Macroeconomics: economic crisis update, by Charles I. Jones Notes: Chapter 4: A model of production The Cobb-Douglas production function: Y = AKαLβ A 1 Exercise: Please try different values for A, α, β, and the values of K and L in line 11 and se α 1/3 When α + β = 1, we have constant returns to scale. In that case, if we triple both K and L, β 2/3 When α + β > 1, we have increasing returns to scale. In that case, if we triple both K and L K L Y MPK MPL 8 6 6.604 0.275 0.734 By comparing this and the next line, note th 24 18 19.812 0.275 0.734 So, if you, say, double both K and L, MPK an ΔK ΔY ΔY/ΔK ΔL ΔY ΔY/ΔL 1 0.264431 0.264431 1 0.714757 0.714757 0.100000 0.027402 0.274022 0.1 0.073174 0.731738 0.010000 0.00275 0.275046 0.01 0.007336 0.733558 0.001000 0.000275 0.275149 0.001 0.000734 0.733741 0.000100 2.75E-05 0.275159 0.0001 7.34E-05 0.73376 0.000010 2.75E-06 0.27516 0.00001 7.34E-06 0.733761 0.000001 2.75E-07 0.275161 0.000001 7.34E-07 0.733762 1E-07 7.34E-08 0.733762 1E-08 7.34E-09 0.733762 1E-09 7.34E-10 0.733761 Note that ΔY/ΔK converges (or, reaches a steady value) as ΔK becomes small. Same is true for ΔY/ΔL as ΔL becomes small. These converged values are the MPK and MPL. MPK = αY/K and MPL = βY/L are also true. So, this is a second way to calculate MPK and MPL, as in lines 11 and 12. values of K and L in line 11 and see what happens to Y and to MPK and MPL. The yellow cells change automatically, according to the for at case, if we triple both K and L, Y will also triple. (Check lines 11 and 12.) hat case, if we triple both K and L, Y will more than triple. When α + β < 1, we have decreasing returns to scale. In that case, if we triple ing this and the next line, note that under constant returns to scale, MPK and MPL depend only on K/L. say, double both K and L, MPK and MPL will be unaffected. e for ΔY/ΔL as ΔL becomes small. MPL, as in lines 11 and 12. utomatically, according to the formulae embedded in them. s to scale. In that case, if we triple both K and L, Y will less than triple. (Check.) Homework Due on Wednesday, February 24, 9:00 am Exercise 5, page 91 Assume Y = AK1/3L2/3 Country k y A w r rk rk/y United States 79865 33293 Canada France Hong Kong South Korea Indonesia Argentina mexico Kenya Ethiopia 328 635 Q: The countries that are poorer than the United States are poorer soley because they have less capital per person. True or F Write your answer in the cell above. Assume Y = AK3/4L1/4 w r rk rk/y s capital per person. True or False?
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