Difficulty: Intermediate | Prerequisites: Cost Minimisation Foundations (Part 1 of these notes), first-order conditions, partial derivatives, Cobb-Douglas production functions.
Part 1 of these notes derived the cost function from the cost minimisation problem. This part picks up two threads from there. First, the profit-maximisation first-order conditions generate factor demand equations, which describe how much capital and labour a firm wants at various input prices. Second, the short-run introduces a fixed factor (typically capital), which simplifies the maths but changes the shape of the cost function. The lecture closes by showing that a cost function, even one you simply write down rather than derive, lets you solve for a supply curve in a few lines.
Factor demand curves come from setting the value of the marginal product equal to the input price. In the short run, one input is fixed, so the cost function becomes a simple expression in output alone. The real payoff: any cost function, however it was obtained, can be differentiated to give price = marginal cost, which is the firm's supply curve.
Factor demand (input demand)
The quantity of an input (capital or labour) that a profit-maximising firm wants to employ, expressed as a function of input and output prices. Think of it as the demand curve, but for hiring workers or renting machines rather than buying goods.
Inverse factor demand
The factor demand relationship written with the input price on the left-hand side. For capital: r = p * (l/k)^(1/2). You plot input price on the vertical axis and the quantity of the input on the horizontal axis, just like a regular demand curve.
Value of the marginal product (VMP)
The output price multiplied by the marginal product of an input: p * MP. A profit-maximising firm hires each input up to the point where VMP equals the input's price.
Short run
A time horizon in which at least one factor of production is fixed and cannot be adjusted. Typically capital (e.g. a factory) is fixed, while labour is variable. In simple terms, the short run is "you are stuck with the factory you have."
Fixed cost
A cost that does not change with the level of output. In the short run, the cost of the fixed factor (r times k-bar) is a fixed cost. It must be paid regardless of how much is produced.
Supply function (supply curve)
The relationship between the output price and the quantity of output a firm chooses to produce. Derived by setting price equal to marginal cost and solving for y.
The two first-order conditions from profit maximisation are:
p * MP_k = r (the value of the marginal product of capital equals the rental rate)
p * MP_l = w (the value of the marginal product of labour equals the wage)
For the production function y = k^(1/2) * l^(1/2), these become:
r = p * (l/k)^(1/2) (inverse demand for capital)
w = p * (k/l)^(1/2) (inverse demand for labour)
These are called inverse demand functions because the input price is on the left-hand side.
Graphing: input price on the vertical axis, quantity of the input on the horizontal axis. The curve slopes downward, just like a standard demand curve.
Each factor demand curve holds the other input constant. If l changes, the demand curve for k shifts.
In the short run, capital is fixed at some level k-bar. The firm can only adjust labour.
Profit becomes: profit = py - wl - r * k-bar.
Only one first-order condition remains: p * MP_l = w, or equivalently MP_l = w/p.
Worked example: y = k^(1/2) * l^(1/2), with k fixed at 100.
MP_l = (1/2) * 100^(1/2) * l^(-1/2) = (1/2) * 10 * l^(-1/2) = 5 * l^(-1/2).
Setting p * MP_l = w gives: p * 5 * l^(-1/2) = w.
Solving for l: l = (5p/w)^2.
This is the short-run labour demand. It increases when output price rises and decreases when wages rise.
Comparative statics (from the lecture's graphs):
Increasing w (the wage) reduces the demand for labour. Graphically, the isoprofit line steepens and the tangent point shifts left.
Increasing p (the output price) increases the demand for labour and therefore increases supply. The isoprofit line flattens and the tangent point shifts right.
With k fixed, the cost minimisation problem simplifies. For y = k-bar^(1/2) * l^(1/2), solving for l gives l = y^2 / k-bar.
The short-run cost function is: c(r, k-bar, y) = r * k-bar + w * y^2 / k-bar.
The first term (r * k-bar) is the fixed cost; the second term (w * y^2 / k-bar) is the variable cost.
Variable cost is quadratic in y, which means marginal cost is increasing. This is the typical short-run shape.
The lecture's closing example shows the real power of the cost function approach. You do not need to start from a production function.
Given: c(y) = (1/5)(y - 5)^3 + 25.
Profit: profit = py - (1/5)(y - 5)^3 - 25.
First-order condition: p - (3/5)(y - 5)^2 = 0.
Solving for y: y = 5 + sqrt(5p/3).
This is the firm's supply curve. It tells you how much the firm produces at each price, and it was derived in three lines from a cost function that was simply stated, not built from a production function.
Factor demand first-order conditions
p \cdot MP_k = r \quad \text{and} \quad p \cdot MP_l = wInverse factor demands (for y = k^(1/2) l^(1/2))
r = p \left( \frac{l}{k} \right)^{1/2} \quad \text{and} \quad w = p \left( \frac{k}{l} \right)^{1/2}Short-run profit (k fixed)
\pi = py - wl - r\bar{k}Short-run labour demand (y = k^(1/2) l^(1/2), k = 100)
l = \left( \frac{5p}{w} \right)^2Short-run cost function (y = k-bar^(1/2) l^(1/2))
c(r, \bar{k}, y) = r\bar{k} + w \frac{y^2}{\bar{k}}Supply curve derivation example
Given c(y) = (1/5)(y - 5)^3 + 25:
\frac{\partial \pi}{\partial y} = p - \frac{3}{5}(y - 5)^2 = 0y = 5 + \sqrt{\frac{5p}{3}}Factor demand is the economics behind every hiring and capital-investment decision. When a tech company decides whether to hire more engineers or invest in better servers, it is comparing the value of each input's marginal product to its price. The short-run constraint is real: a restaurant cannot build a bigger kitchen overnight, so it adjusts staffing to meet demand within its existing space.
Students often forget that each factor demand curve holds the other factor constant. If both inputs change simultaneously, you are moving along a different curve, not the one you drew.
A common error is treating the short-run cost function as if it has no fixed cost component. The term r * k-bar is always there, even when the firm produces zero output.
Students sometimes think supply is derived from the production function directly. It is not. Supply comes from setting price equal to marginal cost, which requires the cost function. The production function is an ingredient, not the final step.
The supply curve derivation example (c(y) = (1/5)(y - 5)^3 + 25) trips students up because the cost function is given, not derived. The point is that you do not always need to go through the full cost minimisation exercise. Any valid cost function will do.
⚠️ Be ready to derive factor demand equations from a production function and identify whether they are direct or inverse demand.
⚠️ Short-run problems are common on exams. You will be given a production function and a fixed level of capital and asked to find the short-run cost function or short-run labour demand.
⚠️ The supply curve derivation is the capstone of this lecture. Expect a problem that gives you a cost function and asks you to find the supply curve. The steps are: write profit = py - c(y), differentiate with respect to y, set equal to zero, solve for y.
⚠️ Understand comparative statics: how does labour demand change when w rises? When p rises? Be able to explain the intuition, not just the maths.
True or false: The inverse demand function for capital has the rental rate r on the vertical axis and k on the horizontal axis. (True.)
Fill in the blank: In the short run, the firm can only adjust ______, because capital is fixed. (Labour.)
True or false: If the output price p increases, short-run labour demand decreases. (False, it increases.)
Fill in the blank: The supply curve is derived by setting ______ equal to marginal cost. (Price.)
True or false: The fixed cost r * k-bar disappears from the first-order condition when you differentiate profit with respect to y. (True, because it is a constant.)
Q: For the production function y = k^(1/2) l^(1/2), derive the inverse factor demand for labour.
A: The first-order condition for labour is p * MP_l = w. The marginal product of labour is (1/2) * k^(1/2) * l^(-1/2) = (1/2)(k/l)^(1/2). So w = p * (1/2)(k/l)^(1/2), or equivalently w = p(k/l)^(1/2) (absorbing the 1/2 depends on which specific form is used; the lecture's version has w = p(k/l)^(1/2)).
Q: Suppose y = k^(1/2) l^(1/2) and k is fixed at 100. Find the short-run cost function.
A: Solve for l from the production function: l = y^2 / k-bar = y^2 / 100. Total cost is c = r * 100 + w * (y^2 / 100). The fixed cost is 100r; the variable cost is (w/100) * y^2.
Q: Given c(y) = (1/5)(y - 5)^3 + 25, derive the supply curve.
A: Profit = py - (1/5)(y - 5)^3 - 25. Differentiate with respect to y: d(profit)/dy = p - (3/5)(y - 5)^2 = 0. Rearrange: (y - 5)^2 = 5p/3. Take the positive square root: y - 5 = sqrt(5p/3). So y = 5 + sqrt(5p/3).
Q: Why does a higher output price shift the short-run factor demand for labour to the right?
A: When p rises, the value of each unit of labour's marginal product (p * MP_l) increases. The firm finds it profitable to hire more workers at the same wage, so the quantity of labour demanded at each wage increases, which is a rightward shift of the demand curve.
Factor demand connects to labour economics and the theory of wages: the idea that firms pay workers according to their marginal product runs through both fields. The short-run cost function reappears in the analysis of market structure, where the shape of marginal and average cost curves determines whether a firm earns economic profit, breaks even, or shuts down. The supply curve derived here is the building block for market supply, which aggregates individual firms' supply curves horizontally.
Factor demand, input demand, inverse demand function, value of marginal product, VMP, short run, long run, fixed factor, fixed cost, variable cost, short-run cost function, isoprofit line, comparative statics, supply curve, supply function, p = MC, price equals marginal cost, Cobb-Douglas, ECN 312, intermediate microeconomics, intermediate macroeconomics, Arizona State University, Lecture 12, market supply