Which of the following is a Cobb–Douglas production function?

I. Q = f(K, L)

II. Q = 5K + 2.5L

III. Q=min{K, 2L}

IV. Q= K50L0.75

Question

Which of the following is a Cobb–Douglas production function?

I. Q = f(K, L)

II. Q = 5K + 2.5L

III. Q=min{K, 2L}

IV. Q= K50L0.75

BlackTom AI · Accepted Answer

The question asks which of the given functional forms is a Cobb–Douglas production function. I will evaluate each option in turn.
Option I: Q = f(K, L). This is a generic statement indicating a production function with capital K and labor L, but it provides no specific functional form. A Cobb–Douglas production function is a precise functional form with multiplicative powers of inputs, typically Q = A K^α L^β. Simply writing f(K, L) does not identify a Cobb–Douglas form, so this option is not necessarily a Cobb–Douglas function.
Option II: Q = 5K + 2.5L. This is a linear additive (sum) form in K and L, i.e., a linear production function. Cobb–Douglas functions are multiplicative in inputs and exhibit constant elasticity of substitution of unity only in log-linear form, not additive in inputs. Therefore, this option does not represent a Cobb–Douglas function.
Option III: Q = min{K, 2L}. This is a Leontief (perfect complements) production function, where output is constrained by the bottleneck input. Cobb–Douglas functions do not take a min form; they are smooth and differentiable with diminishing marginal products. Hence, this is not Cobb–Douglas.
Option IV: Q = K^{0.5} L^{0.75} (written as Q = K50L0.75 with the implied exponents). This is a multiplicative function of K and L with exponents that sum to 1.25, which indicates constant returns to scale only if the sum equals 1; in this case the sum is 0.5 + 0.75 = 1.25, which would imply increasing returns to scale. Nevertheless, the functional form—Q = A K^α L^β with α, β > 0—is the defining characteristic of a Cobb–Douglas production function. The specific powers show the Cobb–Douglas structure, even if the returns to scale are not unit. Therefore, this option matches the Cobb–Douglas form.
In summary, each option was evaluated against the defining characteristics of a Cobb–Douglas production function: a smooth, multiplicative form with inputs raised to constant exponents. Only the fourth option exhibits this multiplicative power structure with K and L raised to positive exponents, characterizing a Cobb–Douglas specification.

Which of the following is a Cobb–Douglas production function? I. Q = f(K, L) II. Q = 5K + 2.5L III. Q=min{K, 2L} IV. Q= K50L0.75单项选择题

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