Question

Using whatever method you like, derive the supply curve for the following production function

f(x1, x2) = 5ln(x1) + ln(x2) and w1 =w2 = 1.

Answer 1

Approach followed - calculate optimality condition, substitute back into production function, calculate value on input in terms of Q, substitute back into Cost curve, calculate MC and finally equate it to price to get the supply curve.

Given,

Q = 5ln(x1) + ln(x2)

w1 = w2 = 1

TC = x1w1 + x2w2 = x1 + x2

at optimum, MRTS = w1/w2

MRTS = MPx1/MPx2

MPx1 = dQ/dx1 = 5/x1, MPx2 = dQ/dx2 = 1/x2

MRTS = (5/x1)/(1/x2) = 5x2/x1 = w1/w2 = 1

5x2 = x1, x2 = x1/5

Substituting back in production function,

Q = 5ln(x1 )+ ln(x1/5) = ln(x1⁵) + ln(x1) - ln5 = ln(x1⁶) - ln5 = 6ln(x1) - ln5

Property of log - [ ln(a) + ln(b) = ln(ab), ln(x^a) = a*ln(x) ]

ln(x1) = (Q + ln5)/6

x1 = exp[(Q + ln5)/6]

Substituting x1 in cost function TC

TC = x1 + x2 = x1 + x1/5 = 6x1/5 = (6/5)*exp[(Q+ln5)/6]

MC = dTC/dQ = (6/5)*exp[(Q + ln5)/6]*(1/6)

MC = (1/5)*exp[(Q + ln5)/6]

MC curve is the firm's short run supply curve, hence

P = (1/5)*exp[(Q+ln5)/6]

is the firm's short run supply function.

Hope the answer helped you :)