How to derive a supply curve

Isoquants and isocosts to fixed, marginal and average costs

[This essay is intended for students in my law and economics and my microeconomics class (for law students). Read this essay one time through, ignoring the queries that are interspersed. Go back to the queries after you feel you have understood the essay. The queries test whether you have understood enough to develop a richer understanding of the topic of the essay (deriving a supply curve). These instructions will also apply to future essays.

I have bolded certain words. These are terms that you should know going forward. If you do not understand them, ask your favorite foundational AI model about them.

I will post the python notebook that generates all the figures in this essay in case you are interested in reproducing them.]

In my last essay, I explained how to use a combination of a budget line and preferences to derive a consumer demand curve. A key feature of the derivation is that a consumer’s optimal choice of what to buy, when she has a limited budget, is to look for the bundle of goods that puts her on the highest indifference curve and that she can afford given her budget.

In this essay, I am going to derive a supply curve for a company. A critical step is that the firm minimize the cost of producing goods. We are going to see analogies to indifference curves and budget lines, so the logic behind a supply curve will resemble the logic behind demand curves.

At the end of the essay I will introduce two concepts, aggregate demand and supply, that we will use to determine the allocations, i.e., the combination of price and quantity, that a market generates.

Cost minimization

Imagine you want to start a coffee shop in a hipster neighborhood. (I have a reasonable amount of experience with this, albeit from the consumer’s perspective.) To begin with you need a space to set up shop. I suggest going to a neighborhood that used to be zoned for factories, but is now mainly residential. Find a building with exposed brick, and large windows—hipsters love that. You need to decorate in that faux-vintage style that is in vogue: subway tiles, antiqued mirrors, curvy wood chairs, marble-top tables. You also need to buy or rent a fancy coffee machine, e.g., a La Marzocco machine (to show people you’re serious about coffee), and mugs with sassy quotes and classic Italian espresso cups. Finally, you’ll need inputs into coffee: beans, sugar, regular milk, skim milk, soy milk, almond milk, coconut milk, rice milk, and, of course, cashew milk. Economists call all these items (physical) capital.

But coffee doesn’t make itself. So you need a few baristas! Just remember that people with tattoos or beards, or both, make more compelling coffee. Experience is optional. Economists call baristas labor.

Your coffee shop will use a combination of capital and labor to produce coffee. But what combination? Should you buy an automated machine so fewer baristas can make more coffee? Or do you want your baristas to labor over each cup, measuring just the right amount of beans, packing the espresso just right, and designing beautiful coffee art with cream?

At the end of the day, you are a for-profit company. And you are in a neighborhood that, in all likelihood, has a half dozen other hipster coffee shops you’ll compete with. If you want to survive, or even make a profit, you have to produce your oat-milk cortados at minimal cost.

The figure below helps us visualize your problem, and the solution. On the x-axis I show the quantity of labor inputs (measured in full time workers). On the y-axis I show the quantity of capital inputs (which, for simplicity I will treat as a single input, measured in a unit like space or weight). The curved line is an isoquant: all the combination of inputs that produces 100 cups of coffee per day—the amount of coffee you anticipate selling each day. (FYI: “Iso” means “same”. And the further an isoquant is from the origin (0,0), the greater the quantity it represents.) An isoquant resembles an indifference curve, except an indifference curve shows the combination of goods (output, not inputs) that produce the same level of utility (not output). Your cafe’s problem is to choose the combination of inputs that produces 100 cups at the lowest cost.

Aside: An isoquant represents a firm’s “technology”. If another firm can produce the same 100 cups of coffee with less of each input, then its isoquant would be closer to the origin. And, as we shall see, it will be able to produce coffee at a lower cost than your firm. Economists would also say that the other firm has a better technology than you. Maybe they have a better machine, or a technique for percolating beans faster, or a process that makes the barista able to make more caramel macchiatos per hour. Technology can encompass many aspects of production, not just complexity of machinery.

To solve this problem, you need to know how much each input costs. You need to know workers’ wages (w) and the price of physical capital, often called a rental price (r). With this information you can draw an isocost curve, which shows different combinations of labor and capital that cost the same number of dollars in total. The isocost curve has a negative slope equal to -w/r, the relative price of labor to capital. This looks a lot like a budget line, except that the slope of the budget line is the relative price of good 1 (since good 1, rather than labor, is on the x-axis) and good 2 (since good 2, rather than capital is on the y axis). Both the isocost and the budget line give you the total dollar cost of different combinations of items, be they inputs or consumption goods. The closer an isocost curve is to the origin (0,0), the less the cost of inputs. Our figure draws a few isocosts, each representing a different level of total cost or expenditure.

So how do you determine which combination of inputs produces 100 cups of coffee per day at minimum cost? Look for the isocost curve that intersects the isoquant for 100 cups of coffee but represents the lowest expenditure of money. That minimal isocost curve is the one that touches the relevant isoquant just once. Unless that one contact point is along one of the axes, that contact point will be a point of tangency between the isocost and isoquant, i.e., a point where both curves have the same slope. That contact point also represents the cost-minimizing combination of inputs that will produce 100 cups of coffee. We know it can produce that much because it is on the 100 cup isoquant. We know it is cost minimizing because it is on the lowest isocost that touches the 100 cup isoquant.

Query. Can you explain why the isocost that intersects the isoquant twice cannot be the cost minimizing isocost?

This should remind you of the optimal choice of consumption goods x1 (apples) and x2 (books) in my “Derivation of the Demand Curve” essay. There the optimal combination was the point on the budget constraint that got you to the highest indifference curve. By contrast, here, we are looking for the point on the isoquant that gets you to the lowest isocost. There we took the budget as given, and considered different indifference curves, because we were maximizing happiness given our fixed budget constraint. Here we took the isoquant (i.e., the analog to the indifference curve) as given, and considered different isocost curves (i.e., the analog to budget lines), because we are minimizing cost given the fixed quantity of output we wanted to produce.

The firm’s supply curve

We took it as given that your coffee shop would sell 100 cups of coffee per day. But it is possible that you do worse, because the market is competitive, or you do well, because you do have a better hipster vibe than the other neighborhood shops. (Or, maybe, you make better coffee?) In any case, we will want to figure out the optimal combination of labor and capital to minimize costs at each level of output. That will be given by the tangency between each relevant isoquant, and the various isocost curves. If we draw them out, we can see the quantity expansion path, i.e., how labor and capital inputs change as your coffee shop produces more and more cups of coffee.

But this expansion path also reveals a firm’s supply curve. Each isoquant is associated with a quantity and, via the cost-minimizing combination of inputs, an isocost that gives the cost of that quantity. Now, let’s make a table with different quantities given by the isoquants in one column. And another column with incremental cost with each additional unit of quantity.

If we make a new plot with quantity on the x-axis and the cost on the y-axis, we see how much each additional unit costs the firm to make. The units here are 10 cups per day, so the graph shows the marginal cost of each additional 10 cups per day your coffee shop makes. This is your coffee shop’s supply curve!

Different types of costs

Let’s revisit your firm’s costs. Recall the different types of physical capital you need to start a coffee shop. On the one hand, there were items like the building, furniture and decor, and machinery. On the other hand there were inputs like beans, sugar and various milks. These differ in an important way. Can you guess how?

The first group has costs that do not increase as you increase the number of cups per day that your shop makes. Even if you rent the building or the machine, your rent does not directly increase with the number of cups you make. The second group, however, is proportional to how many cups you make each day. For every 10 more cups, you need more beans. We call costs that do not rise with quantity fixed costs. And costs that do scale with quantity are called marginal costs.

Query. Physical capital is not the only category of costs that can be sorted into fixed cost and marginal cost subcategories. Labor costs can as well. Are barista’s fixed or marginal costs? Does it matter whether you pay them a salary to work for 40 hours per week or hire them by the hour? How does the productivity of baristas, i.e., how many cups they can make each hour affect how you categorize labor costs into fixed and marginal cost buckets?

The figure below plots the fixed costs (F) and marginal costs (MC) on a figure with quantity on the horizontal axis and dollars on the vertical axis. You will notice that fixed costs are incurred just once, but even before you produce a single cup. By contrast, marginal costs are incurred with every cup you produce. Importantly, marginal costs are different than your fixed cost: marginal cost is the incremental cost of producing another cup of coffee per day, ignoring your fixed costs.

I drew your marginal costs as increasing with the number of cups you produce. It’s not because you need more beans per cup. Rather, it is because more cups have other costs that pile up. For example, the more beans you grind, the more frequently you have to clean your espresso maker and perhaps even have it serviced. And as your barista’s have to make more cups, they are more likely to bump into each other and spill a cup, requiring them to remake the cup. These costs aren’t just proportional to cups, but increasing per cup with each cup. If you need to clean your machine every 10 days at 100/cups per day, you might need to clean your machine every 3 days at 200 cups per day. If you go from 100 to 200 cups per day, your spills don’t go from 10 to 20, they may go from 10 to 25. An upward sloping marginal cost curve is not a feature that is unique to coffee shops. It is a common feature across products and sectors. So this aspect of your coffee shop’s costs, like your fixed cost, is fairly general.

Finally, I want to discuss the concept of average cost and economies of scale. The average cost of producing, say, 100 cups per day, is your total cost (including both fixed and marginal costs) divided by the number of cups per day.¹ You can compute this average cost for each quantity of output, e.g., 100 per day, 200 per day, etc. There are 2 important differences between average and marginal costs. One is that the average cost includes fixed costs. The other is that marginal costs measure the cost of the last unit produced. By contrast, average cost is the (average) cost of every unit until the last unit.

As the figure shows, average costs are U-shaped. Average cost starts above fixed cost for the first unit (the first 10 cups per day) because producing the first unit entails both the fixed cost plus the marginal cost of the first unit. After that, however, average costs fall. Why? The next few units all have similar marginal cost, according to the marginal cost curve, but share the fixed cost. So the average cost falls by 1/q for every unit q you produce. This is a region over which we say the firm has increasing returns to scale, meaning, the more it produces, the less each unit costs on average. After marginal costs start rising, however, they eventually stop the average cost curve from falling. After that average costs start creeping up, completing the “U” shape.²

Revisiting the supply curve

The most important lesson to draw from the figure above is that the firm’s supply curve is the portion of the marginal cost curve that is above minimum average cost. This means that the supply curve has two features.

First, the supply curve begins only when quantity is above the level q* that minimizes average cost. Why? If quantity is below that level, increasing it would obviously lower average cost, and thus increase profit. So you are better off producing more. Profit is equal to revenue minus cost. Revenue is price p times quantity q. Your coffee shop is in a competitive market, so you don’t really control price – it is set by the market.³ Cost is fixed cost, plus per-unit cost c(q) times quantity. (Note that c(q) may be increasing with q, i.e., unit costs are rising.) So profit is

Profit = pq - F - c(q) q

where the “(q)” signifies that per unit cost c changes with quantity. If we divide by q, then average profit is

Average profit = p - (F + c(q) q)/q = p - AC(q)

Because q is below q’, average cost falls with quantity below q. That means average profit rises with more quantity. Total profit is average profit times q, so profit also rises with quantity. Your coffee shop would make more profit if it increased quantity until at least q’, regardless of what the price is.

Second, the supply curve is on the marginal cost curve rather than the average cost curve. Why? A profit maximizing firm thinks on the margin. Suppose you are at some level of quantity q that is above q’. What is the value of increasing quantity by 1 unit? On the upside, you increase revenue by the price of a cup of coffee. But on the downside, your marginal cost increases.⁴ You can ignore fixed cost, because that does not change with quantity. If the market price is above MC(q), then you should produce more, because the marginal benefit of an additional unit is greater than the marginal cost of that unit. And you should produce less if the market price is below MC(q), because that last unit costs more than the revenue it generated. Indeed, you should produce the quantity q** where the marginal cost of production is equal to price – because you want to produce more below that point, and less above that point. Note that this calculation of optimal production did not depend on average cost. It depended only on marginal cost.

Average cost is relevant to determining the minimum quantity with which a firm will enter the market. But the exact amount it produces above that minimum depends on marginal cost.

Before I conclude our derivation, you may want to know how to square our first depiction of the firm’s supply curve, which was derived from isoquant and isocost curves, with the second, which is derived from average cost and marginal cost curves. The answer is that the firm’s first unit of production is at q’ which is the level at which the firm’s average costs are at their minimum. Moreover, the incremental cost going from the isoquant for q to q+1 ignores fixed cost. So the supply curve in both figures represented marginal cost, not average cost.

1. If your rent for the shop is on a monthly or annual basis, then you should change the time scale to be cups per month or year, e.g., 100 cups per day x 30 days/month.

2. Indeed, the marginal cost curve goes through the point where average cost is at its minimum. We can use a little calculus to show that. Suppose that total cost is TC(q) = F + MCq. Then average cost is AC(q) = T(q)/q. To compute the minimum of average cost, we take the derivative of AC with respect to q and set it to set it equal to 0: dAC(q)/dq = AC’(q) = T’(q)/q - T(q)/q^2 = 0, where f’(x) means the derivative of f with respect to x. Bring T(q)/q^2 over to the other side and multiply both sides by q to get: T’(q) = T(q)/q. Realize that T’(q) = MC. And T(q)/q is AC(q). So you get the minimum where MC = AC(q). (For advanced students who wonder what happens when marginal costs increase in q, i.e., TC(q) = F + MC(q)q, note that marginal cost in this case is still TC’(q), except that that marginal cost is MC’(q)q + MC(q), not just MC(q).

3. If you tried to increase your price, consumers would get their coffee at another cafe.

4. For those of you who are following the math, marginal cost is not just c(q). Marginal cost is how much total cost changes with quantity, i.e., MC(q) = d[c(q)q]/dq = c’(q)q + c(q). So if c is a function of q, then the full marginal cost is more than just c(q).