University of Miami Utility Maximization and Expenditure Minimization Questions Need help with these 2 questions (i.e. all the questions on the pdf provide

University of Miami Utility Maximization and Expenditure Minimization Questions Need help with these 2 questions (i.e. all the questions on the pdf provided) that focus on Utility Maximization and Expenditure Minimization from one of my Microeconomics problem sets. This is an undergraduate ECON course (Intermediate Microeconomics) Problem Set 2
ECON205
Summer 2020
Due: 10am July 13 on Sakai
1. Assume all p1 , p2 , and u are strictly positive. Identify whether the function:
f pp1 , p2 , uq “ pp1 ` p2 qu
could be an expenditure function or a compensated demand function. Now if we replace the u
with I so that:
f pp1 , p2 , Iq “ pp1 ` p2 qI
can this function be an indirect utility function or an ordinary demand function?
2. Consider an individual with the utility function:
U px1 , x2 q “ lnpx1 q ` x2 .
(a) Solve for the ordinary demand functions x˚1 pp1 , p2 , Iq and x˚2 pp1 , p2 , Iq. If your solutions
depend on the value of the parameters p1 , p2 , and I, carefully state the conditions for the
Bx˚ pp ,p ,Iq
piecewise functions. Find the marginal effect of increasing income by calculating i BI1 2
and determine if they are normal or inferior goods. Solve for the marginal effect of increasing
Bx˚ pp ,p ,Iq
p1 on the ordinary demand function for good 1 by calculating 1 Bp11 2 .
(b) Solve for the indirect utility function V pp1 , p2 , Iq.
(c) Solve for the expenditure function using the duality result for the indirect utility function and the expenditure function epp1 , p2 , uq. Note you should state the conditions using
parameters of the expenditure function p1 , p2 , and u if it is piecewise.
(d) Use the Slutsky equation to solve for the substitution effect which is the marginal effect of
Bxc˚ pp ,p ,uq
increasing p1 on the compensated demand function for good 1 1 Bp11 2 , which should be
Bxc˚ pp ,p ,uq
Bx˚ pp ,p ,epp ,p ,uqq
represented as a function of p1 , p2 , and u. Compare 1 Bp11 2 to 1 1 2Bp1 1 2
which
you solved in part (a). Note also we derived the Slutsky equation from the duality result
˚
xc˚
1 pp1 , p2 , uq “ x1 pp1 , p2 , epp1 , p2 , uqq. Given these two observations, how would you plot
the ordinary demand and the compensated demand? Do they intersect or coincide?
(e) Solve for the compensated demand function for good 1 xc˚
1 pp1 , p2 , uq and compare with the
ordinary demand function.
(f) Suppose p1 “ 1, p2 “ 2, and I “ 10. Now the government wants to impose a $1 per unit
tax on good 1. Calculate the income and substitution effects caused by this change in p1 .
How much does the government need to give the individual for them to be willing to accept
the tax? How much is the individual willing to pay to avoid the tax? If the government
imposes the tax, what’s the tax revenue? How much the government could have collected
with a lump sum tax? How much is the deadweight loss?
3. Consider an individual with the Bernoulli utility function:
upxq “
1 ´ expp´γxq
,
γ
for γ ě 0. The individual has an initial wealth of I and can choose to invest z ď I in an asset,
which pays out 1 ´ a with probability p and 1 ` a with probability 1 ´ p for every dollar invested.
1

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