CHAPTER 5

Capital and Labor

An important issue that arises in macroeconomics has to do with how the economy performs, given that it is in a classical state of full employment. Recall Alan Blinder’s comment, noted in Chapter 1, about utilizing “inputs more efficiently.” This is called the efficiency or allocative problem in economics and is a problem that economic agents solve in the classical model. The efficiency problem can be seen as one aspect of the overall coordination problem.

In this chapter, we will see how individual economic agents solve this problem as it relates to capital and labor. In the preceding chapter, we considered in detail how economic agents solve the problem of how much to save and therefore how much financial capital to provide to the users of financial capital, that is, investors. Here we will combine that decision with the choice calculus of producers as it relates to the use of financial capital and thereby the acquisition of physical capital.

Decisions to Save and Invest

In previous chapters, we used the letter r to indicate the rate of interest. What we didn’t do is include any discussion of inflation and how that might affect r, given that r represents the inflation-adjusted, real interest rate.

If someone buys an IOU for $100 and if the inflation rate is zero, he will have received a real rate of return of 5% if he can sell or redeem that IOU for $105 a year later. More specifically, if a slice of pizza costs $1.00, he will surrender purchasing power over 100 slices when he buys the IOU and will be able to buy 105 slices when he sells or redeems it.

Suppose, however, that prices rise by 1% over the course of the loan, so that the price of a slice of pizza rises to $1.01. Then the $105 dollars will permit him to buy only $103.96 worth of pizza, or at the new, inflated price, 103.96 slices. His real return on the loan will be only 3.96%. It turns out that the willingness to lend money (and more generally to provide financial capital) depends on his expectation of inflation.

In order to flesh out this point, let’s go to a post-Garden-of-Eden world where Adam wants to borrow $10,000 from Eve for the purpose of buying a pizza oven for Adam’s Pizzeria. We say that Adam wants to sell Eve an IOU or a bond for $10,000. We designate the nominal interest rate that Eve will demand as R.

Adam expects his investment in the oven to yield a certain amount of income per dollar invested. This is the marginal product of capital, which we designate MP_k. Adam also expects the oven to undergo economic depreciation annually at the rate of d. The cost to him of buying the oven in year 1 is P₁Δk, where P₁ is an index of year-1 prices and Δk is the cost of the oven, which in this case is $10,000. By setting P₁ = 1, we set P₁Δk = $10,000.

We can make the underlying concepts more concrete by introducing some more numbers. Thus, suppose that the MP_k equals 7%, so that every dollar sunk into the purchase of an oven yields 7 cents in new revenue. Also let the economic depreciation rate, d, equal 1%, so that the oven annually loses 1% of its value through wear and tear. If prices rise by, say, 4.0% over the course of the year, the resale value of the oven will rise to P₂ΔK = $10,400, ignoring depreciation. The oven will, however, have depreciated by 1%, leaving him with an oven worth $10,296, which is $296 more than he paid for the oven. By the first of the following year, Adam will have added $728.00 (= P₂ΔkMP_k) to his sales for having bought the oven. This adds up to a payout of $1,024 (= $296 + $728) on a purchase of $10,000 and a return of 10.24% (= $1,024/$10,000). We can use this information to provide a formula for the nominal rate of return on Adam’s investment:

Letting equal the rate of inflation,

Equation (5.1) then becomes

or here

Equation (5.1) simplifies to

which simplifies still further to

Next let’s distinguish between Adam’s actual rate of return, calculated in equation (5.4), from his expected rate of return. This will depend in part on what he expects the inflation rate to be. Given that he expects the inflation rate to be , he will expect the period-2 price level to be

His expected nominal return is then written as

or

which simplifies still further to

We can see from equations (5.5) and (5.9) that his expected and actual nominal return are both 10.24% if actual inflation equals Adam’s expected inflation . In reality, the two numbers will seldom be exactly equal, though we will consider in what follows how they would tend to be equal. Suppose they are not equal. Suppose that while as before. Then

and

We see that there are two measures of the return to investment: There is the expected or (as it is sometimes called) the ex ante return, and there is the actual or ex post return. The difference arises from elements in the equation for the nominal return for which the investor does not have perfect foresight. Here we focus on inflation. (The investor might equally lack perfect foresight in estimating the marginal product of capital and the depreciation rate.) In this example, Adam will want to borrow the money needed to finance his investment if Eve is willing to lend it to him at an interest rate not greater than 9.18%. If the actual return turns out to be 10.24%, so much the better for Adam and so much the worse for Eve, in that she underestimated the inflation rate and therefore overestimated the return on her loan in charging Adam 9.18% for the loan.

So now let’s see whether Eve will want to lend Adam the $10,000 he needs for his investment, which is to say whether she will want to buy the IOU Adam wants to sell her. Adam will be willing to sell the IOU as long as she doesn’t demand interest exceeding 9.18% on the loan.

Eve, for her part, has in mind some return on the loan that she expects to receive. She knows that her real return, r, will be lower than her nominal return R, which is the interest rate she will charge Adam, if prices rise over the course of the loan. Suppose she wants to get real return r^E of 6.0%, and suppose she expects inflation to run at 2.0% over the period of the loan, which we set at one year. At what value should she set R?

Let’s attack the problem by writing an equation for what the real value of her bond (which can be an IOU bought from Adam) will be when it matures.

where b₁ is the amount that she pays for the bond, and b₂ is the real value of the bond a year after she buys it. If Eve lends $10,000 to Adam, so that b₁ = $10,000, if she lends that money at a rate of 5%, as specified by the bond, and if the inflation rate is 2%, then that bond will be worth $10,294 in real dollars when it matures a year later.

Now if we designate Eve’s expected inflation rate to be , we can rewrite (5.13) as

where r^E is the real return on the bond that Eve wants to receive.

We can calculate r^E as

or, more simply,

We can then use equation (5.15) to solve for the nominal rate of return on the loan that she must demand, given and given that she wants to receive a real return of r^E.

Or, if r^E = 6%, and if Eve’s expected inflation rate is 2%, then

With a little algebra and simplification, we get

If NRR^E in equation (5.10) comes to 9%, Adam and Eve will find it mutually beneficial for Eve to lend to Adam.

Because Eve cannot predict inflation with perfect accuracy, however, we have to allow that the actual real return on the loan may be different from her intended real return.

With that in mind, her actual real return on the loan becomes

or

where is the actual rate of inflation. If turns out to exceed , Eve will be disappointed to learn that her actual return was less than her intended return.

We can figure that as long as NRR^E exceeds R, Adam will be willing to borrow from Eve and that as long as R permits Eve to enjoy her expected return r^E, she will be willing to lend to Adam. However, the gap between NRR^E and R will shrink as Adam borrows more from Eve, owing to the law of diminishing returns, which will manifest itself in a decline in MP_k and owing to Eve desiring a greater real return as she lends more.

We will spell out that process in greater detail in the following paragraphs. For now, let’s observe that Adam and Eve reach an equilibrium in their transactions when Adam’s NRR^E equals Eve’s R, so that

That is, Adam will continue borrowing from Eve and Eve will continue lending to Adam until his expected nominal return equals hers.

At this point we make a bold assumption, which is that in the long run everyone has the same expectation of inflation. Wrong as this must be in practice, it has an intuitively plausible foundation, based on the idea of rational expectations in economics. Because everyone on both sides of the capital market must make some assessment of future prices in making a decision whether to provide or use financial capital, each actor will use the best information available. As discussed in Volume II, Chapter 1, one model of expected inflation is that inflation equals the growth of the money supply. Thus all that Adam and Eve, right along with Cain and Abel, have to do is watch the growth of the money supply, information on which is available to all, in order to form a rational expectation of . Thus we now have a state of affairs in which so that

Note the assumptions we have to make in order to get this equation. We must assume that Adam and Eve not only have the same expectations of inflation but also that their expectations are accurate and that they have reached equilibrium in their transactions with each other. This is what rational expectations is all about. Proponents of this point of view will argue that if anyone’s foresight is wrong, he will learn from his mistakes and arrive at expectations that are in line with reality.

What we lose here in plausibility, at any rate, we gain in simplification. That’s because equation (5.24) reduces to

How do Adam and Eve reach this equilibrium? Going back to our example involving Adam and his pizza oven, suppose each dollar he spends on an oven yields $.08 in revenue (in “real” dollars). Then

Now what does it cost Adam to get his oven? What is his cost of capital? Well, suppose Eve demands a real return of 6% on her loan and that the depreciation rate for the oven is 1%. In that event his cost of capital, cc, is

Because MP_k > cc, Adam will want to acquire the financing to expand his stock of ovens. If it had turned out that MP_k < cc, then the last dollar he spent on ovens would have cost more than it yielded in new revenue, and he would have wanted to sell at least some of his oven capital to a different pizza maker. In general, Adam will want to expand his capital stock as long as MP_k > cc and contract it as long as MP_k < cc.

Now what about Eve?

In Chapter 3, we imagined that the individual adjusted his consumption–saving decision to a given real interest rate r. The individual adjusts his current-period consumption to the point where his = 1 + r.

In that chapter, we treated the interest rate as a datum, determined by the market. Here we treat it as a magnitude to be determined through interaction of borrower and lender. As long as Eve, the lender, demands only a small amount of future consumption in order to provide another dollar of financing, that is, as long as her is low, she will expect only a low return r^E on a dollar lent to Adam. On the other hand, if her is high, she will expect a high return.

Now think of what this means to the return to Adam on his acquisition of ovens, as opposed to his cost of acquiring them. If Adam has borrowed very little from Eve and if his stock of ovens is therefore low, his MP_k will be high relative to Eve’s r^E. MP_k will exceed cc, and Adam will want to expand his oven stock.

On the other hand, if Adam has already borrowed a great deal from Eve, his MP_k will be low, owing to the law of diminishing returns. For her part, Eve’s , and therefore her r^E will be high, owing to the law of the diminishing marginal rate of substitution of c_t for c_t+1. (If her falls when she expands c_t, it will rise when she contracts it by lending to Adam.) Adam’s cc will exceed his MP_k, and he will want to sell off some of his oven stock. Then in equilibrium, after these adjustments and given that r^E = r,

In this state, also, the equilibrium condition

will be satisfied, as will equation (5.24).

We can think of Adam as wanting to expand his holdings of capital just to the point where the marginal product of capital equals the cost to him of obtaining that capital. We write this condition as

so that his preferences line up with Eve’s:

From Chapter 2, equation (2.26) we get:

If NX and T – G are both zero, we see that every dollar of capital spending (as on pizza ovens) requires a dollar of private saving. From the foregoing analysis, the cost-of-capital (cc) curve for the entire economy will be upward sloping, owing to the law of the diminishing marginal rate of substitution of c_t for c_t+1. The MP_k curve, on the other hand, will be downward sloping owing to the law of diminishing returns. We can therefore portray market equilibrium as in Figure 5.1, where cc* is the equilibrium cost of capital and K* the equilibrium capital stock.

Let’s consider some features of this equilibrium. Let the equilibrium stock of ovens, expressed in Figure 5.1, be $1 million. Now suppose that technological progress or some other factor causes MP_K to shift upward, as in Figure 5.2. Suppose further that at the new equilibrium, the capital stock equals $1.5 million. Net investment will come to $500,000. Given that d = 1%, gross investment is

Figure 5.1 The supply and demand for capital

Figure 5.2 An increase in the demand for capital

which must be matched by an equal amount of saving.

In a static equilibrium, K will reach a level K*’ and r will rise to r*’, at which point ΔK equals zero and saving equals depreciation dK. We will consider what a more realistic “dynamic” equilibrium looks like in the next chapter.

International Capital Movements

It is appropriate at this point to take into account the fact that capital flows across national borders in response to differences in interest rates, expectations of inflation, and exchange rate variations. Now, considering the multiplicity of countries involved, it is necessary to enter an even more rarefied atmosphere, in which there are only two countries. Let’s call them the United States and Europe, which we label as ROW (for the rest of the world). Each country has its own currency, the dollar and the euro, and each country lets its currency fluctuate freely against the other. Borrowers and lenders can move financial capital and goods seamlessly across national borders.

The reader can probably see where this is headed, which is toward the conclusion that there will emerge a single, global real interest rate r. But rather than just give out the ending, let’s see how we get there.

To get there, let’s add another strong assumption, whereby it is just as easy for you or me to buy things in Boston as it is to buy them in, say, Paris. That assumption permits us to assume purchasing power parity, whereby exchange rates will adjust in such a way as to make every good as costly in “real” terms in one country as they are in another. In the context of the current example, there is some exchange rate ॉ, equal to the dollar price of a euro, that satisfies the equation

where P^US is an index for the prices of goods in the United States and P^ROW an index for the prices of goods in Europe (i.e., the rest of the world). To give this further concreteness, let’s return to our pizza-only world and assume that the dollar price of a slice of pizza is 1 and that the euro price of a slice of pizza is 0.5. Then the dollar price of a euro must be

To see what is going on here, imagine that the Boston price of the pizza slice suddenly rose to $2.00. Then Bostonians would convert dollars into euros, buying 0.5 euros on the dollar, and use those euros to buy pizza in Paris, where a dollar’s worth of euros buys them a whole slice of pizza, compared to the half of slice that a dollar now buys in Boston. This would increase the demand for euros and decrease the demand for dollars until the exchange rate depreciated to

So far, so good? Well, if so, let’s break down equation (5.34) to read

shorthand for which is

a relationship that goes by the name of relative purchasing power parity: If exchange rates can fluctuate, the exchange rate must undergo a percentage change equal to the percentage change in U.S. prices minus the percentage change in rest-of-the-world prices (which is zero in the current example). The U.S. price of a pizza slice rose 100% while the rest-of-theworld price stayed constant. As a result, ॉ also rose by 100% from 2 to 4.

Now we introduce yet another assumption called the interest parity condition:

where R^US is the nominal U.S. interest rate, and R^ROW is the nominal rest-of-the-world interest rate. Equation (5.39) is an equilibrium condition, whereby arbitrageurs will move financial capital across national borders in such a way as to equalize the expected nominal return to capital.

Consider an example. Suppose that Eve can buy a bond in Boston for $1,000 and that the interest rate on the bond is 10%. She could also buy a bond in Paris that pays 5%. It seems that the U.S. bond is the better buy. But not so fast. Suppose that Eve expects the dollar to depreciate by 7% before the bond matures in a year. Let’s take to stand for the expected depreciation of the dollar. Then because ॉ is the dollar price of a euro, equals 7%. So what happens if Eve buys the bond in Paris? Well, if ॉ equals 2 when she buys the bond, her $1,000 will get her a bond whose face value is €500. At 5%, that bond will pay out €525. But when the bond pays off, those €525 will be worth $1,124, given that a euro now buys $2.14 in American money. She is better off buying the European bond than the American bond, which will pay out only $1,100. Now

The nominal return to capital in the United States is 10%, while the nominal return to capital in Europe is 12.4%.

This means that money will flow from the United States to Europe, causing R^ROW to fall and R^US to rise until the two sides of the equation are equal. Conversely, if the U.S. bonds had been the better buy, so that 1 + R exceeded , then money would have moved from Europe to the United States until equation (5.39) was satisfied.

Now we have all the information we need in order to establish the existence of a single, global r. We can simplify equation (5.39) to get

whereby the nominal U.S. interest rate will equal the nominal rest-ofthe-world (European) interest rate plus the expected depreciation of the dollar. Now substitute equation (5.41) into equation (5.38) to get

which we can rewrite as

Going back to equation (5.22),

so that

If we assume that the depreciation rate is the same everywhere, then

Voila! The cost of capital is the same everywhere. We have to distinguish again, however, between intentions and outcomes. For actual r^US to equal actual r^ROW, lenders and borrowers across the globe must converge on the same expectations of inflation and, from there, the same expectations of currency movements. Also there must be no difference in risk factors that can influence the direction of capital flows.

Figure 5.3 The supply and demand for capital in a globalized economy

As far-fetched as these assumptions are, we have to figure that the logic of this discussion has major implications for government policy making. In particular, it implies that policies aimed at increasing domestic saving may have no noticeable effect on domestic investment insofar as increased saving will flow into whatever corner of the globe that offers, momentarily, the highest rate of return. Furthermore, policies aimed specifically at pushing the real domestic interest rate down or up may have no effect because the domestic rate is tied into the global rate.

In this world, the cost of capital line becomes horizontal, and the equilibrium capital stock in any country depends only on where the marginal product of capital curve intersects the cc curve, as in Figure 5.3.

Arnold Harberger once summed this up as follows:

The Supply and Demand for Labor

Now it’s time to take up the question of how wage adjustments equilibrate the supply and demand for labor. In Chapter 3, we worked out the condition under which workers adjust their work time to the real wage they get for their efforts. And we have worked out the condition under which income earners adjust their saving to the return on saving. We have also worked out the condition for matching the saving choices of income-earners with the investment choices of firms.

Now it remains to do two further things: First we have to work out the condition for matching the work choices of individuals with the hiring choices of firms. Second, we have to move beyond the static analyses of previous chapters to take up the conditions for economic growth as they pertain to work and saving. We postpone that discussion to the next chapter.

Let’s think again about production in terms of pizza. In fact, let’s assume that the only thing the economy produces is pizza. To keep things simple, suppose further that pizza sells for $1 a slice and that the employment of an additional hour of labor permits the firm to produce two more pizzas or 20 more slices.

We say that the marginal product of labor MP_l is the additional amount of production that is forthcoming per additional unit of labor employed or, in this instance,

If the worker is paid a nominal wage W of $10 per hour, then because the dollar price of a slice of pizza P is 1.00, his real wage w is also $10.

Figure 5.4 The demand for labor curve

The question is how much labor should the firm hire.

Consider Figure 5.4, where the MP_l curve is the demand for labor curve. At point A, the firm is using 100 units of labor per day, at which quantity the MP_l is $20. At point C, where it uses 300 units of labor, MP_l is $5 (the law of diminishing returns pushes down MP_l as the number of workers increases). The pizza shop will want to adjust its use of labor to point B, where the firm is using 200 units of labor and where the real cost of an additional hour of labor just equals the MP_l. At any point to the left of B, the marginal product of labor would exceed the cost of hiring labor, and at any point to the right of B, the cost would exceed the marginal product. Thus B is the profit-maximizing point.

Thus in equilibrium,

Now let’s think about how much labor workers want to supply. Suppose that a worker can be expected to provide just eight hours of work per day if he is paid $5.00 per hour. The $5.00 that he would receive for providing the fifth hour is just high enough to compensate for the leisure that he sacrifices for working that hour. Thus his MRS_LeLay = $5.00. When he provides 10 hours, his MRS_LeLay is $10 and when he provides 15 hours it is $20. If the wage rate is $10, he will provide 10 hours. That is, the reward that he must receive for the last hour of labor provided must just equal the reward that he must receive in order to compensate him for the hour of leisure thus sacrificed. See Figure 5.5.

Figure 5.5 The supply of labor curve

Figure 5.6 The supply and demand for labor

Suppose there are 10 firms of the kind described earlier and 200 workers. Figure 5.6 provides the aggregate supply and demand curves that are derived from the foregoing assumptions. In equilibrium, supply and demand bring the MRS_LeLay for each worker and the MP_L into line with the real wage rate so that

See Figure 5.6.

If w temporarily rises above $10, the quantity of labor supplied will exceed the quantity demanded, and w will fall. If w temporarily falls below $10, the quantity of labor demanded will exceed the quantity supplied, and w will rise. Only when w = $10, is the labor market in equilibrium.

One More Step

Now let’s wrap up this chapter by assuming a particular production function applies to the firm and the economy. A production function that offers convenience and simplicity is the Cobb-Douglas production function, specified as

where Z stands for total factor productivity and is a coefficient that represents the state of existing technology and the general legal and cultural conditions in which business operates. The coefficient α represents the share of income Y going to owners of capital and the coefficient (1 – α) the share going to labor. If we refer to Table 2.7 in Chapter 2, labor compensation was 63% of national income in 2016. If we treat all the other components of national income as income to capital, the capital share was 37%.

When we are using the Cobb-Douglas production function, we can calculate the MP_K and MP_L as follows:

and

We can therefore write

and

This establishes an important link between the return to investing and the ratio of output to capital and between the real wage rate and the ratio of output to labor. Thinking of as the productivity of capital and as the productivity of labor, we see the link between productivity and the reward for saving and for work.

With these fundamentals in place, we can now go on to consider the problem of economic growth.