Question

Your firm is producing water bottles in a perfectly competitive market. You estimated your production function ?(?,?)=(?∙?−1)^1/3 +1. Consider that labor the price ? and capital the price ? are both equal to 1. Based on the assumption for a perfectly competitive market (all firms are the same), what it will be the price at the equilibrium in this market in the long run? If at the equilibrium there are 1000 firms in the market, what it will be to total quantity supply in the market?

Answer 1

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

The Lagrangian function for cost minimization would be . The FOCs would be as below.

or or or or or .

or or or or or .

or or or .

Comparing the first two FOCs, we have or or , which is the required optimal combination of inputs. Putting this in the last FOC, we have or or or or , and since , we have or . These are the required input demands.

Putting this in the cost of production, we have the cost function as or or .

The equilibrium price would be the minimum value of the (long run) average cost. The average cost would be as . The AC would be minimum at where

or

or

or

or

or

or

or

or

or

or

or

or

or

or

or .

The AC is minimum at where , and at this quantity, AC is or or or or or or dollars.

In the long run, the price would be $1.36.

Also, each firm would supply , and the total supply would hence be units (supposing 0.1 bottle would not be produced).