There Are Many Kuhn Trucker Conditions and Optimization Problem Question
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Assignment 4:
Check Course UM Learn page for due date
1.. Jill has $ 500 which can choose to spend on two goods, called x and y. The first good, x, has
a price of $5 and the second good, y, has a price of $2. She has the following utility function.
U = ( x ? 5) 2 + ( y ? 4) 2
a) Set up Jills optimization problem.
b) List all nine Kuhn-Tucker conditions.
c) Does this problem satisfy the Kuhn-Tucker conditions? Explain.
c) Solve for the optimal values of x and y. (Describe your logic, if your answer to c) is no. )
2. Consider the monopolist internet provider. The firm divides its market into two periods:
High demand (H) and Low demand (L)
The monopolist faces inverse demand curves:
???????? = 60 ? 0.04????????
???????? = 40 ? 0.04????????
The firm faces per unit capacity cost c = 10 and per unit production cost b = 4.
In each period, the firm can only produce output up to its capacity, K.
a)
b)
c)
d)
e)
f)
Write the firms choice variables.
Write the firms profit equation (as a function of output — always).
Write the Lagrangian for this problem.
Find all Kuhn-Tucker conditions.
Fine the optimal values of ???????? , ???????? , ???????? , ???????????? ???????? .
Find the firms maximum profit.
3. You are analyzing a firms production. You conclude that the firms production is
characterized by a constant elasticity of substitution between inputs, but the elasticity is
greater than 1.
a) Write, in general form, the equation you would use to analyze this firm.
b) The answer in a) should contain a parameter
?.
Is
?
greater than or less than 0? Explain
briefly.
4. A firm is minimizing cost, given a production function: ???? = ???? 1/4 ????3/4
(Recall cost is rK+wL)
a)
b)
c)
d)
e)
Set up the firms optimization problem.
Solve for the conditional demand curves, K= K(w, r, Q) and L = L(w, r, Q)
solve for the firms cost function C= C(w, r, Q)
Find the firms conditional demand functions K=K(w,r,Q) and L=L(w,r, Q)
Use duality theory to find the firms budget-constrained production function,
Q=Q(w,r,C)
(this is sometimes called a conditional production function because it is conditional on
the firms budget, C)
?1/2 ?1/2
???????? ????
5. Given the indirect utility function ???? = ????????
a) Find the Expenditure function
b) Find the consumers Marshallian demand functions for ???? and ????.
c) Find the consumers Hicksian demand functions for ???? and ????.
d) Draw the optimization problem that yields an indirect utility function as the maximum value
function in a graph (utility maximization subject to a constraint).
e) Draw the optimization problem that yields the expenditure function as the (negative)
maximum value function (its a minimum, but it has the envelope properties) in a 2nd graph.
f) Explain how d) and e) relate to each other, and state whether or not, according to your
graphical analysis, you would expect ????????????? , ???????? , ????? = ????(???????? , ????????, ????). (The value of the Marshallian
demand for x to equal the value of the Hicksian demand for x.)
6. Repeat parts a) b) and c) of 7. above for indirect utility function
(NOTE: Just do the math. This is a Leontief type function.)
???? =
????
(2???????? +????????)
.
7. Jenny lives two periods: 1, and 2. Her utility is describe by:
???? = ????????????1 + 0.97 ln ????2
She earns $200 in period 1 and $250 in period 2.
Jenny has the option to save or borrow at 5% interest.
Ornot is helping Jenny decide the optimal amount of savings to maximize her utility.
Ornot says that Jennys budget constraint is:
????1 + ????2 /1.05 ? 200 + 250/1.05
a) Check that Ornots budget constraint is correct, assuming that Jenny is planning her
consumption for both periods in period 1. Explain why this is correct or not correct.
b) Set up Jennys two-period optimization problem.
c) Find all Kuhn-Tucker conditions.
d) Solve for ????1? ???????????? ????2? .
e) Does Jenny borrow or lend in period 1?
8. Janet is going shopping. Her utility function is ???? = ???? 2 ???? 3 . She has both a budget constraint:
200 ? 3???? + 3???? and a ration constraint: ???? ? 2???? + 4????.
a) Set up the Lagrangian,
b) find the K-T conditions and
c) solve.
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Tags:
optimal values
Optimization functions
KuhnTucker conditions
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