Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), introductory production theory (isoquants, isocost lines), familiarity with Excel Solver.
This lab sits at the intersection of production theory and cost theory. You have already learned that firms combine inputs to produce output; now the question becomes how to combine those inputs at the lowest possible cost for a given output level. The tool that answers that question is the Lagrangian, and the Lagrange multiplier it produces turns out to equal marginal cost, which in turn builds the firm's supply curve. If you missed the lectures on isoquants and isocost lines, revisit those before working through this material.
A firm minimises cost by choosing inputs where the ratio of marginal products equals the ratio of input prices (the isocost-isoquant tangency condition). The Lagrange multiplier from the constrained optimisation equals marginal cost at that output level. When an input price rises, the firm substitutes away from it, total and marginal costs increase, and the profit-maximising output falls.
Cost minimisation
The problem of choosing input quantities to produce a given output at the lowest total cost. Think of it as: "I need to make 200 units, what is the cheapest mix of labour and materials to get there?"
Isoquant
A curve showing every combination of two inputs that produces the same level of output. In simple terms, it is a contour line on a production map.
Isocost line
A line showing every combination of two inputs that costs the same total amount. The slope equals the negative of the input price ratio (PX1 / PX2).
Lagrangian
The function formed by combining the objective (total cost) with the constraint (production function) using a multiplier. It converts a constrained problem into an unconstrained one that standard calculus can solve.
Lagrange multiplier (lambda)
The shadow price of the constraint. In cost minimisation, it tells you how much total cost rises if you are required to produce one more unit of output. In simple terms, it is marginal cost read straight off the optimisation.
Expansion path
The set of cost-minimising input combinations traced out as output increases, holding input prices constant. Each point on it is a tangency between an isoquant and an isocost line.
Marginal cost (MC)
The additional cost of producing one more unit of output. Equal to the Lagrange multiplier at the optimum and equal to the slope of the total cost curve.
Total fixed cost (TFC)
Costs that do not change with output level. In this lab, TFC = $56.00 regardless of how much is produced.
Total variable cost (TVC)
Costs that change with output. Calculated as PX1 * X1 + PX2 * X2.
Supply curve
The relationship between output price and the quantity a profit-maximising firm chooses to produce, traced out by the marginal cost curve above minimum average variable cost.
Reduced gradient
In Solver's sensitivity report, the amount by which the objective function coefficient of a variable would need to improve before that variable would enter the optimal solution at a positive level. A zero reduced gradient means the variable is already in the solution.
Sensitivity report / sensitivity analysis
An output from Excel Solver that reports the Lagrange multiplier and reduced gradients, showing how the optimal solution responds to small changes in the constraint or objective coefficients.
The firm uses two inputs (X1 and X2) to produce output Y. Input prices are PX1 = $2.75 and PX2 = $1.10 in the base case. Fixed costs are $56.00.
The optimisation problem: minimise TVC = PX1 * X1 + PX2 * X2, subject to the production function constraint f(X1, X2) = Y.
Excel Solver finds the cost-minimising input mix at each output level.
At low output (Y = 25), only X2 is used (X1 = 0, X2 = 7.42). The reduced gradient on X1 is positive (1.85), meaning X1 would need to become cheaper before it enters the solution.
As output rises, both inputs enter the solution. At Y = 100: X1 = 14.46, X2 = 27.01. At Y = 225: X1 = 67.29, X2 = 51.38.
TFC stays at $56.00 throughout. TVC and TC rise with output.
Lagrange multipliers from the sensitivity reports:
Output (Y) | Lagrange multiplier | Interpretation |
|---|---|---|
25 | 0.381 | Producing one more unit from Y = 25 adds about $0.38 to cost |
100 | 1.036 | MC at Y = 100 is roughly $1.04 |
150 | 1.236 | MC at Y = 150 is roughly $1.24 |
200 | 1.630 | MC at Y = 200 is roughly $1.63 |
225 | 2.054 | MC at Y = 225 is roughly $2.05 |
The Lagrange multiplier increases with output, confirming increasing marginal cost. Notice that these multiplier values match the MC* column in the cost table.
When PX2 rises from $1.10 to $3.00 (PX1 stays at $2.75):
The firm substitutes away from X2 and uses relatively more X1 at every output level.
At Y = 100: X1 rises from 14.46 to 19.48; X2 falls from 27.01 to 19.85.
At Y = 200: X1 rises from 53.34 to 56.53; X2 falls from 44.94 to 40.40.
TVC, TC and MC all increase at every output level.
At Y = 100: TC rises from $125.48 to $169.12; MC rises from $1.04 to $1.34.
At Y = 225: TC rises from $297.57 to $391.33; MC rises from $2.05 to $2.65.
How the expansion path shifts:
The expansion path plots X2 against X1 across output levels. When PX2 increases, the path rotates toward the X1 axis because the firm uses less of the now-more-expensive input. Both paths start near the origin and curve upward, but the higher-price path sits below the original at every output level.
The tangency condition driving this: at the optimum, the ratio of marginal products (MPX1 / MPX2) equals the ratio of prices (PX1 / PX2). When PX2 rises, the price ratio falls, so the firm must move to a point where MPX1 / MPX2 is also lower, which means using relatively more X1 and less X2.
The marginal cost curve (MC plotted against output Y) is the firm's supply curve above minimum AVC.
With PX2 = $1.10, MC rises from $0.38 (Y = 25) to $2.05 (Y = 225).
With PX2 = $3.00, MC rises from $1.04 (Y = 100) to $2.65 (Y = 225).
The higher-input-price supply curve lies everywhere above the lower-price one. This makes sense: if it costs more to buy an input, it costs more to produce each additional unit of output.
Both curves are upward-sloping, reflecting diminishing marginal returns in the production function.
The firm maximises profit by choosing output where price equals marginal cost (P = MC), then using the cost-minimising input mix to produce that output.
Profit-maximising results under different price scenarios:
Output price (PY) | PX1 | PX2 | X1 | X2 | Y | Profit |
|---|---|---|---|---|---|---|
$1.40 | $2.75 | $1.10 | 42.21 | 39.81 | 175.92 | $30.44 |
$1.95 | $2.75 | $1.10 | 64.43 | 50.06 | 220.33 | $141.41 |
$1.40 | $2.75 | $3.00 | 24.06 | 22.39 | 114.78 | -$28.65 |
$1.95 | $2.75 | $3.00 | 51.40 | 37.55 | 188.82 | $58.19 |
Key observations:
Holding input prices constant, a higher output price leads to more output and higher profit (compare rows 1 and 2, or rows 3 and 4).
Holding output price constant, a higher input price reduces optimal output and profit (compare rows 1 and 3, or rows 2 and 4).
At PY = $1.40 and PX2 = $3.00, the firm makes a loss (-$28.65). The firm would shut down in the long run if this persisted.
The profit-maximising input combinations lie on the expansion path. This confirms that profit maximisation requires cost minimisation as a necessary condition.
The profit-maximising points also lie on the supply curve (MC curve), which follows from the P = MC condition.
\text{Minimise } TC = TFC + PX_1 \cdot X_1 + PX_2 \cdot X_2 \quad \text{subject to } f(X_1, X_2) = Y\mathcal{L} = PX_1 \cdot X_1 + PX_2 \cdot X_2 + \lambda \big[ Y - f(X_1, X_2) \big]First-order conditions:
\frac{\partial \mathcal{L}}{\partial X_1} = PX_1 - \lambda \cdot MP_{X_1} = 0 \implies \frac{PX_1}{MP_{X_1}} = \lambda\frac{\partial \mathcal{L}}{\partial X_2} = PX_2 - \lambda \cdot MP_{X_2} = 0 \implies \frac{PX_2}{MP_{X_2}} = \lambda\frac{\partial \mathcal{L}}{\partial \lambda} = Y - f(X_1, X_2) = 0Tangency condition (derived from equating the first two):
\frac{MP_{X_1}}{MP_{X_2}} = \frac{PX_1}{PX_2}Interpretation of lambda:
\lambda = MC = \frac{\Delta TC}{\Delta Y}Profit maximisation condition:
P_Y = MC \quad \text{(price equals marginal cost)}Cost minimisation is how manufacturing firms decide the mix of labour and capital to deploy on a production line. A car manufacturer choosing between robots (expensive to buy, cheap to run) and workers (cheaper to hire, more expensive per unit over time) solves exactly this kind of problem. When wages rise, the firm shifts toward automation, just as the lab's firm shifts from X2 toward X1 when PX2 increases.
The Lagrange multiplier, as marginal cost, is the number operations managers use to decide whether taking on one more unit of an order is worth the revenue it brings. Supply-chain pricing, make-or-buy decisions, and short-run capacity planning all rely on knowing the marginal cost at the current output level.
Students often think the Lagrange multiplier is just a mathematical curiosity with no economic meaning. It is not. It equals marginal cost, and that number drives every pricing and output decision the firm makes.
Students sometimes confuse the expansion path with a single isoquant. The expansion path connects tangency points across many isoquants (different output levels). A single isoquant shows input combinations for one output level only.
A common mistake is to assume that when an input price rises, the firm simply uses less of everything. In fact, the firm substitutes toward the cheaper input. Total output of the now-expensive input falls, but the other input typically rises.
Students often forget that profit maximisation requires cost minimisation as a precondition. You cannot find the profit-maximising output without first knowing the minimum cost of producing each output level.
⚠️ Be ready to interpret a Lagrange multiplier in words. A common exam question gives you the multiplier value and asks what it means. Answer: it is the marginal cost of output, the additional cost of producing one more unit.
⚠️ Know how to read a Solver sensitivity report. The Lagrange multiplier sits in the Constraints section, not the Variable Cells section.
⚠️ Expect a question on what happens to the expansion path, MC, and profit when an input price changes. The direction of every effect (substitution toward the cheaper input, higher MC, lower optimal output, lower profit) should be second nature.
⚠️ The link between cost minimisation and profit maximisation is a favourite exam topic. Be able to explain why the profit-maximising point lies on both the expansion path and the supply curve.
True or false: The Lagrange multiplier in a cost-minimisation problem equals marginal cost. (True)
Fill in the blank: At the cost-minimising input combination, the ratio of marginal products equals the ratio of ________. (input prices)
True or false: When an input becomes more expensive, the firm uses more of every other input and less of the expensive one. (Roughly true for two inputs, the firm substitutes toward the cheaper input)
True or false: The profit-maximising output level lies on the expansion path. (True)
Fill in the blank: A profit-maximising firm produces where output price equals ________. (marginal cost)
Q: A firm minimises cost using two inputs. The Lagrange multiplier on the output constraint is 1.63 when Y = 200. Interpret this value.
A: The marginal cost of output at Y = 200 is $1.63. If the firm were required to produce one additional unit (Y = 201), total cost would rise by approximately $1.63.
Q: When PX2 rises from $1.10 to $3.00, what happens to the optimal input mix at Y = 100, and why?
A: X1 rises from 14.46 to 19.48 and X2 falls from 27.01 to 19.85. The firm substitutes away from the now-more-expensive X2 toward X1. At the new optimum, the tangency condition MPX1/MPX2 = PX1/PX2 holds at the new price ratio.
Q: Why does the expansion path shift when an input price changes?
A: The expansion path traces cost-minimising input combinations across output levels. Each point is an isocost-isoquant tangency. When PX2 rises, the isocost lines become steeper (slope = -PX1/PX2 gets smaller in absolute value), so the tangency points shift toward more X1 and less X2 at every output level.
Q: Explain why the profit-maximising input combination lies on the expansion path.
A: Profit maximisation requires producing at the output where P = MC, and producing that output at minimum cost. The expansion path is the locus of cost-minimising input combinations, so the profit-maximising point must be on it. Cost minimisation is a necessary condition for profit maximisation.
Q: If the output price is $1.40 and PX2 = $3.00, the firm earns a profit of -$28.65. Should it shut down immediately?
A: In the short run, the firm should continue producing if revenue covers variable costs (P > AVC), because it would lose the full $56.00 in fixed costs by shutting down. In the long run, if this loss persists, the firm should exit the market.
This material connects directly to consumer theory: cost minimisation for the firm (minimise cost subject to output) mirrors expenditure minimisation for the consumer (minimise spending subject to utility). The tangency conditions are structurally identical, with marginal products replacing marginal utilities.
The supply curve derived here feeds into market supply and equilibrium analysis. When you stack individual firms' MC curves horizontally, you get market supply, which intersects market demand to determine equilibrium price and quantity.
The Lagrange multiplier technique reappears throughout economics: in utility maximisation, in resource allocation problems, and in welfare economics (where shadow prices measure the social value of relaxing a constraint).
Cost minimisation, cost minimization, Lagrange multiplier, Lagrangian, shadow price, marginal cost, MC, expansion path, isoquant, isocost, tangency condition, input substitution, factor substitution, supply curve, profit maximisation, profit maximization, P = MC, sensitivity analysis, reduced gradient, Excel Solver, constrained optimisation, constrained optimization, APEC 3002, managerial economics, applied microeconomics, production theory, cost functions, total cost, variable cost, fixed cost, input price change, two-input production.