11. Let the consumer's utility function be f(q1, q2, q3

) = q1q2q3, and her budget constraint y

= p1q1 + p2q2 + p3q3. Consider q1 + (p2/p1)q2 = qc as a composite good. Formulate the

consumer's optimization problems in terms of qc and find the demand function for qc.

12. A consumer who conforms to the von Neumann-Morgenstern axioms is faced with four

situations A, B, C, and D. She prefers A to B, B to C, and C to D. Experimentation reveals

that the consumer is indifferent between B and a lottery ticket with probabilities of 0.4 and

0.6 for A and D respectively, and that she is indifferent between C and a lottery ticket with

probabilities of 0.2 and 0.8 for B and D respectively. Construct a set of von Neumann-

Morgenstern utility numbers for the four situations.

13. A consumer who obeys the von Neumann-Morgenstern axioms and has an initial wealth of

160,000 is subject to a fire risk. There is a 5 percent probability of a major fire with a loss of

70,000 and a 5 percent probability of a disastrous fire with a loss of 120,000. Her utility

function is U = W0.5

. She is offered an insurance policy with the deductibility provision that

she bear the first 7620 of any fire loss. What is the maximum premium that she is willing to

pay for this policy?

14. Determine the domain over which the production function q = 100(x1 + x2) + 20x1x2- 12.5(x1

2 +

x2

2

) is increasing and strictly concave.

ANS: The MPs, f1 = 100 + 20x2 - 25x1 and f2 = 100 + 20x1 - 25x2, are positive over the domain

0.8x1 + 4 > x2 > 1.0x1 - 5, and f11 = f22 = -25 < 0, f11f12 - f12

2 = 225 > 0 throughout two-dimensional

space. It is also necessary to impose the condition that the input values be nonnegative.

15. Assume that an entrepreneur's short-run total cost function is C = q

3 10q

2 + 17q + 66.

Determine the output level at which he maximizes profit if p = 5. Compute the output

elasticity of cost at this output.

16. An entrepreneur uses one input to produce two outputs subject to the production relation x

= A(q1

+ q2

) where , > 1. He buys the input and sells the outputs at fixed prices. Express

his profit-maximizing outputs as functions of the prices. Prove that his production relation is

strictly convex for q1, q2 > 0.

q1

= p1 rAq1

1 = 0

q2

= p2 rAq2

1

= 0

Where, q1 = (

p1

rA)

1/(1)

q2 = (

p2

rA)

1/(1)

The production relation is strictly convex for q1, q2 > 0 if the principal minors of the relevant

Hessian are positive within this domain. The second direct partials are the first-order minors:

2x

q1

2 = ( 1)Aq1

2

2x

q2

2 = ( 1)Aq2

2

These are both positive for q1, q2 > 0 since , > 1 by hypothesis. Finally,

2x

q1q2

= 0 and

2x

q1

2

2x

q2

2 (

2x

q1q1

)

2

> 0

17. An entrepreneur uses two distinct production processes to produced two distinct goods, Q1

and Q2. The production function for each good is CES, and the entrepreneur obeys the

equilibrium condition for each. Assume that Q1 has a higher elasticity of substitution and a

lower value for the parameter than Q2. Determine the input price ratio at which the input

use ratio would be the same for both goods. Which good would have the higher input ratio if

the input price ratio were lower? Which would have the higher use ratio if the price ratio

were higher?

18. Use Sherphard's lemma to find the production function that corresponds to the cost function

C = (r1 +2r1r2 + r2)q, and demonstrate that it is CES.

19. A linear production function contains four activities for the production of one output using

two inputs. The input requirements per unit output are

a11 = 1 a12 = 2 a13 = 3 a14 = 5

a21 = 6 a22 = 5 a23 = 3 a24 = 2

20. Construct a short-run supply function for an entrepreneur whose short-run cost function is

C = 0.04q

3 0. 8q

2 + 10q + 5.