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Economics 9451 Problem Set 5 – Due in Class Tuesday, October 6
1. Determine whether each of these technologies satisfies: i) closed ii) no free
lunch, iii) possibility of inaction, iv) free disposal, v) convexity, and vi) natural
inputs (In (e) and (f) the figures show the typical shape of input requirement
sets; assume V (0) = R2+ in (e)):
2. Consider the production set Z = {(z1 , z2 ) ∈ R2: z2 ≤ 1 − (1 + z1 )2 }.
(a) Demonstrate geometrically whether Z has each of the following properties (HINT: Carefully plot the curve z2 = 1 − (1 + z1 )2 ): i) nonempty ii)
closed iii) no free lunch, iv) possibility of inaction, v) free disposal, vi)
convexity, vii) natural inputs.
(b) Derive the supply/demand correspondence and the profit function.
(c) For what price ratios is z1 an input? For what price ratios is z1 an output?
3. Draw a production set that satisfies no free lunch, possibility of inaction, and
convexity; but violates free disposal.
4. Some production sets Z have a global returns to scale property, defined as
follows:
(a) Z has nonincreasing returns to scale if z ∈ Z ⇒ αz ∈ Z for every α ∈
(0, 1). That is, beginning from any feasible production plan, every linearly
scaled-down version of that production plan is also feasible. If Z is closed
and yet αz is always in the interior of Z when z is in the interior, we say
Z has decreasing returns to scale.
(b) Z has nondecreasing returns to scale if z ∈ Z ⇒ αz ∈ Z for every α ∈
(1, ∞). That is, beginning from any feasible production plan, every linearly scaled-up version of that production plan is also feasible. If Z is
closed and yet αz is always in the interior of Z when z is in the interior,
we say Z has increasing returns to scale.
(c) Z has constant returns to scale if it has both nonincreasing and nondecreasing returns to scale.
Draw a production set that has decreasing returns to scale but that is not convex.
5. Decide whether each of the following technologies has each of the following
properties: i) nonempty ii) closed iii) no free lunch, iv) possibility of inaction,
v) free disposal, vi) convexity, vii) natural inputs. Also characterize the returns
to scale of each.
(a) Z = {(x, y) ∈ R2: x < 1 and y ≤ ln(−x + 1)}.
(b) Z = {(x, y) ∈ R2: x2 + y 2 ≤ 1}.
(c) V (y) = {x ∈ R2+: x21 + x22 ≥ y} for y ≥ 0.
(d) Z = {(x, y) ∈ R2: y ≤ (−x)3 }.
(e) V (y) = {x ∈ R2+: x1 ≥
y
x2 }
for y ≥ 0.
(f)
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