6. Suppose y = f(x) is a production function determining output y as a function of the vector x of nonnegative factor inputs, with f(0) = 0. Show that: (a) If f is concave, then f(x) ≤0 (so each marginal product f(x) is decreasing). (b) If f is concave, then f(x)/λ is decreasing as a function of λ. (c) If f is homogeneous of degree 1 (constant returns to scale), then f is not strictly concave.
6. Suppose y = f(x) is a production function determining output y as a function of the vector x of nonnegative factor inputs, with f(0) = 0. Show that: (a) If f is concave, then f(x) ≤0 (so each marginal product f(x) is decreasing). (b) If f is concave, then f(x)/λ is decreasing as a function of λ. (c) If f is homogeneous of degree 1 (constant returns to scale), then f is not strictly concave.
Chapter1: Making Economics Decisions
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Can you answer this question on pencil and paper. Thank you!
Transcribed Image Text:SM 6. Suppose y =
f(x) is a production function determining output y as a function of the vector x
of nonnegative factor inputs, with f(0) = 0. Show that:
(a) If f is concave, then f(x) ≤ 0 (so each marginal product f(x) is decreasing).
(b) If f is concave, then f(x)/λ is decreasing as a function of λ.
(c) If f is homogeneous of degree 1 (constant returns to scale), then f is not strictly concave.
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I don't believe part (b) was fully answered. Can you please solve it with more detail?
Transcribed Image Text:SM 6. Suppose y =
f(x) is a production function determining output y as a function of the vector x
of nonnegative factor inputs, with f(0) = 0. Show that:
(a) If f is concave, then f(x) ≤ 0 (so each marginal product f(x) is decreasing).
(b) If f is concave, then f(x)/λ is decreasing as a function of λ.
(c) If f is homogeneous of degree 1 (constant returns to scale), then f is not strictly concave.
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