Mr Petersen

Assignment 1

Construction of cost function – two products

Ralph produces two products, denoted ! and ! . For this production, he uses three inputs

! , !  and  ! . All input factors must be larger than or equal to zero. The respective factor prices are

! = 20, ! = 100  and  ! = 60. The technology of the production requires 0.5! + ! ≥

!  and  ! + ! ≥ ! . Thus, ! is a common factor, i.e. it produces both of the two products

simultaneously.

Note that for production of the first product inputs 1 and 3 are perfect substitutes and for the

production of product 2 inputs 2 and 3 are perfect substitutes. Furthermore, one unit of input three

can produce one unit of product 1 and one unit of product 2 simultaneously, a truly remarkable

input!

a) Write down the optimization program to determine Ralph’s cost, ; .

b) Determine the marginal cost of ! if ! < ! .

c) Determine the marginal cost of ! if  ! > ! .

d) Explain your findings intuitively, and in terms of the shadow prices on the technology

constraints.

e) Suppose the first product sells for a price of 45 per unit, while the second product sells for a

price of 105. If there are no constraints on the input factors, what is the optimal choice of

production of the two products?

f) Ralph sets a limit on the purchase of input factors, such that he can only purchase inputs for

a total cost of 40.000. At the same time, the price of the second product drops to 85. What is

the optimal plan of output and input factors and the total revenue when:

1) The prices of the input factors are (P1,P2,P3) = (20,70,100)

2) The prices of the input factors are (P1,P2,P3) = (20,60,100)

3) The prices of the input factors are (P1,P2,P3) = (16,70,100)

Explain the difference between the three optimal plans.

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