Exercise: competitive market

You are the owner of a firm with the following ATC, AVC, and MC:

$\\ ATC= 7+ \frac{0.04Q^2+525}{Q} \\ AVC= 7+ 0.04Q \\ MC= 7+0.08Q$

The firm is in a competitive market, and therefore is a price taker. You also know that the demand for this firm is perfectly elastic and is equal to $30.

Solve for the optimum quantity, and find the firm's economic profit (if any). Will the firm be able to maintain its profit in the long-run? Explain.

$\\ 1). 7+0.08Q=30 \rightarrow Q=287.5 \\ 2). ATC= 7+\frac{0.04(287.5)^2+525}{287.5}=20.326 \\ 3). \Pi= (30-20.326)\cdot (287.5)\approx 2781.3$

First, set MC=MR and solve for Q. We clearly see that the firm is able to maximized its profit when it produces 287.5 units. Since the MC curve is above the ATC curve at this point, the firm is earning an economic profit of about $2781.3. No, the firm will not be able to sustain this economic profit in the long-run. This profit will entice other firms to enter the market, causing a rightward shift in the supply curve, reducing the price until every firm makes a normal profit.

Let's graph that:

If a tax of $20 per units is imposed every firm in the market, the new demand for the firm is $34 (perfectly elastic), should the firm stay open in the short run? Show the effect of the tax on the market with a graph.

add 20Q to the TC formula to represent the tax. Then, the new ATC, AVC, and MC are:

$\\ ATC= 27+ \frac{0.04Q^2+525}{Q} \\ AVC= 27+ 0.04Q \\ MC=27+ 0.08Q$

To find the optimum point and the profit (if any), set MR=MC, solve for Q. Then, plug in this Q into the ATC equation, and multiply the difference in dollars times Q to find the profit/loss.

$\\ 1). 27+ 0.08Q=34\rightarrow Q=87.5 \\ \\ 2). ATC= 27+ \frac{0.04(87.5))^2+525}{87.5}=36.5 \\ \\ 3). (34-36.5) \cdot 87.5= -218.5 \\ \\ 4). AVC<MC \textit{at } Q=87.5$

We see that the firm is earning a negative profit, but since at the optimum quantity the firm is able to pay for its VC, the firm will stay open in the short-run.

Let's graph it:

This is the effect of a per unit tax on the market:

In the long run, some firms will be forced out of the market, further reducing the supply and increasing the equilibrium price up until every firm in this market earns a normal economic profit.

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