# MPK, MPL by eot15664

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```									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
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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