Question

A perfectly competitive market operates in a market with an equilibrium price of P. Its total cost is given by TC = FC + VC(q), where FC (>0) is the fixed cost, VC(q) is the variable cost and and q is the quantity produced by the firm.

1. Write down the optimization problem of this firm.

2. Write down the first order condition. (Assume from now on the equation formed by the first order condition has an interior solution q*>0.)

3. Write down the second order condition(s). What does it say about the shape of the function VC(q).

4. Use the first order condition with the implicit function theorem to calculate ?q*/?FC. Prove that the supply function of this perfectly competitive firm is upward sloping.

Answer 1

1. The profit is the total revenue minus total cost, ie or or . The optimization problem would be as below.

This means that the optimization problem is to maximizing profit with respect to q.

2. The FOC would be as below.

or or or .

Now, for q*, we have .

3. The second order condition would be as below.

or or , ie . Hence, this verifies the SOC for maximization.

From the SOC, we have , meaning that the variable cost curve is convex to origin.

4. The FOC would be as , and putting the optimal q* in profit, we have .

We have , and considering the variables as FC and q*, we have or or or .

Now, since , we may say that is not defined.

The supply function would be as where , ie where or or . The supply would be upward sloping when . We have the slope of the supply curve as or , and since we have from the SOC, hence , ie supply is upward sloping.