Production

Demand has been fully described, and nothing so far says where the goods come from.

The other blade of Marshall's scissors starts with technology rather than with money. Before a firm can be asked what price it will supply at, there has to be a statement of what it is physically able to make, and that statement is the object this lesson builds. Almost everything in it turns out to be the consumer's problem from three lessons ago with different labels on the axes, which is worth noticing, because it means the calculus is already familiar and only the interpretation is new.

The production function

A production function gives the maximum output obtainable from given quantities of inputs. With two inputs, labour $L$ and capital $K$ ,

q = f (L, K)

The word "maximum" is doing quiet work: the function already assumes the firm is not wasting anything, so technical inefficiency has been defined away and the only question left is which efficient point to choose. That is a real restriction, and the empirical literature on firm productivity finds enormous dispersion in output per input among plants making identical products, which the function as written cannot represent.

Capital here means the stock of productive equipment, buildings and machines, not money. This is the most persistent vocabulary trap in the subject. A firm that borrows a million pounds has acquired finance; when it spends the money on a lathe it has acquired capital. The distinction matters because the production function is a physical relationship and money does not appear in it anywhere.

The short run is defined as the period over which at least one input is fixed, and the long run as the period over which all of them vary. These are not calendar durations. For a market stall the long run is a week; for a nuclear plant it is a decade. The definition is about which decisions are still open, and it is why the same firm faces two different cost structures depending on the question being asked.

Total, average and marginal product

Hold capital fixed and vary labour, which is the short-run experiment. Three curves come out of it. Total product is $q$ itself. Average product is ${AP}_{L} = q / L$ , output per worker. Marginal product is ${MP}_{L} = \partial q / \partial L$ , the output added by one more worker with the equipment unchanged.

The relationship between the last two is pure arithmetic and holds for any average and its associated marginal. Differentiate ${AP}_{L} = q / L$ by the quotient rule:

\frac{d}{d L} (\frac{q}{L}) = \frac{L \cdot {MP}_{L} - q}{L^{2}} = \frac{{MP}_{L} - {AP}_{L}}{L}

So average product rises exactly when marginal exceeds average, falls when it is below, and is stationary where they are equal. A new worker better than the current average pulls the average up. This is the same fact as a batting average rising after a good innings, and the identical argument reappears in the next lesson with costs in place of products, which is where it does most of its work.

Example. A workshop with fixed premises has $q = 30 L^{2} - L^{3}$ units per week from $L$ workers. Find total, average and marginal product at $L = 6$ and $L = 10$ , and say what is happening to average product.

At $L = 6$ : $q = 30 (36) - 216 = 864$ , so $AP = 144$ . The marginal product is ${MP}_{L} = 60 L - 3 L^{2} = 360 - 108 = 252$ . At $L = 10$ : $q = 3000 - 1000 = 2000$ , $AP = 200$ , and $MP = 600 - 300 = 300$ . In both cases marginal exceeds average, so average product is still climbing, and indeed it rose from 144 to 200 between the two.

Now you. Same workshop. Find total, average and marginal product at $L = 18$ , and say what is happening to average product there.

Answer

$q = 30 (324) - 5832 = 3888$ , so $AP = 216$ . The marginal product is $1080 - 972 = 108$ , well below the average, so average product is falling. The turning point is where they are equal: $30 L - L^{2} = 60 L - 3 L^{2}$ gives $2 L^{2} = 30 L$ , so $L = 15$ , where both equal 225.

Diminishing marginal returns

In that example the marginal product rises to a peak at $L = 10$ and falls after it, reaching zero at $L = 20$ where total output is at its maximum of 4000. The falling portion is the law of diminishing marginal returns: holding other inputs fixed, the marginal product of a variable input eventually falls.

Three qualifications are essential and routinely dropped.

It is a short-run statement. The whole content of it is that something else is being held fixed, so each additional worker gets a smaller share of the same machines and floor space. Remove the fixed input and the law says nothing.

It says eventually. Early increases in marginal product are entirely normal, and the workshop shows them up to ten workers, usually because specialisation becomes possible once there are enough people to divide the tasks.

And it is not a theorem. It is an empirical regularity, defended by an argument from absurdity: if it failed, the world's wheat could be grown in one flowerpot by adding labour and fertiliser indefinitely. Anne Robert Jacques Turgot stated it for agriculture in 1767, Malthus built his population theory on it in 1798, and it has survived because no counterexample has appeared, not because anyone proved it.

Isoquants and the technical rate of substitution

Now let both inputs vary, which is the long-run problem. Fix an output level $\overline{q}$ and look at the set of input combinations achieving it: $f (L, K) = \overline{q}$ . That level set is an isoquant, and it is the exact analogue of an indifference curve, with one important difference. Its label is not arbitrary. An isoquant marked 4000 units means 4000 units, so unlike utility, output is cardinal and comparable.

Its slope has the same derivation as the marginal rate of substitution. Moving along an isoquant leaves output unchanged, so ${MP}_{L} d L + {MP}_{K} d K = 0$ , giving

TRS = - \frac{d K}{d L} = \frac{{MP}_{L}}{{MP}_{K}}

The technical rate of substitution is how much capital can be released when one more worker is hired, holding output fixed. Isoquants are convex for the same reason indifference curves are: as labour replaces capital, labour gets less productive at the margin and capital more so, and the rate at which one substitutes for the other falls.

Returns to scale

Diminishing marginal returns is about one input. Returns to scale is a different question about all of them at once: multiply every input by $t > 1$ and ask what happens to output. Returns to scale are constant if output multiplies by $t$ , increasing if by more, and decreasing if by less. Confusing the two is the commonest error in this part of the subject, and a production function can perfectly well have diminishing marginal returns to each input separately and increasing returns to scale together.

For a Cobb-Douglas function $q = A L^{α} K^{β}$ the test is immediate. Scaling both inputs by $t$ multiplies output by $t^{α + β}$ , so the sum of the exponents decides it: constant returns at 1, increasing above, decreasing below. Charles Cobb and Paul Douglas fitted exactly this form to United States manufacturing data for 1899 to 1922 and published it in 1928, getting $q = 1.01 L^{0.75} K^{0.25}$ : exponents summing to one, so constant returns to scale, with labour's exponent matching labour's share of national income closely enough to be striking.

Example. A firm has $q = 10 L^{0.3} K^{0.5}$ . What returns to scale does it show, and what happens to output at $L = 100$ , $K = 400$ if both inputs double?

The exponents sum to 0.8, less than one, so returns to scale are decreasing. At $L = 100$ and $K = 400$ , $q = 10 \times 100^{0.3} \times 400^{0.5} = 10 \times 3.981 \times 20 = 796.2$ . Doubling both gives $q = 10 \times 200^{0.3} \times 800^{0.5} = 1386.3$ , a factor of $1.741$ , which is exactly $2^{0.8}$ . Twice the inputs bought 74 per cent more output.

Now you. A firm has $q = 5 L^{0.4} K^{0.8}$ . Classify its returns to scale and find the factor by which output rises when both inputs double.

Answer

The exponents sum to 1.2, so returns to scale are increasing. Doubling both inputs multiplies output by $2^{1.2} = 2.297$ , so twice the inputs give nearly two and a third times the output. Checking directly: $L = K = 32$ gives $q = 320$ , and $L = K = 64$ gives $q = 735.2$ , a ratio of 2.297.

Increasing returns are not a curiosity. They arise from indivisibilities, since half a blast furnace produces nothing, and from geometry, since a tank's capacity grows as the cube of its dimensions while the steel to build it grows as the square. Whenever they persist over the whole relevant range of output, competition among many small firms is impossible, which is a result the market structure lessons will lean on heavily.

Cost minimisation

Technology says what can be made. Choosing among the ways of making it requires prices. Let $w$ be the wage and $r$ the rental rate of capital, so the cost of an input combination is $w L + r K$ , and lines of constant cost, the isocost lines, have slope $- w / r$ .

The problem of hitting a target output at least cost is now visibly the consumer's problem upside down: minimise a linear function subject to reaching a given level set, rather than maximise a level set subject to a linear constraint. The same tangency comes out:

TRS = \frac{{MP}_{L}}{{MP}_{K}} = \frac{w}{r}

or equivalently ${MP}_{L} / w = {MP}_{K} / r$ , the last pound spent on each input buying the same extra output. That second form is Gossen's rule again, and the arbitrage argument is identical: if a pound of labour added more output than a pound of capital, shift a pound.

Example. A firm has $q = \sqrt{L K}$ , faces a wage of £20 and a capital rental of £80, and must produce 100 units. Find the cheapest input combination and its cost.

The marginal products are ${MP}_{L} = 0.5 \sqrt{K / L}$ and ${MP}_{K} = 0.5 \sqrt{L / K}$ , so the TRS is $K / L$ . Setting $K / L = 20 / 80 = 0.25$ gives $K = 0.25 L$ . Substituting into the output constraint, $\sqrt{L \times 0.25 L} = 0.5 L = 100$ , so $L = 200$ and $K = 50$ . The cost is $20 (200) + 80 (50) = £ 4000 + £ 4000 = £ 8000$ . Capital is four times as expensive as labour, so the firm uses a quarter as much of it, and the two bills come out equal, which is the Cobb-Douglas constant-share property appearing on the production side.

Now you. The same technology, but the wage is £45 and the capital rental £5, and the target is 60 units. Find the input combination and the cost.

Answer

$K / L = 45 / 5 = 9$ , so $K = 9 L$ . Then $\sqrt{L \times 9 L} = 3 L = 60$ , giving $L = 20$ and $K = 180$ . The cost is $45 (20) + 5 (180) = £ 900 + £ 900 = £ 1800$ . Expensive labour has pushed the firm to a capital-intensive method, and the expenditure shares are equal again.

That result is the mechanism behind a great deal of observed technology choice. Construction in countries with high wages uses cranes and prefabrication; the same firm building in a low-wage country uses more labour and less equipment, not because of different engineering knowledge but because the tangency sits elsewhere on the same isoquant.

What the production function hides

Three limits deserve stating, because the function is a much stronger assumption than its clean notation suggests.

Aggregating capital is genuinely problematic. Adding a lathe to a lorry to a laptop requires valuing them, and their values depend on the interest rate, which in this theory is determined by capital's marginal product, which requires the aggregate. Joan Robinson pressed exactly this circularity in 1953, and the resulting Cambridge capital controversy ended with Paul Samuelson conceding in 1966 that the aggregate production function cannot be derived from underlying technologies in general. The two-input picture is a useful teaching device and not a theorem about economies.

Most firms make many products, not one. A supermarket's output is not a scalar, and the single-output function cannot express the cost savings from selling bread alongside milk, which economies of scope name and this apparatus cannot represent.

And technology is treated as given and free. In reality it is chosen, purchased, and often the main thing the firm is competing over, which puts research and development outside the model entirely.

With those noted, the machinery does what is needed. Solving the cost-minimisation problem at every output level converts the production function into a cost function, and cost is what the pricing decision actually runs on. That conversion is the next lesson.