4.3. Profit Maximisation

A firm produces and sells a certain amount of a good. The firm’s profit, denoted by π, is defined to be the difference between its total revenue (TR) and its total cost of production (TC). In other words, π = TR – TC. Clearly, the gap between TR and TC is the firm’s earnings net of costs.

Profit is essentially what a firm earns after accounting for all its costs. To calculate profit, firms look at their total revenue, which is what they make from selling their goods, and subtract the total costs they incur to produce those goods. Thus, profit can be thought of as the leftover money after expenses are paid.

Imagine running a lemonade stand. You sell lemonade for $1 a cup. If you sell 100 cups, your total revenue is$100. If your costs for lemons, sugar, and cups amount to $40, your profit is$100 (revenue) - $40 (costs) =$60. This leftover $60 is your profit.

A firm wishes to maximise its profit. The firm would like to identify the quantity q at which its profits are maximum. By definition, then, at any quantity other than q₀, the firm’s profits are less than at q₀. The critical question is: how do we identify q₀?

To maximize profit, a firm needs to find the exact amount of goods (quantity q₀) it should produce. If the firm produces less or more than this quantity, its profits will not be as high as they could be. The firm thus seeks to determine q₀ by analyzing its profits at different levels of production.

Think about baking cookies. If you bake 12 cookies, you might find your profits are high. But if you bake only 6, you lose potential sales; if you bake 24, you might waste ingredients and time if you can't sell them all. The goal is to find that sweet spot of 12 cookies where you maximize your profit.

For profits to be maximum, three conditions must hold at q₀: 1. The price, p, must equal MC; 2. Marginal cost must be non-decreasing at q₀; 3. For the firm to continue to produce, in the short run, price must be greater than the average variable cost (p > AVC); in the long run, price must be greater than the average cost (p > AC).

These three conditions ensure that the firm is not only covering its costs but also maximizing its profit. The first condition means the price of the product equals the cost to make one additional unit (marginal cost). The second condition implies that costs shouldn't be going down as more is produced (otherwise, producing more would still be profitable). The final conditions ensure that the firm is not losing money in the short and long run.

Consider a pizza shop. If they sell a pizza for $15, and it costs$10 to make, they're in profit. But they also need to ensure that as they make more pizzas, the cost does not keep dropping unexpectedly (which could provide false indicators), and they must earn more than their costs overall, ensuring sustainable business.

Profits are the difference between total revenue and total cost. Both total revenue and total cost increase as output increases. Notice that as long as the change in total revenue is greater than the change in total cost, profits will continue to increase. Therefore, we can conclude that as long as marginal revenue is greater than marginal cost, profits are increasing. It follows that for profits to be maximum, marginal revenue should equal marginal cost. In other words, profits are maximum at the level of output (which we have called q₀) for which MR = MC.

To maximize profits, the additional revenue from selling one more unit (marginal revenue, MR) must equal the additional cost of producing that unit (marginal cost, MC). If MR is higher than MC, it indicates that the firm can increase profits by producing more. Conversely, if MR is lower, the firm is losing profits and should decrease production.

Imagine a coffee shop. If selling one additional cup of coffee earns them an extra $5 (MR), but it costs them$4 to make that cup (MC), they're better off making that extra cup, as they're still profiting by $1. The shop should continue this until MR equals MC for optimal profit.

Consider the second condition that must hold when the profit-maximizing output level is positive. The marginal cost curve cannot slope downwards at the profit-maximising output level. This means that if marginal cost is still falling, then the firm has room to increase output and potentially increase profits. Thus, if the marginal cost curve slopes downward at a certain output level, it cannot be at the profit-maximizing point.

If the marginal cost is decreasing as output increases, this means the firm can still produce more at a lower cost, which contradicts the notion of maximizing profit because they haven’t reached their full earning potential. For profit maximization, the marginal cost curve must trend upwards or at least plateau, indicating that firms are fully utilizing their capacity.

Think of a car factory. If the cost to produce more cars decreases as production ramps up, the factory can keep producing. If they hit a point where producing one more car costs more than it did for the last while still making a profit, they’ve reached a peak of their optimal output and must not go beyond it.

Consider the third condition that must hold when the profit-maximising output level is positive. In the short run, the price must be higher than the average variable cost (p > AVC); in the long run, the price must be higher than the average cost (p > AC).

This condition ensures that the firm's revenue from selling goods is enough to cover not only its marginal costs but also its average costs. Specifically, in the short run, firms should at least cover variable costs to keep operating. In the long run, firms should cover all costs, including fixed and variable costs.

Consider a bakery that sells bread. If the price of a loaf is $3 but the variable costs (like flour and water) are$2, the bakery is covering its production costs and making a profit. However, if the price drops below $2, they'll stop baking bread as they cannot cover their ingredient costs, and in the long term, if they cannot cover all costs, they'll shut down.

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