Question

1. Por each of the following production functions (a and b) find the following equations (Hy in terms of Qow and li) MRZSK D) Long-run capital and Tabor demand curve iii) Long-run total cost curve iv) Short-run capital and labor demand curve if the firm is stuck with R = 9 v) Short-run total cost if the firm is stuck with K – 9. (a) Q-12VKA (b) Q=LK +6L 1.2. Meals must be produced with 5 chefs and 2 ovens for each mcal. Tct Q be the number of meals, be the number of chefs, and be the number of ovens. The price of one chef is Pc and the price of one oven is Po (a) What is the long run total cost function for produc ing pricals TC IQ Pc, Po)? (Hint: think of this intuitively rather than using formulas. For exam- ple, what is the cheapest way to produce one meal? Two meals? Q meals?) (b) What is the production function for meals Q(,0% (This tells how many meals can be produced for each l combination of C and o) I

Answer 1

(Question 1)

(a) Q0 = 4L1/2K1/2

(i)

MPL = Q0/L = 4 x (1/2) x (K/L)1/2 = 2 x (K/L)1/2

MPK = Q0/K = 4 x (1/2) x (L/K)1/2 = 2 x (L/K)1/2

MRTS = [2 x (K/L)1/2] / [2 x (L/K)1/2] = K/L

(ii)

Long run cost is minimized when MRTS = w/r

K/L = w/r

K = L x (w/r)

Q0 = 4L1/2[L x (w/r)]1/2

Q0 = 4 x L1/2L1/2 x (w/r)1/2

Q0 = 4 x L x (w/r)1/2

L = (Q0 / 4) x (r/w)1/2

K = [(Q0 / 4) x (r/w)1/2] x (w/r) = (Q0 / 4) x (w/r)1/2

(iii)

Long run total cost (TC) = wL + rK

TC = w x [(Q0 / 4) x (r/w)1/2] + r x [(Q0 / 4) x (w/r)1/2]

TC = (Q0 / 4) x [(wr)1/2 + (wr)1/2]

TC = (Q0 / 4) x 2 x (wr)1/2

TC = (Q0 / 2) x (wr)1/2

(iv)

When K = 9,

Q0 = 4 x L1/2 x (9)1/2

Q0 = 4 x 3 x L1/2

Q0 = 12 x L1/2

L1/2 = (Q0 / 12)

L = (Q0 / 12)2

K = 9

(v)

Short run TC = w x [(Q0 / 12)2] + r x 9

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