14  Perfect Competition

For more information on these topics, see Allen, Doherty, Weigelt, and Mansfield Chapter 7: Perfect Competition.

No reading required!

14.1 Classwork 14: Perfect Competition

So far, we’ve explored two key aspects of a firm’s operations:

  1. Production functions: How firms turn inputs into outputs

  2. Cost functions: How production choices affect a firm’s costs

Now, we’ll examine the final piece contributing to a firm’s decision-making process: the competitive environment. A firm in a highly competitive market will make different choices about how much to produce compared to a monopolist (single seller) or an oligopolist (market with few sellers).

In perfectly competitive markets, firms are price-takers, meaning they cannot influence the market price. The market price is determined by the intersection of market supply and demand curves. Individual firms can sell as much as they want at the market price, and they’ll make that decision about \(q\) based on the market price and their own cost function.

  1. Suppose market demand is given by: \(p = 75 - 0.25 q\) and market supply is given by: \(p = 0.5 q\). Find the equilibrium price and quantity exchanged.

  2. Assume the firm has a total cost function given by \(TC = 100 + 2 q^2 + 10 q\). Calculate the firm’s marginal cost function.

  3. Write down the (price-taking) firm’s profit as a function of its total cost and the equilibrium price of the output good.

  4. What is the profit-maximizing choice of output for the firm?

  5. Use stat_function to plot the firm’s profit on the y-axis and \(q\) on the x-axis to verify your answer to part d was correct. You may want to zoom in or out, setting limits to the axis using xlim or ylim.

  6. Verify that at the firm’s profit-maximizing \(q\), the marginal cost is equal to the price of the output good.

  7. Fill in the blanks to interpret part f: In perfect competition where the firm is a price-(_____), the firm’s quantity choice which maximizes its profit is always where the marginal cost is equal to the price of the output. That’s because marginal cost often looks like a/an (_____)-facing parabola: at some point, it starts increasing and doesn’t stop increasing. Note that the (_____) is the amount by which selling the last unit decreased profit, and the price of the output is the amount by which selling the last unit (_____) profit. When marginal cost is less than the price, we’re adding to our profit and we should keep increasing \(q\). We should keep on increasing \(q\) until the marginal cost (_____) the price: that’s a signal we’ve sold too many, because that extra sale subtracts from our profit more than it adds to it. All the gains to profit are captured when the marginal cost (_____) the price of the output.

  8. There’s one condition on the \(MC = P\) rule: the firm can always choose \(q = 0\) if its profits are never positive. However, in the short-run, the firm’s fixed costs are sunk. What are the fixed costs in our example where \(TC = 100 + 2 q^2 + 10 q\)? In the short-run, if the market price is low enough, the firm will not be able to cover its variable costs, and it should produce 0. If the market price increases, then at some point, the firm starts being able to cover its variable costs and it should produce where \(MC = P\). That “shutdown point” is when the price equals the minimum average variable cost, because at that point, revenue equals total variable cost. In our example, what would the market price need to be for the firm to shut down?