# The production set is bounded because y 2 4 and y 0

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The production set is bounded because y 2 4 and y 0 imply 0 y 2 and 0 x 4. b ) Since the utility is continuous (we know all polynomials are continuous) and the production set is compact (closed and bounded), the Weierstrass Theorem applies to yield a maximum. 4. Suppose a firm’s production function is Q = K 1 / 3 L 2 / 3 and that K = 1000 and L = 125. a ) How much can the firm produce? b ) What are the marginal products of capital ( K ) and labor ( L )? c ) Suppose that the available capital falls by 2 units, while labor increases by 5 units. Without plugging the new numbers for K and L into the production function, compute approximately how much the firm can now produce. Answer: a ) Maximum production is Q = (1000) 1 / 3 (125) 2 / 3 = 250. b ) Now MP K = ∂Q/∂K = 1 3 K - 2 / 3 L 2 / 3 and MP L = ∂Q/∂L = 2 3 K 1 / 3 L - 1 / 3 . Using K = 1000 and L = 125 yields MP K = 1 12 and MP L = 4 3 . c ) The change in production is MP K Δ K + MP L Δ L = - 1 / 6 + 20 / 3 = 6 . 5. The resulting output level is 256 . 5.

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