Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), introductory microeconomics (supply, demand, cost curves), familiarity with exponents and power functions.
Production economics sits at the heart of managerial decision-making: given a technology (a production function) and a set of market prices, how much of each input should a firm use, and how much output should it produce, to make the most money? This lab applies that logic to a specific functional form, the Cobb-Douglas production function, and walks through profit maximisation when some inputs are variable and others are fixed. If you are comfortable with the idea that firms choose input quantities to maximise profit, and that this involves setting marginal value product equal to input price, the rest follows naturally.
A firm with a Cobb-Douglas production function maximises profit by choosing variable input levels so that each input's marginal value product equals its price. When an input price rises, the firm uses less of that input, produces less output, and earns lower profit. Doubling a fixed input does not double profit, because returns to scale and the interaction between inputs matter.
Production function
A mathematical relationship showing the maximum output obtainable from any given combination of inputs, holding technology constant.
Think of it as: the recipe that tells you how much stuff you can make with a given mix of ingredients.
Cobb-Douglas production function
A specific functional form: Y = Beta x X1^Alpha1 x X2^Alpha2, where Beta is a scaling (efficiency) parameter and Alpha1, Alpha2 are output elasticities of each input.
In simple terms, this means each input contributes to output in a multiplicative way, and the exponents tell you how responsive output is to each input.
Variable input
An input whose quantity the firm can adjust in the decision period. In this lab, X1 is the variable input.
Think of it as: the lever the manager can actually pull right now.
Fixed input
An input whose quantity is predetermined and cannot be changed in the decision period. In this lab, X2 is fixed (at 18 or 36 units).
In simple terms, this is the constraint you are stuck with in the short run, such as land or a factory.
Marginal product (MP)
The additional output produced by one more unit of an input, holding all other inputs constant. For a Cobb-Douglas function, MP of X1 = Alpha1 x Beta x X1^(Alpha1 - 1) x X2^Alpha2.
Think of it as: how much extra output you get from adding one more worker (or one more unit of whatever input you are varying).
Marginal value product (MVP)
The marginal product of an input multiplied by the output price. MVP = MP x PY.
In simple terms, this is the extra revenue the firm earns from using one more unit of the input.
Profit maximisation condition (single variable input)
The firm maximises profit where MVP of the variable input equals the input's price: MVP_X1 = PX1.
Think of it as: keep hiring until the last worker's contribution to revenue exactly equals their wage.
Fixed cost
The cost of the fixed input: Fixed Cost = PX2 x X2. This does not change with the level of variable input used.
Variable cost
The cost of the variable input: Variable Cost = PX1 x X1. This changes as the firm adjusts X1.
Total cost
Fixed cost plus variable cost: TC = PX1 x X1 + PX2 x X2.
Revenue
Output multiplied by output price: Revenue = PY x Y.
Profit
Revenue minus total cost: Profit = PY x Y - PX1 x X1 - PX2 x X2.
The functional form is Y = Beta x X1^Alpha1 x X2^Alpha2
Parameter values throughout: Beta = 5.2, Alpha1 = 0.25, Alpha2 = 0.5
X1 is the variable input (the firm chooses how much to use)
X2 is the fixed input (set at either 18 or 36 units across exercises)
Output price PY = $1.50 in all scenarios
Sum of exponents = Alpha1 + Alpha2 = 0.25 + 0.5 = 0.75
Because 0.75 < 1, this production function exhibits decreasing returns to scale
If both inputs were doubled, output would increase by a factor of 2^0.75 (roughly 1.68), not 2
This matters: it means the firm cannot simply scale up proportionally and expect proportional gains
Step 1: Write out the profit function. Profit = PY x Beta x X1^Alpha1 x X2^Alpha2 - PX1 x X1 - PX2 x X2
Step 2: Take the first derivative of profit with respect to X1 (the only variable the firm controls) and set it equal to zero
Step 3: This gives the first-order condition: PY x Alpha1 x Beta x X1^(Alpha1 - 1) x X2^Alpha2 = PX1
Step 4: Solve for X1*. Rearranging: X1* = (PY x Alpha1 x Beta x X2^Alpha2 / PX1)^(1 / (1 - Alpha1))
Step 5: Plug X1* back into the production function to get Y*, then compute revenue, costs, and profit
The left-hand side is the marginal value product of X1 (the extra revenue from one more unit of X1)
The right-hand side is the price of X1 (the cost of one more unit)
At the optimum, the last unit of X1 contributes exactly as much revenue as it costs
Using more X1 beyond this point would cost more than it earns; using less would leave money on the table
X2 enters the first-order condition through X2^Alpha2
A larger X2 raises the marginal product of X1 (the inputs are complements in this function)
So changing X2 shifts the optimal X1* and the resulting profit, even though X2 itself is not a choice variable in the short run
Scenario | PX1 | X2 (fixed) | X1* (optimal) | Y* (output) | Revenue | Total Cost | Profit |
|---|---|---|---|---|---|---|---|
E.1 | $0.80 | 18 | 22.53 | 48.07 | $72.10 | $46.82 | $25.27 |
E.2 | $0.80 | 36 | 35.77 | 76.30 | $114.45 | $86.21 | $28.24 |
E.3a | $1.10 | 18 | 14.74 | 43.22 | $64.84 | $45.01 | $19.83 |
E.3b | $1.10 | 36 | 23.39 | 68.61 | $102.92 | $83.33 | $19.59 |
Optimal X1* rises from 22.53 to 35.77 (an increase of about 59%)
Output Y* rises from 48.07 to 76.30 (about 59% increase)
Profit rises from $25.27 to $28.24 (only about 12% increase)
The lesson: doubling the fixed input increases output substantially, but profit rises by much less because fixed costs also double ($28.80 to $57.60) and variable costs rise too
Optimal X1* falls from 22.53 to 14.74 (a 35% decrease)
Output falls from 48.07 to 43.22
Revenue falls from $72.10 to $64.84
Profit falls from $25.27 to $19.83
The firm responds to a more expensive input by using less of it, which means less output and lower profit
Optimal X1* falls from 35.77 to 23.39 (a 35% decrease, the same proportional response)
Output falls from 76.30 to 68.61
Profit falls from $28.24 to $19.59
The percentage drop in X1* is the same regardless of X2 level, which follows from the Cobb-Douglas functional form
Higher input prices always reduce optimal input use, output, and profit
A larger fixed input raises output and revenue but also raises costs, so the net effect on profit is smaller than you might expect
The firm with X2 = 36 earns higher profit at the low input price ($28.24 vs $25.27), but when PX1 rises to $1.10 the two scenarios produce nearly identical profit ($19.59 vs $19.83), because the larger operation is more exposed to input cost increases
Y = \beta \cdot X_1^{\alpha_1} \cdot X_2^{\alpha_2}Production function. In this lab: Beta = 5.2, Alpha1 = 0.25, Alpha2 = 0.5.
MP_{X_1} = \alpha_1 \cdot \beta \cdot X_1^{\alpha_1 - 1} \cdot X_2^{\alpha_2}Marginal product of X1. Obtained by taking the partial derivative of Y with respect to X1.
MVP_{X_1} = P_Y \cdot \alpha_1 \cdot \beta \cdot X_1^{\alpha_1 - 1} \cdot X_2^{\alpha_2}Marginal value product of X1. This is what the firm sets equal to PX1 to find the optimum.
X_1^* = \left( \frac{P_Y \cdot \alpha_1 \cdot \beta \cdot X_2^{\alpha_2}}{P_{X_1}} \right)^{\frac{1}{1 - \alpha_1}}Optimal variable input. Derived by solving MVP_X1 = PX1 for X1.
\pi = P_Y \cdot Y - P_{X_1} \cdot X_1 - P_{X_2} \cdot X_2Profit equation. Revenue minus total cost (variable cost plus fixed cost).
This is the framework behind input decisions in agriculture (how much fertiliser or labour to apply given crop prices and input costs), manufacturing (how many shifts to run given wage rates), and any setting where a manager controls some inputs while others are fixed in the short run. The comparative statics here, where a rise in input price causes the firm to cut back, is exactly the mechanism behind input demand curves used in labour economics and resource markets.
Students often think that doubling a fixed input will double profit. It does not. Profit depends on the net of revenue and cost, and both move when the fixed input changes. In this lab, doubling X2 from 18 to 36 raised profit by only $2.97 (from $25.27 to $28.24) at the low input price.
Students sometimes confuse marginal product with marginal value product. Marginal product is in physical units (extra output). Marginal value product is in dollars (extra revenue). The profit-maximising condition equates marginal value product to input price, not marginal product.
It is a common error to think fixed costs are irrelevant to profit. Fixed costs do not affect the profit-maximising choice of X1 (they drop out of the first-order condition), but they absolutely reduce the level of profit. A firm can maximise profit and still lose money if fixed costs are high enough.
Students sometimes believe that if an input price rises, the firm should just pass the cost through to higher output prices. In a competitive market (price-taker assumption), the firm cannot set its own output price. It must absorb the hit by adjusting input use.
⚠️ Be able to derive X1* from the first-order condition for a Cobb-Douglas function. This is the central calculation of the lab.
⚠️ Know how to interpret the profit-maximisation condition MVP = input price in words, not just symbols.
⚠️ Expect questions asking you to predict the direction of change in X1*, Y*, and profit when PX1 rises or X2 changes. The qualitative reasoning matters as much as the numbers.
⚠️ Understand why the sum of exponents (Alpha1 + Alpha2) determines returns to scale, and how that affects the firm's response to scaling up.
⚠️ Do not confuse the effect of fixed inputs on the optimisation condition (they shift the MVP curve) with the effect of fixed costs on profit (they reduce the profit level but do not change the optimal X1).
True or False: A Cobb-Douglas function with exponents summing to 0.75 exhibits increasing returns to scale. (False. It exhibits decreasing returns to scale.)
Fill in the blank: The profit-maximising condition for a single variable input is ______ = input price. (Marginal value product, or MVP.)
True or False: Doubling the fixed input in a decreasing-returns-to-scale production function will more than double output. (False. Output increases, but by less than double.)
True or False: Fixed costs affect the optimal choice of the variable input. (False. Fixed costs drop out of the first-order condition. They affect profit level, not the optimal X1.)
Fill in the blank: When PX1 rises from $0.80 to $1.10, the firm uses ______ of X1. (Less.)
Q: Given the production function Y = 5.2 x X1^0.25 x X2^0.5, with PY = $1.50, PX1 = $0.80, and X2 = 18, write the first-order condition for profit maximisation with respect to X1.
A: Set MVP_X1 = PX1. That is: 1.50 x 0.25 x 5.2 x X1^(0.25 - 1) x 18^0.5 = 0.80. Simplify: 1.50 x 1.3 x X1^(-0.75) x 4.243 = 0.80, which gives 8.274 x X1^(-0.75) = 0.80. Solve for X1.
Q: If the price of X1 increases from $0.80 to $1.10 while everything else remains constant, what happens to the optimal quantity of X1, output, and profit? Explain the economic intuition.
A: X1* falls, output falls, and profit falls. The intuition is that a more expensive input has a higher hurdle to clear: the firm stops using X1 at the point where the last unit's revenue contribution equals its (now higher) cost, which occurs at a lower quantity of X1. Less X1 means less output and less revenue, while the per-unit cost is higher, so profit is squeezed from both sides.
Q: Why does doubling X2 from 18 to 36 increase profit by only about $3 (from $25.27 to $28.24) even though output rises from 48.07 to 76.30?
A: Revenue rises substantially (from $72.10 to $114.45), but costs also rise sharply. Fixed cost doubles from $28.80 to $57.60, and variable cost rises from $18.02 to $28.61 because the firm optimally uses more X1 when X2 is higher. The net gain in profit is the difference between the revenue increase and the total cost increase, which is modest given decreasing returns to scale.
Q: Explain why fixed costs do not appear in the first-order condition for optimal X1, yet they still matter for the firm's profit.
A: Fixed costs are constant with respect to X1, so when you take the derivative of the profit function with respect to X1, the fixed cost term drops out (its derivative is zero). This means fixed costs do not influence the choice of X1. However, fixed costs are subtracted from revenue when computing profit, so they directly reduce the profit level. A firm could be optimising correctly and still make a loss if fixed costs exceed the margin between revenue and variable cost.
Q: In a Cobb-Douglas production function, what does the sum of the exponents tell you about the technology? What is the sum in this lab, and what does it imply?
A: The sum of the exponents indicates returns to scale. If the sum equals 1, constant returns to scale. Greater than 1, increasing returns. Less than 1, decreasing returns. In this lab, 0.25 + 0.5 = 0.75, indicating decreasing returns to scale. Scaling up all inputs proportionally yields a less-than-proportional increase in output.
This material connects directly to cost theory: the total cost, variable cost, and fixed cost breakdown here is the foundation for deriving short-run cost curves (ATC, AVC, MC) covered later in the course. The first-order condition MVP = input price is the demand-side counterpart to marginal cost pricing on the supply side, and understanding both is essential for general equilibrium.
The Cobb-Douglas form reappears in consumer theory (utility functions with similar structure), growth theory (the Solow model uses a Cobb-Douglas aggregate production function), and econometrics (log-linear estimation of production functions from data).
Production economics, Cobb-Douglas production function, profit maximisation, profit maximization, marginal product, marginal value product, MVP, first-order condition, FOC, variable input, fixed input, short-run production, returns to scale, decreasing returns to scale, input demand, comparative statics, APEC 3002, managerial economics, applied microeconomics, output elasticity, factor demand, cost minimisation, production optimisation