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Two questions about Microeconomics: (The answer should be more specific)
Q1: Suppose that an individual’s utility function for consumption, C, and leisure, L, is given by
U(C, L) = C^0.5L^0.5
This person is constrained by two equations: (1) an income constraint that shows how consumption can be financed,
C = wH + V,
where H is hours of work and V is nonlabor income; and (2) a total time constraint (T = 1)
L + H = 1
Assume V = 0, then the expenditure-minimization problem is
minimize C − w(1 − L) s.t. U(C, L) = C^0.5L^0.5 = U
(a) Use this approach to derive the expenditure function for this problem.
(b) Use the envelope theorem to derive the compensated demand functions for consumption
and leisure.
(c) Derive the compensated labor supply function. Show that ∂Hc/∂w > 0.
In working following parts it is important not to impose the V = 0 condition until after
taking all derivatives.
(d) Assume V 6= 0, determine uncompensated supply function for labor and compare with
the compensated labor supply function from part (c).
(e) Determine maximum utility, U, using the expenditure function derived in part (a), assume
V = E,
(f) Use the Slutsky equation to show that income and substitution effects of a change in the
real wage cancel out.
Q2: An investor with a total wealth of $100 is faced with the following opportunities. First, he
may invest $100 now and receive $144 if there are good times, but receive $64 if there are bad
times. The investor estimates that good times happen with 50% probability.He can also buy
an investor newsletter whether good times or bad times will occur.
(a) Draw the decision tree that illustrates the options available to the investor and the payoffs
to the different options. Define P as the price of the newsletter.
(b) If the investor is risk-neutral with U(M) = M, where M is income, how much would he
be willing to pay for the subscription to the newsletter?
(c) If the investor is risk-averse with utility U(M) = M0.5, where M is income, how much
would this investor be willing to pay for the subscription to the newsletter?
(d) Suppose that the owner of the newsletter estimates that there are 75 risk-averse investors
like those of part (c) and 25 investors like those of part(b). If it costs zero to produce
the newsletter, how should the newsletter be priced assuming (i) that the owner wishes to
maximize the profits of the news letter and (ii) that this is the only newsletter available
to investors.
Sample Solution
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