A firm uses two factors of production. Its production function is x = - (A³₁ A³₂)^1/2+22 A₁ A₂+ 15 (A₁ A₂)^1/2 , where A1 and A2 are quantities of factors 1 and 2 and x is the output.

a) In the short runA₂ = 100

• Derive the equations for average and marginal products for the other factor
• Over what ranges of output does the firm experience diminishing average and marginal

products of factor 1 ?

b) Repeat question a) treating A₂ as an arbitrarily given constant

c) Both factors are now variable. Derive the equation of Marginal Rate of Technical

Substitution of factor 1 for factor 2

d) If the price of factor 1 is 1 and the price of factor 2 is 4, derive an equation in the

form A₁ =f(A₂) for the firm's expansion path.

e) Along the expansion path of d) calculate the ranges of output for which the firm

experiences increasing , decreasing and constant returns to scale.

f) Are the answers to e) dependent on input prices

g) Derive the equation in the form A₁ =f(A₂) when x = 1350

h) Calculate the firm's maximum possible output

i) Assume that the prices of factors 1 and 2 are 4.44 and 9.99 respectively and output

price is 0.12. The firm is a profit maximizer. Calculate

1) the amount of factors used

2) the amount of output produced

3) the firms profit.

j) Let x = 4y and A₁ = 4a₁ and A₂ = 4a₂. The write the production function in terms

of y, a₁ and a₂

k) Return to the original production function. Assume that the price of both factors is

10. The firm's expenditure on the two factors is 320. What is the maximum possible output?

l) With prices and expenditure as in k) , suppose that the firm has to buy 22 units of

factor 1, what is then the maximum output?