4.3.4. The Profit Maximisation Problem: Graphical Representation

Using the material in sections 3.1, 3.2 and 3.3, let us graphically represent a firm’s profit maximisation problem in the short run. Consider Figure 4.6. Notice that the market price is p. Equating the market price with the (short run) marginal cost, we obtain the output level q₀. At q₀, observe that SMC slopes upwards and p exceeds AVC. Since the three conditions discussed in sections 3.1-3.3 are satisfied at q₀, we maintain that the profit-maximising output level of the firm is q₀.

In this chunk, we are looking at how we can visualize the process by which a firm maximizes its profits. The graph essentially plots the relationship between price, marginal costs, and average variable costs. The firm will continue to produce as long as the price they receive for their product (p) is greater than their average variable costs (AVC) and marginal cost (MC) equals the price. If we draw the demand curve (which is horizontal for a price-taker) at price p and compare it with the marginal cost curve (which generally slopes upwards), we can find the quantity q₀ where these two curves intersect, showing the optimal output level for profit maximization. This is an important concept in economic theory because it shows firms will not produce if they cannot at least cover their variable costs, which is essential for staying in business in the short run.

Think of a pizza shop deciding how many pizzas to make for the evening. The shop knows that it can sell pizzas at $10 each. They calculate their costs, and it turns out that the cost to make each additional pizza is$7. To maximize their profit, they will continue to make pizzas as long as they can sell them for $10 (which is greater than their$7 cost). If they reach a point where the cost to make another pizza exceeds $10, they will stop making more because they would be losing money. The graphically identified quantity (q₀) ensures they generate the highest profit under these conditions.

What happens at q₀? The total revenue of the firm at q₀ is the area of rectangle OpAq₀ (the product of price and quantity) while the total cost at q₀ is the area of rectangle OEBq₀ (the product of short run average cost and quantity). So, at q₀, the firm earns a profit equal to the area of the rectangle EpAB.

In this chunk, we delve into the financial aspect of profit maximization. At the profit-maximizing output level (q₀), we can calculate the total revenue and total costs geometrically based on rectangular areas in the graph. Total revenue (TR) is calculated by multiplying the price (p) by the quantity sold (q₀), which represents the area of rectangle OpAq₀. On the other hand, total cost (TC) takes into account the average cost (AC) of production times the same quantity, giving us the area of rectangle OEBq₀. Profit is then derived by subtracting total costs from total revenues, visually represented as the area EpAB in the graph. Profit maximization occurs where this area is largest.

Imagine the same pizza shop again. If the shop sells 15 pizzas at $10 each, their total revenue will be$150 (15 * $10). If it costs them$5 per pizza for ingredients and labor (amounting to $75 for 15 pizzas), they realize a profit of$75. In terms of our area metaphor, if the revenue rectangle is $150 and the cost rectangle is$75, the area that represents profit would be everything in between, coloring in a visual rectangle of size $75, signifying the shop's profit.

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