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:

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:

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

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:

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):

*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.

*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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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.

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

(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)

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.

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)

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…