A price-taking firm in the X industry has the following (pre-tax) total short-ru

Question

A price-taking firm in the X industry has the following (pre-tax) total short-run cost function: TC = 50 +30Q + Q^2. Price equals $130. The government is considering the following taxation policies: (a) A per-unit tax of $20/unit, (b) A tax of profit of 50% or (c) A lump-sum tax of $10. Fill in the blanks below. (Assume the firm still faces a price of $130 after a tax is imposed).

Profit-maximizing SR quantity prior to any tax (show the pre tax total profit function) = ?

Profit-maximizing SR quantity under the profit tax = ?

Profit-maximizing SR quantity under the excise tax = ?

Profit-maximizing SR quantity under the lump sum tax = ?

Explanation / Answer

a. A price taking firm is one which takes the market price as given and produces an output where P = MC, where MC is the marginal cost.

1. TC = 50 +30Q + Q2

The pre-tax profit function is 130Q - (50 +30Q + Q2) = -50 - Q2 +100Q

MC = 2Q + 30

P = $130

P = MC
2Q + 30 = 130
Q = 50 units are produced before any tax is levied.

b. Under profit tax, the profit function is 0.5( 130Q - (50 +30Q + Q2)) = 0.5 (-50 - Q2 +100Q) = -25 - 0.5Q2 +50Q

Differentiating the profit function wrt Q, we arrive at the first order condition -

-Q + 50 =

Q = 50 units. We see it does not change the optimal quantity.

c. A $20 per unit excise tax is basically a cost on the producer.

The cost function looks like - TC = 50 +50Q + Q2

MC = 2Q + 50

P = $130

P = MC
2Q + 50 = 130
Q = 40, the optimal quantity reduces here now.

d. In the event of a lump sum tax of $10, the TC = 60 +30Q + Q2 .

Since MC remains the same, it won't change my optimal quantity.

MC = 2Q + 30

P = $130

P = MC
2Q + 30 = 130
Q = 50 units are produced after any lump tax is levied.

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