The Cobb-Douglas Aggregate Production Function



The Cobb-Douglas Aggregate Production Function

Output “Y” is a function of labor or employment “N” the stock of capital “K” and the level of technology “A”. “Alpha” is the exponent on employment. It is the elasticity of output with respect to labor.

“Alpha” is the exponent on employment. It is the elasticity of output with respect to labor.

This proposition could be more elegantly derived using logs:

(Where I happen to have substituted 0.7 for alpha.)

a) Finding the level and rate of change of technology from Y, N and K: Ch3N1

(This is also what one does in the Macrodata exercises for chapters 10 and 11)

To find the growth of total factor productivity, you must first calculate the value of A in the production function. This is given by A = Y / (K.3N.7). The growth rate of A can then be calculated as

[(Ayear 2 - Ayear 1)/ Ayear 1] x 100%. The result is:

| |real Y |K |N |A |% chg A | | |

|1960 |2380 |2610 |65.8 |11.99021 | | | |

|1970 |3580 |3850 |78.8 |14.14739 |17.99124 |< 60's growth rate |

|1980 |4900 |5570 |99.3 |14.74263 |4.207408 |< 70's growth rate |

|1990 |6710 |7420 |118.8 |16.33916 |10.82935 |< 80's growth rate |

|2000 |9190 |9850 |135.2 |18.77591 |14.91352 |< 90's growth rate |

Note that the denominator for A is simply an exponentially weighted average of the inputs.

I rounded off the Y and K series above so my A is deliberately a bit off. We still see that the 90’s don’t quite match the 60’s.

Note as well that these are decadal and not annual growth rates!

If we want annual rates we use the idea that (1+g)10=1+G, where G is the decade proportional growth rate and g is the annual proportional growth rate.

|% chg A | | |proportional decade rate | |

| | | | |proportional annual rate |

|17.99124 |< 60's growth rate |0.179912 |0.016682 | | |

|4.207408 |< 70's growth rate |0.042074 |0.00413 | | |

|10.82935 |< 80's growth rate |0.108294 |0.010335 | | |

|14.91352 |< 90's growth rate |0.149135 |0.013998 | | |

So at an annual rate the 60’s had a 1.6% rate of tech change while the 70’s had less than half a percent and the 90’s had 1.3%.

b) Calculate the marginal product of labor by seeing what happens to output when you add 1.0 to N; call this Y’, and the original level of output Y. [A more precise method is to take the derivative of output with respect to N; dY / dN = 0.7A(K / N)0.3 .]

|real Y |K |N |A |N' |Y' |Y'-Y |MPn | |1960 |2380 |2610 |65.8 |11.99021 |66.8 |2405.262 |25.26181 |25.31916 | |1970 |3580 |3850 |78.8 |14.14739 |79.8 |3611.742 |31.74182 |31.80202 | |1980 |4900 |5570 |99.3 |14.74263 |100.3 |4934.49 |34.48984 |34.54179 | |1990 |6710 |7420 |118.8 |16.33916 |119.8 |6749.487 |39.4873 |39.53703 | |2000 |9190 |9850 |135.2 |18.77591 |136.2 |9237.529 |47.52874 |47.58136 | |

b) Finding the optimal level of capital and investment: Ch4N7 first part

Here we are given information on the marginal product of capital and cost of using capital. From that information we determine the optimal (that is profit maximizing level of capital and how much investment the economy should undertake given its current capital stock.

The cost of using capital depends on the following “given” constants r =0.2, d=0.2 (depreciation, this means that 20% of a unit of capital wears out each year). pK =1 (the price of capital) and τ=0.5 (the tax rate on the output produced by capital).

Also given MPKf = 20.4 - .01Kf and that the current capital stock is 1875.

N7a.

Plug the cost variables into the chapter’s formula for the unit cost of capital uc=(r + d)pK. If we purchase a unit of capital with borrowed funds we need to pay (r)pK in interest we also need to set aside (d)pK to cover the cost of depreciation. We also divide the unit cost by (1 - τ) where τ (“tau”) is the rate of taxation on capital’s product to include the cost of taxes that will be paid to the government. uc/(1 - τ) represents the unit costs of capital including taxes. In this particular example with τ=0.5 we are multiplying the unit cost of capital by 2. Each unit of capital will have to produce enough revenue to pay twice the pre tax cost of capital.

uc/(1 - τ) = (r + d)pK / (1 - τ)

= [(.2 + .2) x 1] / (1 - .5) = 0.8.

Now set the tax inclusive cost of capital equal to the marginal product of capital.

MPKf = uc/(1 - τ), so 20.4 - .01Kf = .8; solving this gives Kf = 1960.

Since Kf - K = I - dK, I = Kf - K + dK = 1960 - 1875 + (.2 x 1875) =460 .

b. i. Solving for this in general:

uc/(1 - τ) = (r + d)pK / (1 - τ) = [(r + .2) x 1] / (1 - .5) = .4 + 2r.

MPKf = uc/(1 - τ), so 20.4 - .01K = .4 + 2r; solving this gives Kf = 2000 - 200r.

I = Kf - K + dK = 2000 - 200r - 1875 + (.2 x 1875) = 500 - 200r.

From the first part of this problem we would know that

Id=500-200r this will be used in the second half of the problem which is found on NI.doc.

c) Finding the equilibrium wage and employment: Ch9N4 first part, Ch10N1

(See ISLM.doc for the second part of Ch9N4)

Ch10N1 and the first part of Ch9N4 are essentially the same.

^ will indicate exponentiation

Part 1 of Classical Model: THE LABOR MARKET AND PRODUCTION (the real eco)

Labor demand:

Let Y=A(5N-0.0025N^2)

We are given the marginal product of labor, but could take the derivatives ourselves.

So MPN=A(5-2(0.0025)N)=5A-0.005AN

If A=2, then MPN=10-0.01N where N is labor demanded by firms

Note that in order to maximize profits firms should hire labor (set N) so that w=MPN where w is the real wage.

Labor supply:

Let N=55+10(1-t)w, where N is labor supplied

Let t=0.5, then N=55+5w

Equilibrium N

Substitute the MPN supply side equation for w in the supply equation.

We assume that labor supply equals labor demand because we are looking for an equilibrium where this must be the case.

N=55+5(10-0.01N)

N=55+50-0.05N

1.05N=105

N=100

W=10-0.01N=10-0.01(100)=9

Let’s check “N” and “W” on the demand side: plug them into the demand for labor.

N=55+5w

100=55+5(9)=55+45 yes it checks

OUTPUT (plug N into the production function)

Y=A(5N-0.0025N^2)=2(5*100-0.0025*10000)=2(500-25)=950

Now we are ready for the second half of problem Ch9N4 which is found on ISLM.doc.

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