Question

A firm sells its product in a perfectly competitive market where other firms charge a price of $\mathbf{\$}\mathbf{90}$per unit. The firm's total costs are $\mathit{C}\left(Q\right)\mathbf{=}$$\mathbf{50}\mathbf{+}\mathbf{10}\mathit{Q}\mathbf{2}{\mathit{Q}}^{\mathbf{2}}$.

a.How much output should the firm produce in the short run?

b. What price should the firm charge in the short run?

c. What are the firm's short-run profits?

d. What adjustments should be anticipated in the long run?

Short answer

Answer

1. 20 units

2. $90.

3. $1510

4. In the long-run, the firm will continue to operate on the same way as in the short-run.

Step-by-step solution

Calculating an optimal output in the short-run:

a.

At first, we will calculate an optimal output in the short-run.

Given,

Price in the market $\left(P\right)\mathbf{=}\mathbf{90}\mathbf{}\mathit{v}$dollars

Total cost function: $\mathit{C}\mathbf{\left(}\mathbf{Q}\mathbf{\right)}\mathbf{=}\mathbf{50}\mathbf{+}\mathbf{10}\mathit{Q}\mathbf{+}\mathbf{2}\mathit{Q}$

Fixed cost $\mathbf{=}\mathbf{50}$dollars

Variable cost$\mathbf{=}\mathbf{10}$

We then can determine marginal cost function from the total cost function

$$\begin{gathered} \mathit{C}\mathbf{\left(}\mathbf{Q}\mathbf{\right)}\mathbf{=}\mathbf{50}\mathbf{+}\mathbf{10}\mathit{Q}\mathbf{+}\mathbf{2}{\mathit{Q}}^{\mathbf{2}} \\ \mathit{M}\mathit{C}\mathbf{\left(}\mathbf{Q}\mathbf{\right)}\mathbf{=}\frac{\mathbf{dC}\mathbf{\left(}\mathbf{Q}\mathbf{\right)}}{\mathbf{Q}}\mathbf{=}\mathbf{10}\mathbf{+}\mathbf{4}\mathit{Q} \end{gathered}$$

So, if$\mathit{P}\mathbf{=}\mathit{M}\mathit{C}$, this means that $\mathbf{90}\mathbf{=}\mathbf{10}\mathbf{+}\mathbf{4}\mathit{Q}$

or,$\mathit{Q}\mathbf{=}\mathbf{(}\mathbf{90}\mathbf{-}\mathbf{10}\mathbf{)}\mathbf{/}\mathbf{4}\mathbf{=}\mathbf{20}$

Thus, the optimal level of output is 20 units.

Finding the price of the short run:

b.

The firm should charge the same price as other firms in the perfectly competitive market.

So, the required price is $90

Calculating the firm's profit:

c.

When we know total cost, price and optimal output we can calculate firm's profit.

$$\begin{gathered} \mathit{P}\mathit{r}\mathit{o}\mathit{f}\mathit{i}\mathit{t}\mathbf{=}\mathit{T}\mathit{R}\mathbf{-}\mathit{T}\mathit{C} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left(P\times R\right)\mathbf{-}\mathit{T}\mathit{C} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left(90\times 20\right)\mathbf{-}\left(50+10.20+2.20\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1800}\mathbf{-}\left(50+200+40\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1510} \end{gathered}$$

Thus, the firm will gain a profit of $1510.

Explaining the long-run:

d.

In the long-run, the firms should cover their total variable cost. Here, the firm is covering total cost in the short-run, because it gained a profit. So, this firm should continue to operate on the same way as in the short-run.