Sol-PS ECON 290

Problem Set 2 ECON 290, Models of Choice in Competitive Markets Jean Guillaume Forand∗ Fall 2016, Waterloo 1. Suppose that a firm has a the production function f(z1, z2) =    2z1 + z2 if z1 ≤ 2, 0 if z1 2. (a) Show that this production technology satisfies the no-free lunch property. Solution. Working with the production set, we need to show that (0, 0, q) ∈ Q only if q = 0. To see this, note that (0, 0, q) ∈ Q implies that q ≤ f(0, 0) = 0, so that because q ≥ 0, it must be that q = 0. Working with the production function, we need to show that f(0, 0) = 0, which is true (see above). Working with input sets, we need to show that (0, 0) ∈ Z(q) only if q = 0. To see this, note that if q 0, then for any (z1, z2) ∈ Z(q), we have 2z1 + z2 ≥ q 0, so that we must have z1, z2 0. (b) Does this production technology satisfy the free disposal property? Solution. No, this production technology does not satisfy the free disposal property. To show this, we only need to find examples of production plans that violate the definition of the property. Working with the production set, consider production plans (1, 1, 1) and (4, 4, 1/2). Note that (1, 1, 1) ∈ Q (because 1 ≤ 2 · 1 + 1) and that production plan (4, 4, 1/2) has more inputs and less outputs than (1, 1, 1). However, we have that (4, 4, 1/2) ∈/ Q (because q = 1/2 0 = f(4, 4)). Working with the ∗Room 131, Department of Economics, University of Waterloo, Hagey Hall of Humanities, Waterloo, Ontario, Canada N2L 3G1. Office phone: x. 33635. Email: . Website: http://arts. 1 production function, consider inputs (1, 1)