Today we revisited some of the concepts we touched on last week, when we looked at economic growth when the labor force and efficiency are constant (that is, they do not grow).

**The Cobb-Douglas production function**This function gives us a behavioral relationship between the three key variables of interest: capital stock per worker, output per worker and labor efficiency.**The equilibrium condition**We saw that the equilibrium condition that exists for the Cobb-Douglas production function is the condition that holds the capital stock constant. People, foreigners and the government all save money, which is reinvested to build the capital stock. Depreciation reduces the capital stock. When the investment and the depreciation match up – this is the equilibrium condition.**An example**If the government suddenly decides to increase expenditure and reduce taxes, then governmental savings will decrease. Since governmental savings decrease, this causes a decrease in overall savings. Because the savings have decreased (but depreciation is still the same) the overall capital stock decreases up to the point at which the new lower level of savings can maintain it. This new lower level of capital stock results in a lower output per worker, so the equilibrium point shifts. The figure below shows what happens:

*A decrease in **governmental savings decreases the savings rate. This allows capital to depreciate without being replenished and changes the equilibrium line, making it steeper (shift left). This causes a decrease in equilibrium output per worker (or real GDP per capita).*

So far, we assumed that the labor stock is constant, and that the labor efficiency is also constant. Now we remove that assumption and assume that they grow at a constant rate every year. So what’s the new equilibrium point? Well, the *principle* is the same. To calculate the new equilibrium point, we must have capital intensity (or capital stock per unit output) constant. For no-growth, the capital intensity was stable at the savings rate-depreciation rate ratio.

If **labor **is growing at a constant rate (call this rate *n*% per year), then capital stock must increase to keep up with the increase in labor. To have the same capital stock per worker before and after the increase in labor, capital must grow by that same *n*% per year.

If **efficiency** is also growing at a constant rate (call this rate *g*% per year), then capital stock must increase to keep up with the increase in efficiency. Again, capital stock must grow at the same rate as the efficiency. That means capital also grows at *g*% per year.

Don’t forget that we still have **depreciation** in the capital stock, and this must be replaced.

Now, to maintain all this growth, and to replace depreciated capital, the money has to come from somewhere. And it does – from *investment* (or savings). So the total savings per year increase to maintain that additional growth in capital stock.

So, to tally everything up – investment has to cover:

- Growth in capital stock coming from
*labor*(growing at*n*% per year) – so capital must increase by*n*K per year. - Growth in capital stock coming from efficiency (growing at
*g*% per year) – so capital must increase by*g*K per year. - And depreciation of capital stock (depreciating at
*delta*% per year) – so capital must increase by*?*K per year.

The total increase in capital that investment must cover? *n*K + *g*K +* delta*K

Since total savings are the same as total investment, we can say that: sY = *n*K + *g*K +*delta*K.

And it so happens that since this is the rate at which capital is constant, it is also the equilibrium condition! Do a little mathematical manipulation, and we get:

The same logic as before applies here that we used to justify the no-growth equilibrium.

If the savings rate decreases without corresponding decreases in output, the right hand side (RHS) will become smaller than the left hand side (LHS). But total capital stock will also decrease since it’s not being replenished, making the LHS smaller. This will cause the two to be equal again. The same applies for increases in *n*, *g* or *delta*.

If the savings rate increases without corresponding decreases in output, the RHS will become larger than the LHS. But an increase in the savings rate means an increase in investment into capital, which would cause an increase in capital, which would cause an increase in the LHS, making them equal again. The same applies for decreases in *n*, *g* or *delta*.

That’s enough for now. In another post, I’ll talk about the production function sequel.

If you enjoyed this post, make sure you subscribe to my RSS feed! You can also follow me on Twitter here.

If you enjoyed this post, make sure you subscribe to my RSS feed! You can also follow me on Twitter here.