Costs, Revenue and the Profit-Maximising Output

The short run and the long run

Before measuring costs, we need to know what a firm can and cannot change. Economists split time into two periods. They are not measured in days or months.

Take a small restaurant in Cork. Next week it cannot build a bigger kitchen or move to a new building, so the size of the premises and the ovens are fixed. It can hire an extra waiter or buy more ingredients, so labour and ingredients are variable. Over several years it can expand, move or install a new kitchen. That is the long run.

Total, average and marginal revenue

Revenue is what the firm receives from selling its output.

A firm that sells at a constant price. An ice-cream stall in Sligo sells cones at €3. If it sells 200 cones, then $TR = 3 \times 200 = €600$ and $AR = 600 \div 200 = €3$. Selling a 201st cone brings in another €3, so $MR = €3$. When the price is constant, P = AR = MR. This is the case for a price taker, a firm so small that it cannot influence the market price.

A firm that must cut its price to sell more. Many firms face a downward-sloping demand curve. To sell an extra unit, they must lower the price on all units. Then MR is less than the price.

The 2nd unit sells at €9, but the firm also loses €1 on the first unit, which now sells at €9 instead of €10. So its extra revenue is $9 - 1 = €8$, not €9.

Costs: fixed, variable, total, average and marginal

In the short run a firm has two kinds of cost.

How the cost curves behave

Average fixed cost always falls as output rises, because the same fixed cost is spread over more units. MC usually falls at first and then rises. This comes from the law of diminishing marginal returns: in the short run, adding more and more of a variable factor to a fixed factor (more chefs in one small kitchen) gives smaller and smaller extra output. So each extra unit costs more to make.

The MC curve cuts the average cost curve at its lowest point. The reason is the same as for exam marks. If your next test mark is below your average, your average falls. If it is above, your average rises. So:

• When MC < AC, the average cost is falling.

• When MC > AC, the average cost is rising.

• When MC = AC, the average cost is at its minimum.

Profit and break-even

In economics, the cost of running a firm includes normal profit, the minimum reward the owner needs to stay in business. You will study this further in later work. For the calculations on this page, profit simply means $TR - TC$.

When the price is constant and variable cost per unit is constant, each unit sold contributes $P - AVC$ towards covering fixed costs. So:

The profit-maximising output: MC = MR

Back to the bakery. It should bake the 151st loaf if that loaf adds more to revenue than to cost, that is, if $MR > MC$. It should stop when the next unit adds more to cost than to revenue, that is, if $MC > MR$.

• If $MR > MC$, the next unit adds more to revenue than to cost. Profit rises, so produce more.

• If $MR < MC$, the next unit adds more to cost than to revenue. Profit falls, so produce less.

• At $MC = MR$, there is no gain from changing output.

Using a table

When MR and MC are not exactly equal in a table, choose the last unit for which MR ≥ MC.

Using equations (Higher Level)

When costs and revenue are given as formulae, MC and MR are the derivatives:

• Constant price $P$: $TR = PQ$, so $MR = P$.

• Falling price, such as $P = a - bQ$: $TR = aQ - bQ^2$, so $MR = a - 2bQ$. The MR line is twice as steep as the demand line.

The second-order condition shows that MC = MR gives a maximum and not a minimum. Profit is $\pi = TR - TC$. At the maximum, $\dfrac{d\pi}{dQ} = MR - MC = 0$ and $\dfrac{d^2\pi}{dQ^2} < 0$. That means MC must be rising faster than MR, or in other words MC cuts MR from below. At a point where MC is falling, MC = MR would be a minimum of profit. Producing a little more would raise profit.

Try it: the widget shows $MC = 10 + 2Q$ and $MR = 50$ (the same cost function as the Waterford firm in the exercise below, $TC = 200 + 10Q + Q^2$). Find where they cross.

Should the firm stay open?

A loss does not always mean the firm should shut down in the short run. Fixed costs must be paid whether or not it produces. So:

• If $P \ge AVC$, the firm covers its variable costs and has something left towards fixed costs. It should keep producing in the short run, even at a loss, because the loss is smaller than the fixed costs.

• If $P < AVC$, each unit loses money before fixed costs are counted. The firm should shut down, and the loss is then just the fixed costs.

In the long run, a firm that cannot cover all its costs should leave the industry.

Scale in the long run. In the long run a firm can change its scale. Economies of scale are falling long-run average costs as output grows (for example, buying ingredients in bulk or spreading marketing costs). Diseconomies of scale are rising long-run average costs when a firm becomes too big (for example, poor communication and slow management). These help explain the size of firms and why some aim for growth.

Is profit the only objective?

MC = MR assumes the firm wants maximum profit. Many firms do not behave this way, and you should be able to discuss why.

Critique of the profit-maximising assumption. In favour: firms that ignore profit for long cannot survive, shareholders and lenders demand returns, and competition punishes inefficiency, so profit is a good first model. Against: in large firms owners and managers are different people with different aims (the principal-agent problem), managers rarely know MC and MR exactly, and many firms value reputation, staff and sustainability. So MC = MR is a useful benchmark, but it does not describe every firm.

Comparison with profit maximisation. Take a firm with $P = 100 - 2Q$. Then $TR = 100Q - 2Q^2$ and $MR = 100 - 4Q$. A revenue maximiser sets $MR = 0$, so $Q = 25$. A profit maximiser with, say, a constant $MC = 20$ sets $100 - 4Q = 20$, so $Q = 20$. The revenue maximiser always produces more and sells at a lower price, with lower profit, as long as MC is positive.

Back to the beginning: the Galway bakery should bake the 151st loaf only if its marginal revenue (€4 at a constant price) is at least its marginal cost. It stops at the output where MC rises to meet MR.

Summary

• Short run: at least one factor is fixed. Long run: all factors are variable.

• Revenue: $TR = P \times Q$, $AR = TR \div Q = P$, $MR = \Delta TR \div \Delta Q$. For a price taker, $P = AR = MR$. If the firm must cut its price to sell more, $MR < P$.

• Costs: $TC = FC + VC$, $ATC = AFC + AVC$, $MC = \Delta TC \div \Delta Q$. MC rises because of diminishing marginal returns and cuts AC at its minimum.

• Profit $= TR - TC$. Break-even output (constant price and AVC) $= FC \div (P - AVC)$.

• Profit maximisation: produce where $MC = MR$ with MC rising. In the short run, keep producing if $P \ge AVC$.

• Other objectives: sales maximisation, growth, satisficing, social and ethical goals and survival can lead a firm to produce a different output from the profit-maximising one.

Key terms: short run, long run, fixed cost, variable cost, total revenue, average revenue, marginal revenue, marginal cost, break-even, price taker, diminishing marginal returns, profit maximisation, sales maximisation, satisficing.