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class handout

# class handout - MRTS = L K We need to substitute for...

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Class Handout (October 2) 100B, Prof. Jacobsen 1. Suppose that q = f ( L , K ) = L 2 + K 2 ( ) 1 2 , and q = 20, and K = 16 (in the short run). a. Find the amount of labor, L , employed by the firm. L = 12 b. Calculate the marginal product of labor. Does the production function satisfy diminishing marginal returns to labor? MP L = L L 2 + K 2 ( ) 1/2 The function does not satisfy diminishing marginal returns to labor: ! MP L ! L = K 2 L 2 + K 2 ( ) 3/2 > 0 c. Derive the marginal rate of technical substitution (MRTS) in the long run ( K is no longer fixed at 16). MRTS = ! L K

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d. Does the production function have a diminishing |MRTS|? Take absolute value first:
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Unformatted text preview: MRTS = L K We need to substitute for K (because it is a function of L along an isoquant). From the definition of an isoquant: q = L 2 + K 2 ( ) 1/2 q 2 = L 2 + K 2 K = q 2 ! L 2 ( ) 1/2 Substituting: MRTS = L q 2 ! L 2 ( ) 1/2 ! MRTS ! L = q 2 q 2 " L 2 ( ) 3/2 > This function does not have a diminishing |MRTS|. The isoquants are concave: e. Does the production function satisfy increasing returns to scale, constant returns to scale, or decreasing returns to scale? The function is constant returns to scale....
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