be efficient. Explain Samuelson’s argument.
(b) How did Coase find that lighthouses got around the problem identified by Samuelson?
3. In society A, each person i = 1, . . . , n gets to consume all she produces. In society B, all
production is shared equally. Net utility is u(x) − cxi , where c is the cost to the individual
of producing one unit of xi . In society A, x = xi ; in society B, x = [xi + (n − 1)x−i ]/n where
x−i is each other person’s production. Assume marginal utility is decresing in x.
(a) Compare the equilibrium production in society A and with the symmetric Nash equilibrium production in society B.
(b) Which society is more egalitarian? Which society is richer? Explain.
4. Suppose that a society has R rich people and P poor people. The rich are altruistic, so
their utility is UiR = ui (yR − P xi ) + αP v(yP + (R − 1)xj + xi ), and the poor get utility
UiP = v(yP + (R − 1)xj + xi ). In these equation, yP < yR are incomes to rich and poor,
respectively, and xi is the amount of rich person i’s income that is donated to each poor
person, and (R − 1)xj is the amount all other rich people donate. Assume that 0 < α < 1.
(a) Characterize the symmetric Nash equilibrium to this problem, where each rich person
chooses how much to donate to the poor. In your solution, assume that each rich person
takes the contribution of the other rich as given.
(b) Explain why the rich face a public goods problem which results in too few donations to
the poor. That is, compare the solution to Nash equilibrium to the solution toP
choosing
R
the xi and xj that maximizes the the sum of the rich people’s utility, WR = R
i=1 Ui .
Assume that in your solution, the choice of all other xj is fixed when choosing xi , and
assume that in equilibrium, xi = xj = x.
P
R
(c) Now, find the solution to the xi and xj when the goal is to maximize W = R
i=1 Ui +
PP
P
i=1 Ui . Explain what is different about these two solutions. In which case would you
expect the xi = xj = x to be larger? Explain why these solutions differ in the direction
that they do.
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