EconS 301 â Intermediate Microeconomics Review Session #5

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( ) ( ) 1 2 Qλ = 50 λMλL + λM + λL 1 1 1 2 2 2 ( ) = 50λ λ ML + λM + λL 1 2 = 50λ ML + λM + λL 1 ⎡ ⎤ = λ 50( ML) 2 ⎢ + M + L ⎣ ⎥ ⎦ = λQ Thus, by increasing all inputs by a factor λ , output goes up by a factor of λ . Since output goes up by the same factor as the inputs, this production function exhibits constant returns to scale. b) Is the marginal product of labor ever diminishing for this production function If so, when Is it ever negative, and if so, when Answer 1 2 ⎛M ⎞ Recall MPL = 25⎜ ⎟ + 1. Suppose M > 0. Holding M constant, increasing L will decrease the MP L . ⎝ L ⎠ The marginal product of labor is decreasing for all levels of labor. The MP L , however, will never be negative since both components of the equation will always be greater or equal to zero. In fact, for this production function, MP L ≥ 1. Exercise 3 2 1 2 2 1 1 3 3 ⎛5⎞ − 3 3 ⎛10⎞ − 3 3 A firm’s production function isQ = 5L K with MPK = ⎜ ⎟L K and MPL = ⎜ ⎟L K . ⎝3⎠ ⎝ 3 ⎠ a) Does this production function exhibit constant, increasing, or decreasing returns to scale Answer Thus, we have constant returns to scale. 2 1 ( ) 3 ( ) 3 Qλ = 5 λL λK 2 1 2 1 3 3 3 3 = 5λ λ LK 2 1 3 3 = 5λLK 2 1 ⎡ ⎤ 3 3 = λ ⎢5LK ⎥ ⎣ ⎦ = λQ 2
) What is the marginal rate of technical substitution of L for K for this production function Answer 1 1 ⎛10 ⎞ − 3 3 L K MP ⎜ ⎟ L 3 2K MRTSLK , = = ⎝ ⎠ = 2 2 MPK ⎛5 ⎞ − L 3 3 ⎜ ⎟LK ⎝3 ⎠ c) What is the elasticity of substitution for this production function Answer The formula for the elasticity of substitution is given by, ⎛ ⎛ K ⎞ ⎞⎛ ⎞ ⎛ K ⎞ d ⎛ ⎞ percentage change in capital to labor ratio ⎜ Δ ⎜ ⎟ L ⎟⎜ MRTSLK , ⎟ ⎜ ⎟ L ⎜ MRTSLK , ⎟ σ = = ⎜ ⎝ ⎠ ⎟ ⎝ ⎠ ⎜ percentage change in MRTS K ⎟ = ⎜ K ⎟ LK , ΔMRTSLK , ⎜ ⎟ dMRTSLK , ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ L ⎠ ⎝ L ⎠ So from part (b) we know, ⎛K ⎞ d 2K MRTSLK , K MRTS ⎜ ⎟ MRTSLK , = → = 2 LK , and L ⎛K ⎞ L = 2 . So, ⎝ L ⎠ dMRTS , ⎜ ⎟ ⎝ L ⎠ ⎛K d ⎜ ⎞ L ⎟ ⎛ ⎞ MRTSLK , 1 Thus, σ = ⎝ ⎠ ⎜ ⎜ ⎟ ⎛ ⎞ 2 1 dMRTS K ⎟= ⎜ ⎟ = . LK , ⎜ ⎟ ⎝2 ⎠ ⎝ L ⎠ LK 1 = . 2 Exercise 4 Suppose a firm’s production function initially took the formQ 500( L 3 K) of a manufacturing innovation, its production function is now Q = 1000(0.5L+ 10 K) . = + . However, as a result a) Show that the innovation has resulted in technological progress in the sense defined in the text. Answer We simply need to show that given a fixed combination of inputs, the quantity produced will increase as a result of the innovation. Assume a fixed level of labor and capital at 2 and 2 units respectively. Now we just calculate the quantities produced before the innovation and after the innovation using these input levels. Before the innovation, Q = 500( L+ 3 K) = 500(2 + 3(6)) = 4000 . And after the innovation, Q = 1000(0.5L+ 10 K) = 1000(1 + 20) = 21000 . So we obviously have more output after the innovation, thus we have technological progress. 3
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Page 5: 3. The MRTS LK , = a. MPK MPL b.

EconS 301 â Intermediate Microeconomics Review Session #5

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EconS 301 â Intermediate Microeconomics Review Session #5