Cobb Douglas II: This Time It’s With Intensity!

So far when we talk about the behavioral relationship between the variables of interest, we are talking about the Cobb-Douglas production function. This relates output per worker (real GDP per capita, Y/L), capital stock per worker and labor efficiency (or technology). That function looks something like this:

Cobb Douglas Production Function

However, whenever we talk about equilibrium, we always talk in terms of the capital-output ratio. For instance, in the no-growth model, the equilibrium condition was shown to be:

Equilibrium 2

And in the constant-growth case, the equilibrium condition was:

Equilibrium with Labor and Efficiency Growth

So it might help us if we rewrote the production function relating output per worker (or real GDP per capita) with capital intensity or the capital-output ratio. That production function looks like this:

Cobb Douglas - The Sequel

Let’s call this production function the sequel, since like a Hollywood sequel, it has the same principle characters (but in different form), it is basically the same formula, and you get the same thing from both.

Now let’s take a look at what this function looks like, and what happens when we vary alpha, (the diminishing-returns-parameter):

Varying Alpha in the Cobb Douglas Sequel
The Cobb Douglas “sequel” for increasing alpha. Note that increasing alpha increases overall output for any given capital intensity. Also, a lower alpha causes a slower increase in output as capital intensity increases, compared to a higher alpha (the line is flatter and “less curvy” for lower alpha).

As you can see, increasing alpha gives higher capital per worker for a given level of capital intensity. Also, a higher alpha will cause the line to be steeper and increase output per worker faster for a given increase in capital intensity. Below is the change in Cobb Douglas: The Sequel, for increases in labor efficiency, E.

Cobb Douglas - The Sequel for Varying levels of Efficiency
Output per worker increases with increases in efficiency (E), and the growth is proportionate. That is (for instance), a five-fold increase in E will cause a corresponding five-fold increase in output per worker.

Thus, increases in labor efficiency cause corresponding increases in output per worker.

All the datasets that created the graphs above have been uploaded to the ‘Handouts’ section to the right. I encourage you to explore the data, change the numbers and see how that affects the shape and level of the curve.

About the relationship of Cobb Douglas: The Sequel with g (the growth rate of efficiency) This relationship is nonexistent. Since the production function is a “snapshot in time” and talks only about a specific time-period, we cannot have a single production function showing changes g. In class, I meant to have the slide show the relationship with the steady-state capital-labor ratio and g, NOT the Cobb Douglas Sequel and g. Apologies for the confusion. I’ve removed that slide and will replace it for the next class.

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6 comments for “Cobb Douglas II: This Time It’s With Intensity!

  1. Andrea Ringer
    February 6, 2008 at 1:04 pm

    Changingthe production function from one focusing on capital labor ratio to a function focusing on capital output ratio:

    multiply K/Y by Y/L

    regroup K/Y and Y/L by raising them seperately to alpha

    divide out both sides by (Y/L) raised to alpha

    raise both sides now to the 1/(1-alpha) power, which gets Y/L alone on the left side.

    The result is the capital output ratio form of the production function.

  2. Arnav
    February 6, 2008 at 5:35 pm

    Unfortunately, that’s not right, Andrea. Someone else still has a chance…

  3. Aaron Ordower
    February 6, 2008 at 10:35 pm

    (Y/L)=(K/L)^(alpha)(E)^(1-a)
    Multiply (K/L)^a(E)^(1-a) by [1/(1-alpha)/1/(1-alpha)]
    Then you get the new:
    (Y/L)=(K/Y)^(a/1-a)(E)

  4. Nathaniel Futterman
    February 6, 2008 at 11:00 pm

    First, set the two equal to each other, since both equal Y/L.
    (K/L)^alpha * E^(1-alpha) = (K/Y)^(alpha/(1-alpha)) * E

    Next, divide both sides by E. This simplifies to:
    ((K/L)^alpha)/(E^alpha) = (K/Y)^(alpha/(1-alpha))

    Then, if you take the alpha-root of each side, you get:
    (K/L)/E = (K/Y)^(1/(1-alpha))

    Next, multiply both sides by E. You get:
    (K/L) = (K/Y)^(1/(1-alpha)) * E

    Once you have this equality, you can substitute this into the original equation for K/L.
    Thus, you have:
    (Y/L) = ( (K/Y)^(1/(1-alpha)) * E )^alpha * E^(1-a)

    When you multiply this all out, you are left with:
    Y/L = (K/Y)^(alpha/(1-alpha)) * E^alpha * E^(1-alpha)

    which equals
    Y/L = (K/Y)^(alpha/(1-alpha)) * E

    which is the second equation.

  5. Christopher Desvernine
    February 7, 2008 at 2:31 pm

    Y/L = (K/L)^alpha (E)^(1-alpha)

    First step: Raise both sides to 1/(1-alpha)

    (Y/L)^(1/(1-alpha)) = (K/L)^(alpha/(1-alpha)) * (E)

    Second step: Find a multiplicative factor that will yield Y/L. So, multiply both sides by (Y/L)^(-alpha/(1-alpha)).

    Y/L = (K/L)^(alpha/(1-alpha)) (Y/L)^(-alpha/(1-alpha)) * (E)

    Final step: Regroup.

    Y/L = (K/Y)^(alpha/(1-alpha)) * (E)

  6. Arnav
    February 7, 2008 at 8:59 pm

    Excellent jobs – Nathaniel and Christopher. Extra kudos to both of you for not mentally “copy-pasting” from the book. I think for that you both deserve the bonus half-point.

    Sorry Aaron, I’m not quite clear about that multiplicative factor of yours…

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