Exercise 5.b.1 from microeconomics theory by Andreu Mas-Colell

Draw two production sets: one that violates irreversibility and one that satisfies this property.

1. non increasing returns to scale
2. non decreasing returns to scale
3. constant return to scales

return Exercise 5.B.1: Draw two production sets: one that violates irreversibility and one prope that satisfies this property. (a) N return (vii) Nonincreasing returns to scale. The production technology Y exhibits (b) N return nonincreasing returns to scale if for any ye Y, we have aye Y for all scalars a E [0, 1]. In words, any feasible input output vector can be scaled down (see Figure 5.B.5). Note that nonincreasing returns to scale imply that inaction is possible [property (iv) ]. (viii) Nondecreasing returns to scale. In contrast with the previous case, the production process exhibits nondecreasing returns to scale if for any ye Y, we have aye Y for any scale a 2 1. In words, any feasible input-output vector can be scaled up. Figure 5.B.6(a) presents a typical example; in the figure, units of output (good 2) can be produced at a constant cost of input (good 1) except that in order to produce at all, a fixed setup cost is required. It does not matter for the existence of nondecreasing returns if this fixed cost is sunk [as in Figure 5.B.6(b)] or not [as in Figure 5.B.6(a), where inaction is possible]. (ix) Constant returns to scale. This property is the conjunction of properties (vii) and (viii). The production set Y exhibits constant returns to scale if ye Y implies aye Y for any scalar a 2 0. Geometrically, Y is a cone (see Figure 5.B.7)