Question

You are the manager of a firm that produces a product according to the cost function C(qi )=$\mathbf{160}\mathbf{+}\mathbf{58}\mathit{q}\mathit{i}\mathbf{-}\mathbf{6}\mathit{q}\mathit{i}\mathbf{+}\mathbf{2}\mathbf{}\mathit{q}\mathit{i}\mathbf{}\mathbf{3}$. Determine the short-run supply function if:

a. You operate a perfectly competitive business.

b. You operate a monopoly.

c. You operate a monopolistically competitive business

Short answer

Answer:

1. In perfect competition, the short-term supply function is MC curve above AVC curve.

2. The monopoly market doesn’t have a supply function.

3. In a monopolistic competition, determining the supply function in the short run is impossible

Step-by-step solution

Determining the short-term supply function in perfectly competitive business

a.

When the marginal cost curve is above the trough of the average variable cost curve in the short run, the notion of the perfectly competitive supply curve is applied.

In the short run, the firm's manager produces with the following cost function:

$\mathit{C}\left(q_1\right)\mathbf{=}\mathbf{160}\mathbf{+}\mathbf{58}{\mathit{q}}_{\mathbf{1}}\mathbf{-}\mathbf{6}{\mathit{q}}_{\mathbf{1}}^{\mathbf{3}}$

We can determine the marginal cost by deriving the cost function with respect to

$$\begin{gathered} \mathit{C}\mathbf{\left(}{\mathbf{q}}_{\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathbf{160}\mathbf{+}\mathbf{58}{\mathit{q}}_{\mathbf{1}}\mathbf{-}\mathbf{6}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathit{q}}_{\mathbf{1}}^{\mathbf{3}}\mathit{n} \\ \frac{\mathbf{dC}\mathbf{\left(}{\mathbf{q}}_{\mathbf{1}}\mathbf{\right)}}{{\mathbf{dq}}_{\mathbf{1}}}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{12}{\mathit{q}}_{\mathbf{1}}\mathbf{+}\mathbf{3}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathit{n} \end{gathered}$$

Thus,$\mathit{M}\mathit{C}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{12}{\mathit{q}}_{\mathbf{1}}\mathbf{+}\mathbf{3}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

The variable average cost can be calculated using the same initial cost function, with fixed costs (BLUE) accounting for 160 and variable costs accounting for the rest of the term (RED).

$\mathit{C}\mathbf{\left(}{\mathbf{q}}_{\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathbf{160}\mathbf{+}\mathbf{58}\mathit{q}{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathit{q}{\mathbf{1}}^{\mathbf{3}}$

Total Variable Cost: $\mathbf{58}{\mathit{q}}_{\mathbf{1}}\mathbf{-}\mathbf{58}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathit{q}}_{\mathbf{1}}^{\mathbf{3}}$

role="math" localid="1657020057437" $$\begin{gathered} \mathit{A}\mathit{V}\mathit{C}\mathbf{=}\frac{\mathbf{TVC}}{\mathbf{q}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{58}{\mathbf{q}}_{\mathbf{1}}\mathbf{-}\mathbf{6}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{q}}_{\mathbf{1}}^{\mathbf{3}}}{{\mathbf{q}}_{\mathbf{1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{6}{\mathit{q}}_{\mathbf{1}}\mathbf{+}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}\left(2\right) \end{gathered}$$

Now we can equate equations (1) andto satisfy the condition MC = AVC at its minimum level and solve for $\mathit{q}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}$:

$$\begin{gathered} \mathbf{58}\mathbf{-}\mathbf{12}{\mathit{q}}_{\mathbf{1}}\mathbf{+}{\mathbf{3}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{6}\mathit{q}\mathbf{+}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{3}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{-}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{=}\mathbf{12}{\mathit{q}}_{\mathbf{1}}\mathbf{-}\mathbf{6}{\mathit{q}}_{\mathbf{1}} \\ \mathbf{2}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{=}\mathbf{6}{\mathit{q}}_{\mathbf{1}} \\ \mathbf{2}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{-}\mathbf{6}{\mathit{q}}_{\mathbf{1}}\mathbf{=}\mathbf{0} \\ {\mathit{q}}_{\mathbf{1}}\mathbf{(}\mathbf{2}{\mathbf{q}}_{\mathbf{1}}\mathbf{-}\mathbf{6}\mathbf{)}\mathbf{=}\mathbf{0} \\ \mathbf{=}\frac{\mathbf{6}}{\mathbf{2}} \\ \mathbf{=}\mathbf{3} \end{gathered}$$

Therefore theMCandAVCcurve will intersect at q = 3

By substituting the obtained value of production ( q = 3 ) in the HCV equation:

$$\begin{gathered} \mathit{A}\mathit{V}\mathit{C}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{6}{\mathit{q}}_{\mathbf{1}}\mathbf{+}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}} \\ \mathit{A}\mathit{V}\mathit{C}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{6}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{+}{\mathbf{3}}^{\mathbf{2}} \\ \mathit{A}\mathit{V}\mathit{C}\mathbf{=}\mathbf{49} \end{gathered}$$

In short, in a completely competitive market, the supply function will be controlled in the near term by the marginal cost curve (MC), which is above the average variable cost (AVC).

Therefore, the short-term supply function in a perfectly competitive business is

$\mathit{P}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{12}{\mathit{q}}_{\mathbf{1}}\mathbf{+}\mathbf{3}{\mathit{q}}_{\mathbf{1}}^{\mathbf{2}}$

We can check this function by substituting the quantities q = 3 and it should give us the same price as the AVC

$$\begin{gathered} \mathit{p}\mathbf{=}\mathbf{58}\mathbf{-}\mathbf{12}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{+}\mathbf{3}{\mathbf{\left(}\mathbf{3}\mathbf{\right)}}^{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{49} \end{gathered}$$

Determining the short-term supply function in monopoly business

b.

Prices are decided by the inverse demand function in the case of a monopoly. Thus, this condition doesn’t require a supply function. The condition is governed by the relationship in which marginal revenues equal marginal expenses.

Determining the short-term supply function in monopolistically competitive business

c.

Despite the fact that monopolistic competition has characteristics of both perfect and monopolistic competition, firms maximize profits when marginal revenues equal marginal costs. The prices that maximize profit are determined by setting in the inverse demand function where the profit-maximizing output is also found.

Hence, in a monopolistic competition, determining the supply function in the short run is impossible.