Question

A firm wants to know how to minimize its costs for its new production plant, for any desired level of output. Suppose the production is given by y = f(x₁, x₂) = 2 x₁^1/2 x₂^1/2, where x₁ and x₂ are inputs, and the input prices are w₁ and w₂.

i. Sketch the cost minimization problem. Label your axes and graph appropriately. (1 point)

ii. Find the optimal choices of x₁ and x₂. (2 points)

iii. Find the cost function. (1 point)

iv. Now suppose input prices are given. Let the price of each input be equal to 4. In this case, what would be the firm's equation for marginal cost?

Answer 1

The production function is given as , and the cost of production would be .

i. The graph is as below, to minimize cost for a given quantity.

As can be seen, the C'' is less than required cost for producing y0 units, while C' is more than required cost for producing y0 units. At cost C, where the isocost and isoquant is tangent, the amount of cost is minimum cost required to produce y0 units. Curves used here are y=10 ; w1=w2=1 ; C''=7.5, C'=12.5 and C=10.

ii. The optimal choices would be where or or or or , which is the cost minimizing combination of inputs. Putting it in the production function, we have or or , and since , we have or or . These are the required optimal choices of x1 and x2.

iii. The cost function would be or or or . This the the cost function.

iv. The cost function for given price is , and the marginal cost would be hence or . This is the equation of firm's marginal cost.