a. Suppose that a firm’s production function is q = 9x1/2in the short run, where there are fixed costs of \(1000, and x is the variable input whose cost is \)4000 per unit. What is the total cost of producing a level of output q? In other words, identify the total cost function C(q).

b. Write down the equation for the supply curve.

c. If price is $1000, how many units will the firm produce? What is the level of profit? Illustrate your answer on a cost-curve graph.

The value of x is calculated with the help of the production function.

$$\begin{gathered} q=9x^{\frac{1}{2}} \\ Onsquaringbothsides, \\ {q}^2=81x \\ x=\frac{q^2}{81} \end{gathered}$$

The total cost function is the sum of the firm's fixed cost and variable cost. The value fixed cost is $1000. The value of variable cost is the product of cost on each variable input. Thus, the total cost of the firm is:

$$\begin{gathered} C\left(q\right)=FC+VC \\ =1000+4000X \\ =1000+4000\left(\frac{q^2}{81}\right) \end{gathered}$$

The total cost function of the firm is c(q) = 1000+4000/81q²

The equation for the supply curve will be given by equating the price with the firm's marginal cost.The supply curve equation is formulated below:

$$\begin{gathered} P=MC \\ =\frac{d\left(TC\right)}{dq} \\ =\frac{d\left({\displaystyle \frac{4000q^2}{81}}\right)}{dq} \\ =\frac{8000q}{81} \end{gathered}$$

The equation for the supply curve is P = 8000q/81.

The firm will maximize its profit at the quantity where the marginal revenue from it marginal cost of it.The value of the price is the firm's marginal revenue for that quantity. The quantity produced by the firm is calculated below by equating MC with the MR.

$$\begin{gathered} MR=MC \\ 1000=\frac{8000q}{81} \\ q=\frac{1000\times 81}{8000} \\ q=10.13 \end{gathered}$$

The firm will produce 10.12 units of output.

The profit of a firm is the difference between the total revenue earned at this quantity and the total cost.

$$\begin{gathered} Profit\left(\$\right)=TR-TC \\ =\left(1000\times 10.13\right)-1000+\left(4000\times \frac{10.{13}^2}{81}\right) \\ =10130-1000+5067.5 \\ =14197.5 \end{gathered}$$

The firm will earn a profit of $14197.5.

The following graph shows the output and profit level of the firm at the price level of $1000.The gap between the total cost curve (TC) and the total revenue curve (TR) is the profit of the firm in the graph.

In the graph, the output is shown on the x-axis and the price on the y-axis. MR is the total revenue curve (TR), and TC is the total cost curve. Both the curves are derived by putting the value of Q = 1, 2, 3, 4, and so on. The gap between A and B shows the profit level at the price level of $10000. The output at this price level is marked on the x-axis.