Question

Consider the following Cobb-Douglas production function for a firm that uses labor hours (L), capital (K), and energy (E) as inputs:

Q = (0.0012L^0.45)(K^0.3)(E^0.2)

1. Determine the labor, capital and energy production elasticities.
2. Suppose that worker hours are increased by 2 percent holding other inputs constant.  What would be the resulting percentage change in output?
3. Suppose that capital is decreased by 3 percent holding other inputs constant.  What would be the resulting percentage change in output?
4. What type of returns to scale appears to characterize this production function?

Answer 1

page-01 see and oure 0.45, 0.3 and 0.2 Hence = 0 3 = 0.2 Answeg cobb- douglas production function for a fixm that uses labor howns (L), Capital (k), and energy (E) inputs: (0.001920.4536 03160:2) a) labor, capital energy production elasticities Accoding to the above function clearly that exponents over labos, capital energy srespectively labor production elasticity = 0.45 capital production elasticity & Energy produition elasticity - b) Resulting percentage change in output labor prodution elasticity = 0.45 means that 11. increase in working hours keeping all other inputs constant, will increase output by 0.45-1. and hence a 1. increase increase product or output by 0.90-1. c) Resulting porcentage changeoutput capital pooduitton elastiáty means that I'l. increase in working hours keeping all other inputs will decrease output by 0.3.1. and hence 3:1. decease in labor hours will decene poodution o output by ) 3x0.3 = 0.90-1. fence it change in output = -0.91 where negative sign laboy hours will -) ex 0.45 = 0.3 constant 1

) F, ) indicates that output will decrease. page-02 (d) what type ist returns to scale appears to charcuterize this prodution function A function exhibit decreasing returns to scale if f(th, th, te) < tx f(K, L, E) for all t>/ A function exhibit increasing returns to scale if Alth, th, tE) >txfCK, L, E ) for all t>/ A function exhibit constant sietwins to scale if f (tk, th, te ) = txt ck, L,E) for all t >| Here fck, L, E) 0.45k F (tk, TL, EE) = 0.0012(+4) 0.49(+2) 0.3 (tEjº.2 -) F(tk, TL,ZE) = 10:45+0-3+0.2 ť 13 0 for all to 760.001220.45 0.3 0.3) = 4 x4(k:28) returns Hence this function 0.3 0.2 = 0.0012 L 02 0.45 0.3 0:0012L к 0.95 (0.0012_00454-363) <t As +0.95 기 K 2) - 1 exhibit decreasing to scale.