Question

A firm’s revenue is given as: R= 100q + 8q². The firm’s total cost of production is given as: C= 250 + 500q.

1. Find out the firm’s profit-maximization level of output (q*).
2. What will be the market price at which the firm sells the output, q*?

Answer 1

Answer

The firm's revenue (R) = 100q + 8q²

The firm's total cost of production, C = 250 + 500q

The firm maximizes its profit at the quantity of output(q^*), for which marginal revenue(MR) is equal to the marginal cost (MC).

MR = change in total revenue(R) due to change in one unit of output(q)

Or, MR =  dR/dq = R/ q , where 'd' or, '' stands for change.

R = 100q + 8q²

Now differentiating R with respect to q , we get,

dR/dq = 100 + 16q

MR = 100 + 16q

MC = change in total cost (C) due to change in one unit of output(q).

Or, MC = dC/dq = C/q , where 'd' or, '' stands for change.

C = 250 + 500q

Now differentiating C with respect to q , we get,

dC/dq = 500

MC = 500

Now the profit maximization quantity (q^*) requires MR = MC

Putting the values of MR, and MC, we get,

100 + 16q = 500

Or, 16q = 500-100

Or, 16q = 400

Or, q = 25

  the firm's profit maximization level of output or quantity (q^* ) is 25 units.

Now, Revenue (R) = Price (P) * Quantity (q)

Or. P = R / q

Or, P = (100q + 8q² )/q

Or, P = 100q/q + 8q²/q

Or, P = 100 + 8q

Now, putting the value of q in the above equation, we get,

P = 100 + 8 * 25

Or, P = 100 + 200

Or, P = 300

the market price the firm will set for output q^* , is 300 money unit.

So, the firm's profit maximization level of output ( q^* ) = 25 units

The market price at which the firm sells the output, q^* = 300 money unit.

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