Assumptions of the Neoclassical Model
Economics wouldn't get anywhere without assumptions. The neoclassical model has many.
Start with a production function. It assumes the Cobb–Douglas form.
$$\displaystyle Y = AF(K,L) = AK^{\alpha} L^{1 - \alpha} $$
Here, _Y_ is output, _A_ is total factor productivity, _K_ is capital, and _L_ is labor. It also assumes diminishing marginal returns to both capital and labor.
How would you interpret these assumptions of diminishing returns if you take either of these variables (say, capital) to infinity?
No, that's not right.
Total income will continue to grow as more capital is added.
Actually, no.
Income per person will continue to grow as more capital is added.
Exactly!
The change in income becomes smaller and smaller. Total income grows, and income per person grows, but the change becomes very small as additional capital is made available. That's the idea.
Once you add in the depreciation rate of capital, δ, assuming that some capital wears out every year, then logically there has to be a limit to the capital that can be added, and a steady state can be found. The change in capital here can be thought of as
$$\displaystyle \Delta K = sY - \delta K $$.
This is really the same thing put into an equation. The change in capital is some exogenously given saving rate, _s_, multiplied by output each period, which is total savings (which is also investment), minus the amount that wears out each period, which is the depreciation rate multiplied by the capital stock.
There's a little more to it than just that, though. Assume that population is growing at rate η, which means
$$\displaystyle \frac{\Delta L}{L} = \eta $$.
Then go back to the "change in capital" equation and think of how it could be converted into $$ \Delta k$$ instead of $$ \Delta K $$. In other words, think now in terms of capital per capita. Recognizing η as population growth, do you think that the η will work to increase $$ \Delta k$$, or decrease it?
No, it will decrease it, actually.
Right!
Since the question here is about capital per worker, more workers means less capital per worker. That's logical. When some math happens, the resulting percentage change in capital per capita then makes sense.
$$\displaystyle \frac{\Delta k}{k} = \frac{sY}{k} - \delta - \eta $$
Working toward a steady state of capital per worker, now consider that this is really a steady level of effective capital per worker; that is, the output-to-capital ratio is constant. With this imposed, the output growth per worker has to equal the capital growth per worker. Since output growth from the production function is
$$\displaystyle \frac{\Delta y}{y} = \frac{\Delta A}{A} + \frac{\alpha \Delta k}{k} $$,
this leads to the condition
$$\displaystyle \frac{\Delta y}{y} = \frac{\Delta k}{k} = \frac{\Delta A}{A} \times \frac{1}{(1 - \alpha)} $$.
One last substitution: Let θ be the change in technology, and you're left with a growth rate of output per capita of
$$\displaystyle \frac{\theta}{1 - \alpha} $$.
Then the growth rate of output has to be that plus some extra to cover the additional workers in the population that are added each year, or
$$\displaystyle \frac{\theta}{1 - \alpha} + \eta $$.
How would you interpret what happens to capital in the steady state when one additional worker enters an economy, assuming some amount of technological progress?
Yes!
This follows from the steady state condition.
$$\displaystyle \frac{\Delta y}{y} = \frac{\Delta k}{k} = \frac{\Delta A}{A} \times \frac{1}{(1 - \alpha)} $$
If technological progress is ignored, or assumed to be 0, then there's no change in capital per capita. Total capital grows, and capital per worker remains constant. But with technological progress, even the capital per worker grows in the steady state.
This is often confusing, so think of it like this: A worker has a hammer. If another worker is added, another hammer is needed. More workers, more capital, same capital per capita. But technology makes that hammer into a nail gun. Now when more workers are added, you don't need to add just more hammers, but more "super hammers," the new nail gun. This is additional capital on top of additional capital, so that capital per capita grows as well. That's the only way to maintain the output-to-capital ratio now.
Not exactly, although this is a tempting answer.
Capital per worker remains the same in a sense, but consider the technological progress as well. This growth rate is shown in the prior equations, and it mandates some change in capital per worker.
No.
These couldn't simultaneously hold if an additional worker were added to the economy. Total capital has to increase in a steady state.
The steady state ratio of output to capital can now be pulled together from some of these pieces:
$$\displaystyle \frac{Y}{K} = (\frac{1}{s})[(\frac{\theta}{1 - \alpha}) + \delta + \eta] \equiv \Psi $$.
That's a lot of Greek, and a little tough to interpret. So rearrange this by multiplying _s_ and _K_ on both sides, and converting to a per capita basis for the steady state.
$$\displaystyle sf(k) = [\delta + \eta + \frac{\theta}{(1 - \alpha)}]k $$
This is nice. The left-hand side is just the portion of output that is invested. It has to equal the right-hand side, which is the amount of capital needed to replace what wears out, cover any new workers, and adjust to new technology. Think carefully about each piece of this now; what graphical shape would match each side of this equation?
No.
One side isn't linear. Don't forget about diminishing returns that exist in this model.
No, the left side isn't linear.
Notice that each piece is just a numerical parameter multiplied by capital per worker.
Absolutely!
The left side is the concave production function, showing diminishing returns to capital again. On the right side, each piece is just a numerical parameter that's exogenously input to the model, each multiplied by capital per worker. This leads to a graphical representation of the neoclassical model showing the steady state as the intersection.
To summarize:
[[summary]]
As _K_ approaches infinity, income approaches 0
As _K_ approaches infinity, income per person approaches 0
As _K_ approaches infinity, incremental income approaches 0
Increase it
Decrease it
Capital increases, and capital per worker increases
Capital increases, and capital per worker remains the same
Capital remains the same, and capital per worker increases
Both sides are linear
The left side is linear, and the right side is not
The right side is linear, and the left side is not
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