Question

ABC Corp wants to produce whatever output it wishes to sell at minimum cost. Its production function is y = z1/2z1/2, where y is output and z ≥ 0 and z ≥ 0 are inputs. It 1212 is a price taker in input markets so its total cost, C, equals w1z1 + w2z2, where w1 > 0 and w2 > 0 are input prices.

(i) Fix output at y0 > 0 and use the production function to write z2 as a function of y0 and z1. And then use this function to write C as a function of z1 and the parameters w1, w2, y0.

(ii) Use the above algorithm to find ABC’s cost-minimizing input demand func- tions, zi(w1, w2, y0), i = 1, 2.

Answer 1

Given : Y = z₁^1/2z₂^1/2

C = w₁z₁ + w₂z₂

setting Y = Y₀, we get, Y₀ = z₁^1/2z₂^1/2

squaring both sides, we get: Y₀² = z₁z₂ => z₂ = Y₀²/ z₁

Therefore, z₂ as a function of Y₀ and z₁ is given by z₂ = Y₀²/ z₁

substituting the value in cost function, we get,

C = w₁z₁ + w₂ (Y₀²/ z₁) = 1/z₁ [w₁z₁² + w₂Y₀²]

C as a function of z₁ and the parameters w₁, w₂, Y₀ is given by C = 1/z₁ [w₁z₁² + w₂Y₀²]

ii) cost minimizing input demand function is given by differentiating C w.r.t. z₁ to get the minima.

=> w₁ = w₂(Y₀²/z₁²)

rearranging and solving for z₁ we get, z₁² = w₂Y₀²/w₁

z₁ = w₂²Y₀/w₁² = (w₂/w₁)²Y₀

ABC’s cost-minimizing input demand func- tions, zi(w1, w2, y0), i = 1, 2 is given by z₁ = (w₂/w₁)²Y₀.