A firm produces output using two factors of production according to technology f

Question

A firm produces output using two factors of production according to technology f(x1, x2) = X1^1/2*X2^1/4. The price of output is 4. Factors 1 and 2 cost to the firm w1 and w2 per unit, respectively.

(i) Write an expression for the firm’s profit.

(ii) Write an equation having on the left hand side the marginal product of input 1 times the price of output and on the right hand side its wage (w1). Do the same for input 2.

(iii) Solve the two equations you found in (ii) in the two unknowns, x1 and x2. (This will give the amounts of factors 1 and 2 that maximize the firm’s profits.)

(iv) If w1 = 2 and w2 = 1, how many units of factors 1 and 2 will the firm demand? How much output will it produce? How much profit will it make?

Explanation / Answer

Output, Q = X10.5X20.25

(i)

Total revenue, TR = P x Q = 4 x X10.5X20.25

Total cost, TC = w1.X1 + w2.X2

Profit, Z = TR - TC = 4 x X10.5X20.25 - w1X1 - w2X2

(ii)

MP(X1) = dQ / dX1 = 0.5. X20.25 / X10.5

P x MP(X1) = w1

P x 0.5. X20.25 / X10.5 = w1 .....(1)

Again,

MP(X2) = dQ / dX2 = 0.25. X10.5 / X20.75

P x MP(X2) = w2

P x 0.25. X10.5 / X20.75 = w2 ....... (2)

(iii)

Dividing (1) by (2),

2. X2 / X1 = w1 / w2

w1X1 = 2. w2X2 .......(3)

Substituting in total cost,

TC = w1X1 + w2X2 = 2.w2X2 + w2X2 = 3w2X2

X2 = TC / 3w2

X1 = 2.w2X2 / w1 [From (3)]

= 2.w2. (TC / 3w2) / w1 = (2/3) x (TC / w1)

(iv) w1 = 2, w2 = 1

X1 = (2/3) x (TC / 2) = 3 x TC

X2 = TC / 3w2 = TC / 3

Q = X10.5X20.25 = (3 x TC)0.5(TC / 3)0.25 = (TC)0.75 x 30.25

Profit = 4 x X10.5X20.25 - w1X1 - w2X2 = 4 x Q - TC

= 4 x [(TC)0.75 x 30.25] - TC

NOTE: Exact value can be determined if: Either w1 & w2, or TC value is given.

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