Home / Courses / Intermediate Microeconomics / Unit 5

Intermediate Microeconomics Unit 5: Production Functions

A production function describes how inputs become output; marginal products, isoquants and returns to scale are the tools that later determine a firm's costs and supply.

Covers: Production functions and the short run, Marginal and average product, Diminishing marginal returns, Isoquants and the MRTS, Returns to scale, Common technologies and elasticity of substitution, Technical change

Study guide

A production function q = f(K, L) gives the most output a firm can make from capital K and labor L with its current technology. In the short run at least one input, usually capital, is fixed, so output changes only through the variable input. The marginal product of labor, MPL = ∂q/∂L, is the extra output from one more worker; the average product, APL = q/L, is output per worker. The law of diminishing marginal returns says that as more labor is added to fixed capital, MPL eventually falls. When MPL is above APL, the average rises; when it is below, the average falls, so MPL crosses APL at the peak of APL.

In the long run the firm can vary all inputs. An isoquant joins every input combination that produces the same output, much like an indifference curve, except that the output level has a real meaning. The absolute slope of an isoquant is the marginal rate of technical substitution, MRTS = MPL/MPK: the amount of capital that one more worker can replace while keeping output constant. Isoquants are usually convex, with a diminishing MRTS, because it becomes harder to replace capital with labor as capital gets scarce.

Returns to scale describe what happens when all inputs are multiplied by the same factor t. If output rises by exactly t, returns are constant; by more than t, increasing; by less, decreasing. For Cobb-Douglas q = AK^a L^b, the sum a + b tells the story: a + b = 1 is constant, above 1 increasing, below 1 decreasing. Returns to scale are not the same as diminishing marginal returns: a Cobb-Douglas function with exponents 0.6 and 0.6 has diminishing MPL and MPK but increasing returns to scale. For non-Cobb-Douglas forms, check by scaling the inputs directly.

Three technologies appear constantly. Fixed proportions, q = min{aK, bL}, have L-shaped isoquants and no substitution between inputs. Perfect substitutes, q = aK + bL, have straight isoquants and a constant MRTS of b/a. Cobb-Douglas has smooth convex isoquants with MRTS = (b/a)(K/L). The elasticity of substitution σ summarizes the curvature: 0 for fixed proportions, 1 for Cobb-Douglas, and infinite for perfect substitutes. Cobb-Douglas exponents are also output elasticities, and under competitive factor pay they equal each input's share of revenue. Technical progress shifts the production function up and moves every isoquant toward the origin.

Key ideas

• The short run has at least one fixed input; in the long run every input is variable.
• MPL is the derivative of output with respect to labor; APL is output per worker.
• MP above AP pulls AP up; MP equals AP at AP's maximum.
• Isoquants show equal-output input combinations, and their slope is the MRTS = MPL/MPK.
• Returns to scale concern scaling all inputs; diminishing returns concern one input.
• For Cobb-Douglas, returns to scale depend on the sum of the exponents.
• Fixed proportions, Cobb-Douglas and perfect substitutes have elasticities of substitution 0, 1 and ∞.

Formulas and rules

Common mistakes

• Confusing diminishing marginal returns with decreasing returns to scale.
• Using the sum-of-exponents rule on functions that are not Cobb-Douglas.
• Computing average product when the question asks for marginal product.
• Inverting the MRTS as MPK/MPL with labor on the horizontal axis.
• Assuming extra labor raises output in a fixed-proportions technology when capital is the binding input.

Before the test, make sure you can

• Calculate marginal and average products from a production function or a table described in words.
• Explain the relationship between MP and AP and find where AP peaks.
• Calculate the MRTS and explain why isoquants are convex.
• Determine returns to scale for Cobb-Douglas and for other functional forms.
• Distinguish diminishing marginal returns from returns to scale.
• Identify fixed-proportions, perfect-substitute and Cobb-Douglas technologies and their isoquants.
• Interpret Cobb-Douglas exponents as output elasticities and factor shares.

Words to know

Production function

The maximum output obtainable from each combination of inputs with a given technology.

Short run

A period in which at least one input is fixed.

Long run

A period long enough that all inputs can be varied.

Marginal product

The extra output from one more unit of an input, holding other inputs fixed.

Average product

Total output divided by the quantity of an input.

Diminishing marginal returns

The tendency for an input's marginal product to fall as more of it is used with other inputs fixed.

Isoquant

A curve showing all input combinations that produce the same output.

MRTS

The rate at which one input can replace another while keeping output constant; MPL/MPK.

Constant returns to scale

Scaling all inputs by t scales output by exactly t.

Increasing returns to scale

Scaling all inputs by t raises output by more than t.

Decreasing returns to scale

Scaling all inputs by t raises output by less than t.

Fixed-proportions technology

Inputs must be used in a set ratio; q = min{aK, bL}.

Perfect-substitutes technology

Inputs can replace each other at a constant rate; q = aK + bL.

Cobb-Douglas technology

q = AK^a L^b, with smooth convex isoquants and constant factor shares.

Elasticity of substitution

Percentage change in the input ratio K/L for a 1% change in the MRTS.

Output elasticity

Percentage change in output from a 1% change in one input; the exponent in Cobb-Douglas.

Technical progress

An improvement that lets the firm produce more with the same inputs, shifting isoquants inward.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.