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1
Managerial Economics (ARE) 100A
University of California, Davis
Spring Quarter, 2011
Instructor: John H. Constantine
Homework 3: Due Friday October 14, 2011
Problem 1
:
Rita receives utility from consuming food (X) and clothing (Y) according to the utility function:
U(X,Y) = X
0.8
Y
0.2
.
Rita has $5,000 per month to spend, and food costs $5 per unit and clothing cost $10 per unit.
How much
food and clothing will Rita purchase to maximize utility, and what is her total utility?
(a)
Solve for the optimal (numeric) values of X and Y by using the Lagrangean method.
(b)
What is the numeric value for
?
Interpret this result.
(c)
What are the numeric values for MU
X
and MU
Y
when evaluated at the optimal solution?
(d)
Numerically verify that MRS
XY
= P
X
/P
Y
at the optimal solution.
You are to verify MRS
XY
*
with
numbers
.
(e)
What is the total utility value at the optimal solution?
Suppose the government sets a consumption quota of 700 units X.
(f)
What is the new optimal solution?
(g)
Does the MRS equal the price ratio at your solution in part (f)?
(h)
What is the value of utility given the restriction?
Does this result make sense
?
Suppose the government sets a consumption quota of 900 units X.
(i)
What is the new optimal solution?
(j)
Does the MRS equal the price ratio at your solution in part (i)?
(k)
What is the value of utility given the restriction?
Problem 2
:
Joe has a monthly income of $200 which he allocates among two goods: meat and potatoes.
We also
know that Joe’s utility function is given by the equation:
U(M,P) = 2M + P,
where M is pounds of meat consumed and P is pounds of potatoes consumed.
(a)
Suppose meat costs $4 per pound and potatoes cost $2 per pound.
Draw Joe’s budget constraint
(place potatoes on the Xaxis and meat on the Yaxis.)
(b)
Given Joe’s utility function above, what is his utility maximizing consumption bundle (show the
utility function and the optimal consumption bundle on the graph in part a.)
(c)
An outbreak of potato rot raises the price of potatoes to $4 per pound.
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