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ECONOMICS 207
SPRING 2008
FINAL EXAM
For your information, the Hessian matrix in the profit maximization problem written as
π(x1, x2) = pf(x1, x2) − w1x1 − w2x2
is given by
H(π(x1, x2)) =
∂2π(x1, x2)
∂x1∂x1
∂2π(x1, x2)
∂x1∂x2
∂2π(x1, x2)
∂x2∂x1
∂2π(x1, x2)
∂x2∂x2
The bordered Hessian in the constrained optimization problem written as
L(x1, x2, λ) = f(x1, x2) − λg(x1, x2)
is given by
HB =
2
6
6
6
6
6
6
6
4
∂2L(x1, x2, λ)
∂x1∂x1
∂2L(x1, x2, λ)
∂x1∂x2
−
∂2L(x1, x2, λ)
∂x1∂λ
∂2L(x1, x2, λ)
∂x2∂x1
∂2L(x1, x2, λ)
∂x2∂x2
−
∂2L(x1, x2, λ)
∂x2∂λ
−
∂2L(x1, x2, λ)
∂λ∂x1
−
∂2L(x1, x2, λ)
∂λ∂x2
0
3
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
4
∂2L(x1, x2, λ)
∂x1∂x1
∂2L(x1, x2, λ)
∂x1∂x2
∂g(x1, x2)
∂x1
∂2L(x1, x2, λ)
∂x2∂x1
∂2L(x1, x2, λ)
∂x2∂x2
∂g(x1, x2)
∂x2
∂g(x1, x2)
∂x1
∂g(x1, x2)
∂x2
0
3
7
7
7
7
7
7
7
5
where we use the equivalencies
∂g(x1, x2)
∂x1
= −
∂2L(x1, x2, λ)
∂x1∂λ
∂g(x1, x2)
∂x2
= −
∂2L(x1, x2, λ)
∂x2∂λ.
Hints:
i. MPi =∂f(x1, x2)
∂xi
ii. 242 = 576iii. 2 × 11 × 24 = 528iv. 256× 3 = 28
× 3 = 768v. 25 × 32 = 52
× 25 = 800vi. 975
75= 13
vii. 9 × 27 = 9 × 128 = 1152viii. 12 × 32 = 3 × 4 × 25 = 3 × 27 = 384ix. 5 × 64 = 5 × 26 = 320
Date: 7 May 2008.
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Problem 1. [40 points]Below you are given a production function for a competitive firm. You are also given the price of the firm’s output and the prices of the
two inputs used by the firm. Output price is represented by p, the price of the first input by w1 and the price of the second input by w2.
f(x1, x2) = 30x1 + 20x2 − x2
1 + 2x1x2 − 3x2
2
p = 10
w1 = 40, w2 = 20
a. Write an equation that represents profit as a function of the two inputs x1 and x2. Simplify the expression.
b. Find all first and second partial derivatives of the function.
∂π∂x1
∂π∂x2
∂2π∂x1∂x1
= ∂2π∂x1∂x2
=
∂2π∂x2∂x1
= ∂2π∂x2∂x2
= −60
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Figure 1. Maximum Profit
0
6
12
1824
x1
0
3
7
11
16
x2
-5000
0ΠHx1,x2
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c. Find potential profit maximizing levels of x1 and x2.
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d. Fill in the elements of the Hessian matrix of the profit equation evaluated at the critical values of x1 and x2 and then verify that the levelsof x1 and x2 you found are either maximum, minimum or saddle points.
H =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∂2π∂x1∂x1
= ∂2π∂x1∂x2
=
∂2π∂x2∂x1
= ∂2π∂x2∂x2
=
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
=
e. Show that the optimal level of output is 529?
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Problem 2. [40 points]Consider a consumer with a utility function given by
v = u(x1, x2) = 10x1 + 20x2 − 2x2
1 + 2x1x2 − x2
2
The consumer faces prices and an income constraint given by
p1 = 30, p2 = 30, m0 = 1050
Find potential levels of x1 and x2 to maximize utility for this consumer given the income constraint and the stated prices. Verify thatthese consumption levels maximize utility.
a. Set up the objective function for this problem and find all first and second partial derivatives of the function with respect to x1 and x2.
L(x1, x2, λ) =
∂L(x1, x2, λ)
∂x1
∂L(x1, x2, λ)
∂x2
∂2L
∂x1∂x1
∂2L
∂x1∂x2
∂2L
∂x2∂x1
∂2L
∂x2∂x2
= −2
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b. What is the derivative of the objective function in this problem with respect to λ?
c. Find the partial derivatives of the constraint equation with respect to x1 and x2.
∂g(x1, x2)
∂x1
∂g(x1, x2)
∂x2
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In figure 2 you can see a maximum utility point and how it is unattainable given the budget constraint.
Figure 2. Utility and the Budget Constraint
0
4
8
1316
19
x1
0
5
11
1822
29
x2
0
150
300
uHx1,x2
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In figure 3 you can see the the tangency between one indifference curve and the budget line.
Figure 3. Tangency Between Indifference Curve and Budget Line
0 5 10 15 200
5
10
15
20
25
30
x1
x 2
10
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d. Use the information from 2a and 2b to find critical values for x1, x2 and λ. Note that λ = 1
15.
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e. Use the answers from part 2d and the expressions from parts 2a and 2c to fill in the bordered Hessian matrix for this problem. Thendetermine whether the critical values indicate a maximum, a minimum, or a saddle point.
HB =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∂2L
∂x1∂x1
= ∂2L
∂x1∂x2
=∂g(x1, x2)
∂x1
=
∂2L
∂x2∂x1
= ∂2L
∂x2∂x2
=∂g(x1, x2)
∂x2
=
∂g(x1, x2)
∂x1
=∂g(x1, x2)
∂x2
= 30 0
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
=
f. If income went up by $1.00, by how much would utility rise?
12
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Problem 3. [40 points]Below you are given a production function for a competitive firm. You are also given the price of the firm’s output and the prices of the
two inputs used by the firm. Output price is represented by p, the price of the first input by w1 and the price of the second input by w2.
f(x1, x2) = x2/5
1x
1/3
2
p = 60
w1 = 12, w2 = 5
a. Write an equation that represents profit as a function of the two inputs x1 and x2. Simplify the expression.
b. Find all first and second partial derivatives of the function.
∂π∂x1
= ∂π∂x2
=
∂2π∂x1∂x1
= −72
5x−8/5
1x
1/3
2
∂2π∂x1∂x2
=
∂2π∂x2∂x1
= ∂2π∂x2∂x2
=
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Figure 4. Maximum Profit
0
20
40
60
x130
60
90
x2
-100
0
100
200
ΠHx1,x2L
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c. Show that the potential profit maximizing levels of x1 and x2 are x1 = 32, x2 =?.
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d. Fill in the elements of the Hessian matrix of the profit equation evaluated at the critical values of x1 and x2 and then verify that the levelsof x1 and x2 you found are either maximum, minimum or saddle points.
H =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∂2π∂x1∂x1
= −−9
40
∂2π∂x1∂x2
=
∂2π∂x2∂x1
= ∂2π∂x2∂x2
= −5
96
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
=
16
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e. What is the optimal level of output?
f. Show that the marginal value product of x2 at its optimal value is equal to w2?
g. Explain in words why the value of the marginal product for each input for this firm is equal to the price of that input at the profitmaximizing level of input use for that input.
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Problem 4. [40 points]Consider a firm with a production function given by
f(x1, x2) = x2/5
1x
1/3
2
The firm faces prices and a cost constraint given by
w1 = 12
w2 = 5
c0 = 704
Find potential levels of x1, x2 and λ to maximize output for this firm given the cost constraint and the stated prices. Verify that theseinput levels maximize output.
18
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a. Set up the objective function for this problem and find all first and second partial derivatives of the function with respect to x1 and x2.
L(x1, x2, λ) =
∂L(x1, x2, λ)
∂x1
=∂L(x1, x2, λ)
∂x2
=
∂2L
∂x1∂x1
= −6
25x−8/5
1x
1/3
2
∂2L
∂x1∂x2
=
∂2L
∂x2∂x1
= ∂2L
∂x2∂x2
= −2
9x
2/5
1x−5/3
2
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b. What is the derivative of the objective function in this problem with respect to λ?
c. Find the partial derivatives of the constraint equation with respect to x1 and x2.
∂g(x1, x2)
∂x1
=∂g(x1, x2)
∂x2
=
20
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d. Use the information from 4a and 4b to find critical values for x1, x2 and λ. Hint: λ = 1
60
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In figure 5 you can see how the cost constraint restricts the levels of output than can be produced.
Figure 5. Output with a Cost Constraint
08
16
32
48
x1
0163248648096x2
0
4
8
12
16
fHx1,x2
22
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In figure 6 you can see the the tangency between one indifference curve and the budget line.
Figure 6. Output Maximization Subject to a Cost Constraint
0 10 20 30 40 50 600
20
40
60
80
100
x1
x 2
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e. Use the answers from part 4d and the expressions from parts 4a and 4c to fill in the bordered Hessian matrix for this problem. Thendetermine whether the critical values indicate a maximum or a minimum.
Hint: The determinant of the bordered Hessian is 11
32
HB =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∂2L
∂x1∂x1
= ∂2L
∂x1∂x2
= 1
960
∂g(x1, x2)
∂x1
=
∂2L
∂x2∂x1
= ∂2L
∂x2∂x2
=∂g(x1, x2)
∂x2
=
∂g(x1, x2)
∂x1
=∂g(x1, x2)
∂x2
= 5 0
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
=
24
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f. How much output can this firm produce given it spends only $704?
g. What is the marginal product of x1 at its optimal value?
h. What is the marginal product of x2 at its optimal value?
i. Show that the ratio of the marginal products is equal to the input price ratio?
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Problem 5. [45 points]Consider a producer with a production function given by
f(x1, x2) = x2/5
1x
1/3
2
The firm faces prices and an output target given by
w1 = 12, w2 = 5, y0 = 16
Find potential levels of x1 and x2 to minimize the cost for this producer to reach the target level of output given the stated prices. Verifythat these input levels minimize cost.
a. Set up the objective function for this problem and find all first and second partial derivatives of the function with respect to x1 and x2.
L(x1, x2, λ) =
∂L(x1, x2, λ)
∂x1
=∂L(x1, x2, λ)
∂x2
=
∂2L
∂x1∂x1
= ∂2L
∂x1∂x2
= −2
15λx
−3/5
1x−2/3
2
∂2L
∂x2∂x1
= ∂2L
∂x2∂x2
=
26
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b. What is the derivative of the objective function in this problem with respect to λ?
c. Find the partial derivatives of the constraint equation with respect to x1 and x2.
∂g(x1, x2)
∂x1
= 2
5x−3/5
1x
1/3
2
∂g(x1, x2)
∂x2
=
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Figure 7. Cost Minimization Subject to an Output Constraint
0 10 20 30 40 50 600
20
40
60
80
100
x1
x 2
28
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d. Use the information from 5a and 5b to find critical values for x1, x2 and λ.
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e. Substitute the values for x1, x2 and λ into the bordered Hessian matrix. Show that the determinant of this matrix is -11
1920.
HB =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∂2L
∂x1∂x1
= 9
40
∂2L
∂x1∂x2
= −1
16
∂g(x1, x2)
∂x1
= 1
5
∂2L
∂x2∂x1
= ∂2L
∂x2∂x2
=∂g(x1, x2)
∂x2
=
∂g(x1, x2)
∂x1
=∂g(x1, x2)
∂x2
= 0
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
=
Here are some hints:
i. cofactor
[
∂g(x1, x2)
∂x1
]
= −1
64
ii. 5 × 64 = 320iii. 160 = 5 × 32 = 5 × 25
iv. 12 × 32 = 384v. 320 = 26
× 5vi. 384 = 27
× 3vii. 27
× 5 × 3 = 1920
30
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f. Do the critical values from part d minimize cost?
g. How much does this firm spend on inputs?
h. How much output does it produce?
i. What is the marginal product of x1 at its optimal value?
j. What is the marginal product of x2 at its optimal value?
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k. Show that the ratio of the marginal products is equal to the input price ratio?
l. Interpret the condition in part k.
m. Why did you not need to do any of the work in problem 5 given you had already worked problems 3 and 4?
