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Chapter 3 / Resource Constraints and Production Possibilities

3.4 Shifts in the PPF


All of our analysis thus far has dealt with production functions of a single input, labor: for example, \(\begin{aligned}x_1 &= f_1(L_1) = 10\sqrt{L_1}\\ x_2 &= f_2(L_2) = 6\sqrt{L_2}\end{aligned}\)

One interpretation of this is that the two goods are produced using only labor. However, a better way of thinking about it might be that what we’ve been analyzing is the short run PPF: that is, the PPF given a fixed allocation of capital between the two goods.

For example, suppose that Chuck’s production functions for producing fish and coconuts are actually \(\begin{aligned}f_1(L_1,K_1) &= \tfrac{5}{4}\sqrt{L_1K_1}\\ f_2(L_2,K_2) &= \sqrt{L_2K_2}\end{aligned}\) and that Chuck has both $\overline L = 100$ hours of labor and $\overline K = 100$ units of capital — for example, stone tools or pieces of wood that could be used to either help him fish or help him break coconuts open. If Chuck is currently using $\overline K_1 = 64$ units of capital for fishing, and $\overline K_2 = 36$ for coconuts, then we get back our original production functions \(\begin{aligned}f_1(L_1| \overline K_1 = 64) &= \tfrac{5}{4}\sqrt{64L_1} = 10\sqrt{L_1}\\ f_2(L_2| \overline K_2 = 36) &= \sqrt{36L_2} = 6\sqrt{L_2}\end{aligned}\) But what happens if he adjusts his capital? Try shifting his capital allocation in the lower-right diagram below to see what happens to his short-run PPF as he reallocates capital between his two goods:

As you can see, while reallocating labor results in a movement along Chuck’s short-run PPF, reallocating capital shifts his short-run PPF: for any division of labor, the more capital he has devoted to fish, the more fish he’ll produce.

The long-run PPF

Of course, Chuck still has a capital resource constraint, so there are still combinations of fish and coconuts that he can’t produce. Even if he devotes all his labor and all his capital to producing fish, he can only produce $x_1^{max} = f_1(100,100) = 125$ fish; and if he devotes all his resources to coconuts, he can only produce $x_2^{max} = f_2(100,100) = 100$ coconuts. Chuck’s long-run PPF shows the combinations of output he can choose if he varies all his inputs:

The diagram below illustrates the relationship between the short-run and long-run PPFs: each short-run production possibility sets corresponding to some specific allocation of capital lies within the long-run production possibility set. Put another way, the long-run PPF represents the outer envelope of the short-run PPFs: every point in the long-run production set lies within the short-run production set for some allocation of capital.

Improvements to technology

The last thing we should note about PPFs is that they constrain people, and people don’t like being constrained.

There are two ways the PPF could shift out: an increase in resources, or an improvement in technology.

An increase in resources could mean increasing the capital stock. A lot of the plot of “Cast Away” concerned Chuck fashioning new spears with which to fish, or drills to get into coconuts more easily.

An improvement to technology could mean Chuck’s production functions themselves shift: for example, in addition to having better spears with which to fish, Chuck also becomes a better fisherman, which allows him to catch more fish for any given amount of labor and capital devoted to fishing.

We can represent improvements to Chuck’s technology by treating Chuck’s production functions as functions not only of labor and capital, but also some “level of technology” (like human capital) which augments both capital and labor:

\[\begin{aligned}f_1(L_1,K_1) &= A_1\sqrt{L_1K_1}\\ f_2(L_2,K_2) &= A_2\sqrt{L_2K_2}\end{aligned}\]

Use the sliders in the graph below to see how shifts in $A_1$ and $A_2$ affect his long-run and short-run PPFs:

You’ll see a lot more of this kind of analysis when you study macroeconomics; but in the meantime, since we’ve established what combinations of goods Chuck can produce, let’s turn our attention to what he should produce: that is, which combination of fish and coconuts in his feasible set is his “optimal choice.” To do that, we need to think about Chuck’s preferences.

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