Question

Suppose you are the manager of a watchmaking firm operating in a competitive market. Your cost of production is given byC= 200 + 2q2, whereqis the level of output andCis total cost. (The marginal cost of production is 4q; the fixed cost is \(200.)

a. If the price of watches is \)100, how many watches should you produce to maximize profit?

b. What will the profit level be?

c. At what minimum price will the firm produce a positive output?

Short answer

1. The manager should produce 25 watches to maximize profit.

2. The profit level would be $1050.

3. At a minimum price of $50, the firm will produce a positive output.

Step-by-step solution

The optimal quantity of watches for the firm

The firm will produce that quantity of watches for which the profit is maximum. The profit is maximum when marginal revenue is equal to the marginal cost. The marginal cost of production is 4q, the marginal revenue is calculated below:

$$\begin{gathered} \mathrm{MR}=\frac{\mathrm{d}\left(\mathrm{TR}\right)}{\mathrm{dq}} \\ =\frac{\mathrm{d}\left(\mathrm{p}\times \mathrm{q}\right)}{\mathrm{dq}} \\ =\frac{\mathrm{d}\left(100\mathrm{q}\right)}{\mathrm{dq}} \\ =100 \end{gathered}$$

The optimal quantity is determined by equating MC with MR,

$$\begin{gathered} \mathrm{MC}=\mathrm{MR} \\ 4\mathrm{q}=100 \\ \mathrm{q}=25 \end{gathered}$$

The optimal quantity which the manager should produce is 25 units.

Calculating the profit level

The total cost (TC) is given by 200 + 2q2. By substituting the value of q in the equation, the total cost is determined below:

$$\begin{gathered} TC\left(\$\right)=200+2q^2 \\ =200+2{\left(25\right)}^2 \\ =200+2\times 625 \\ =1450 \end{gathered}$$

The total revenue (TR) is calculated by multiplying the quantity produced with the price.

$$\begin{gathered} TR=p\times q \\ =\$100\times 25 \\ =\$2500 \end{gathered}$$

The profit is determined by subtracting the TC from TR.

$$\begin{gathered} Profit=TR-TC \\ =\$2500-\$1450 \\ =\$1050 \end{gathered}$$

The firm will generate a profit level of $1050.

The minimum price at which the firm would produce positive output

The firm will produce a positive output equal to the price of its average variable cost.Average variable cost (AVC) is calculated by dividing the total variable cost (TVC) by the quantity produced.

$$\begin{gathered} \mathrm{TVC}=\mathrm{TC}-\mathrm{FC} \\ =\$1450-\$200 \\ =\$1250 \\ \mathrm{AVC}=\frac{\mathrm{TVC}}{\mathrm{q}} \\ =\frac{\$1250}{25} \\ =\$50 \end{gathered}$$

The minimum price at which the firm would produce positive output is $50.