Question

Short run profit max Consider y - 11/2k1/3 Find the optimal input level, optimal output level, and profit of the 11/2k1/3 following production function: y - Let price of the output $10, wage $2, price of capital $2, capital fixed at k-1

Answer 1

As per the question the short run production function Y=L^1/2K^1/3

Price of output = $10, price of labour or Wage (w) = $2 and Price of capital (r) = $2

Capital is Fixed at K=1

As production function Y=L^1/2K^1/3

Marginal Product (MP) of Labour =dY/dL = K^1/3/2L^1/2

Marginal Product (MP) of Capital =dY/dK = L^1/2/3K^2/3

Marginal rate of technical substitution (MRTS) = MP of Labour / MP of Capital

MRTS = (K^1/3/2L^1/2) / (L^1/2/3K^2/3)

MRTS = (K^1/3/2L^1/2 ) X (3K^2/3/ L^1/2) = 3K/2L

At equilibrium level of output = MRTS = Price of labour (w) / Price of capital (r)

3K/2L   = 2/2

3K/2L   = 1

3K=2L   and L=3K/2

Though capital K is fixed and K=1

So L=3K/2            L =3/2 = 1.5

The optimum input level of Labour (L) = 1.5 and Capital (K) = 1

As per production function Y=L^1/2K^1/3

The output at optimum level of Labour (L) = 1.5 and Capital (K) = 1

Y=(1.5)^1/2(1)^1/3 = 1.2247

The optimum output level Y= 1.2247

Total Revenue = output X price of output = 1.2247 X 10 = $12.247 or approx $12.25

Total cost = (Units of labour X wage) + (units of capital X price of capital) = (1.5 X 2) + (1 X 2) = $5

Total Profit = Total Revenue – total cost = $12.25 – $5 = $7.25

Total profit obtained from the production is $7.25