Refer to the following figure when answering the following questions.
Figure 4.1: Production Function

Consider Figure 4.1. For given and fixed
ˉ
L
, the shape of this production function suggests that α in the production function Y=Kα
ˉ
L
1−α is ______.

Question

Refer to the following figure when answering the following questions.
Figure 4.1: Production Function

Consider Figure 4.1. For given and fixed 
ˉ
L
, the shape of this production function suggests that α in the production function Y=Kα
ˉ
L
1−α is ______.

BlackTom AI · Accepted Answer

First, note that the question analyzes a production function of the form Y = K^α L̄^{1−α}, where L̄ is fixed (given). With L̄ held constant, the only variable input in this expression is K, so the shape of Y as a function of K is driven by the exponent α on K.

Option exploration (without assuming any specific choice from the set):
- If α were equal to 1, Y would be proportional to K^1 L̄^0, i.e., Y = K, up to a constant factor from L̄, implying a linear (not concave) relationship in K when L̄ is fixed. This would not exhibit diminishing returns to capital with respect to K, given a fixed L.
- If α were greater than 1, the K-part would exhibit increasing returns in K (Y would rise more than proportionally with K), which is inconsistent with the conventional interpretation of a Cobb-Douglas-like form where the capital exponent is less than one to ensure concavity in K when other inputs are fixed.
- If α were between 0 and 1, the function Y = K^α L̄^{1−α} is concave in K (for fixed L̄), exhibiting diminishing returns to capital: each additional unit of K raises output by a smaller amount than the previous one, given the fixed L̄. This is the standard interpretation of a production function with α ∈ (0,1).
- If α were less than 0, the function would behave oddly with respect to K (output would rise as K decreases, which is implausible in this context and contradicts typical production-function shapes).

In the context of the question, the most natural and conventional interpretation is that α must be less than one to maintain diminishing marginal returns to capital when labor is fixed. The language of the prompt invites recognizing that, with L fixed, the exponent on K determines the concavity with respect to K, and concavity requires 0 < α < 1 (in particular, α < 1).

Therefore, the statement that α is less than one is consistent with the implied shape of the production function when L is fixed and we consider the marginal behavior with respect to capital.

FA25 ECON 302 002 Homework #2 (Model of Production)

Refer to the following figure when answering the following questions. Figure 4.1: Production Function Consider Figure 4.1. For given and fixed ˉ L , the shape of this production function suggests that α in the production function Y=Kα ˉ L 1−α is ______.

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