Here's a topic that obviously has to be covered in a class on economic development but, ultimately, you might find unsatisfying. I'll try to explain the history (briefly), the core models that you will probably run across in your professional life (not necessarily to use, but definitely to know about) and ways of using the concepts of the core model in a way that are applicable to poor countries. This topic is covered in every development and macroeconomics textbook but none deals with it with the level of rigor (usually too much) or relevance (usually too little) I'd like to get across.

History

You might remember the Keynesian model of income determination. This model was developed to explain the depression of the 1930's and even though it's been superseded by later work, still underlies many peoples' thinking about macroeconomics. In the context of growth and development, though, it had a very peculiar implication. It seemed to say that being frugal and saving was a bad thing. One simple diagram shows this:

Keynesian equilibrium requires S = I. Here, the more you save, the lower is total income. Worse, if there is an accelerator (higher income stimulates higher investment), you get this weird paradox -the “paradox of thrift” - that if everyone tried to save more, the total amount of saving in the economy would go down (from S' to S"). This is because saving more reduces income so much that it more than balances a higher savings rate. So, all this stuff about the frugality of the middle class and whatnot is all completely wrong. That didn't seem right. Saving must get you something, after all. After the Second World War, Evsey Domar put together a simple model that showed that the savings should turn into productive capacity and that was a good thing.

The Harrod-Domar Model

Actually, the model here is just Domar's. The British economist Sir Roy Harrod came up with another model of savings and investment even earlier but it was very odd and no one ever used it outside of Cambridge, England. For some reason, the two names have been forever attached to the following model.

Domar started with saying that production was not just a demand phenomenon as assumed by Keynes who modeled an economy with a fixed (and underused) capital stock. It is possible for productive capacity to change over time as a result of investments. And the source of investment in a closed economy would be savings. Domar started with the simplest possible production function for the economy: Y =音 or, there is a constant capital/output ratio 9. Increases to capital (and, therefore, income) would come from investment and investment came from savings. He then assumed that savings was a fixed percentage of income (really just one minus the marginal propensity to consume as in Keynes). Stringing this all together we would have capital stock next year (K+i) is the current capital stock (Kt) plus investment:

K_t+1 = K_t + I_t (1)

But It is investment, which is equal to savings-a constant fraction, s, of income (Yt =音).Substituting all this in equation (1) we get:

Kt+i - Kt = It

Kt+i - Kt = sYt

^K+i ^{— K}t = ^s 9

K+i — K = s

K= 9

Equation (2) says that the proportional change in capital over time is a constant and equals the savings rate divided by the capital to output ratio: 9. What is missing? Equation (2) doesn't take into account, first, the fact that machines wear out and, second, that the labor force is likely to grow as well. Real growth in the economy is best presented in net (of depreciation) and per capita, or more precisely, per worker terms.

So, let's do it again with a constant fraction of machines wearing out each period at rate 8. Then equation (i) must be adjusted for this depreciation:

^Kt+i = Kt⁺ It ^{— 8} Kt = It + ^{(i — 8}) Kt

Solving further:

Kt+i — Kt = It — 8 Kt Kt+i — Kt = sYt — 8 Kt

Kt+i — Kt = s* — 8 K ^Kt+i ^— Kt = s 8

_Kt_ = 9 ^—

Now, in reality there is also population growth such that the amount of capital that each person has to work with is a little less than the one above. To look at this in per worker terms, we just devide equation (4) by L_t (for the labor force), and get:

Suppose population grows at a constant rate n, such that Lt+1 = Lt (1 + n) and capital per capita grow at rate g, such that kt+1 = kt (1 + g) Also small letters will be used to denote “per-capita” terms. i.e., k = K is capital per capita. Then the above equation can be written as:

Kt+1 Lt+1 Kt、8 Kt

^Lt+1 ^Lt ^Lt ^{9 L}t

_s

kt+1 (1 + n) = kt + kt — 8 kt

_s

^kt^{(1 + g)( 1 + n)} = ^kt ⁺ 9^kt ^{— 8k}t

_s

⁽¹⁺g^{)(1 + n)} =^{1 +} 9 - ⁸

1 + g + n + gn = 1 + ^S — 8

^kt +1 — ^{k S}

三 ^g 一 9 ^{— (8 + n)}

The last line follows from the fact that capital and population grow at low rates like 0.02 or 0.05, such that their product, gn, is a very small number. The last equation should be intuitive. The number of machines each worker has at his or her disposal next year is related to the gross investment that equals savings from income sY =普 or, in per worker terms sy =普 minus the depreciation of machines that wear out minus the rate of growth of the labor force since every new worker would have to have the capital stock spread out over more people if machines didn't grow at least as fast as the workforce.

This is often presented graphically:

Since the savings rate, the capital output ratio, the depreciation rate and the growth rate of labor are all constants, the equation for the change in k can be split into two pieces, the gross investment 9 and the “depreciation with re-equipment of new worker^ term (8 + n) both of which are rays from the origin. So, if gross investment is more than enough to cover depreciation and new workers, then capital per worker will grow. In the diagram, if we start at k°, then gross savings and investment are read off the steeper line and the required investment that would maintain the status quo is read off the flatter line. Since there is excess, the difference between the lines is translated into new capital per worker and we end up in the next year at 灯.In that year, savings is again higher than replacement needs (has to be since both are determined by the straight lines) and capital stock in the next, next year is k》.This will go on indefinitely and, in fact, there will be exponential growth at rate (9 — 8 — n) forever.

Domar did not really think of this as a real reflection of long term trends in the economy, he just didn't want to think that savings disappeared from the economy. This was the simplest way he could think of to get something good out of savings in the Keynesian model - it gets transformed into new productive capacity (which makes sense) that leads to future growth. Note that since income per worker is a constant (音)times the capital/labor ratio k, then income per worker grows at the same exponential rate.

A Small Digression: Traps in the Harrod Domar Model

We know from the Harrod Domar Model that growth rate is:

g =(言一8)— n = A — n

Let's call the term in the brackets A, or growth not accounting for population growth. The model assumes a constant rate of population growth, n, but that doesn't have to be the case. In the data it seems that population first rises with income levels and then falls (we'll see this in the topic 9 on fertility). Then we can get traps!

Figure 3: Traps in Harrod-Domar Model

Solow Model

The Domar model, if taken as a model of growth (for which it wasn't really intended) had an extreme implication. It said that the more machines (K) you gave to workers, the more output, IM, you'd get without bound. But that didn't square with ordinary production functions which, while they could have constant returns to scale, couldn't have constant returns to just one factor of production (machines = capital = K). So, Solow used the above equation and simply substituted the standard production function for a constant capital output ratio. So, instead of K = we had y = F(^,L) with Y being subject to constant returns to scale (you double both K and L and you get output to double, too). When we write this in "intensive" form, that is, per worker, we get y = f 侬, 1) or y = f 侬)(recally =匕 and k = £). But f 侬)no longer is a straight line, it is an ordinary production function that displays decreasing returns to the single factor of capital (or machines) per worker. You give one worker a drill press and she handles it pretty well. You give her two and she increases production but not twice as much (she's running back and forth between them). You give her three and you get more but not in proportion. Pretty soon she looks like Charlie Chaplin in "Modern Times" (the eighth best movie ever made) running all over the place with very little to show for the sixth machine you give her.

There is an equation for this, of course, called the "Modern Times" equation (just kidding) which replaces 音 with a more realistic production function. This gives us the change in the capital stock per worker, Solow's core equation, otherwise similar to the Domar equation (5):

k = k_t+\ —k_t=sf 侬)—(3 + n) k (6)

here's nothing wrong with the (3 + n) term — there is still depreciation and labor force growth.

The graph turns into the familiar Solow diagram:

So, the argument is the same as with Domar, if gross investment, that is, the amount you save out of income, sf(k), is more than enough to replace worn out machinery and accommodate new workers, the amount of capital per worker, k, will increase as will output per worker y = f (k). But this time, instead of gross investment always being above the replacement needs, diminishing returns means that it will be more than sufficient when k < k* and you'll grow (move to the right) but if you happen to be at a k〉k), then you won't be saving enough to maintain the capital stock given depreciation and labor force growth and the economy will shrink (move to the left) until the two curves intersect at k).

To give a concrete, algebraic example we'll choose the most conventional production function possible: the Cobb-Douglas. Here, output Y is determined by capital and labor according to Y = F(K)L) = K^aL⁽]_^a)which happens to have all kinds of useful features. It has constant returns to scale so if you double the inputs you get double the output:(2K)^a(2L)⁽¹~^a)= 2^a2^(1_a)K^aL⁽¹~^a)= 2K^aL(]_^a)= 2Y. It so happens (trust me on this one) that the term a represents the share of total income that accrues to owners of capital and 1 — a goes to workers. This is called the “functional distribution of income” - going to factors of production - as opposed to the "size” or personal distribution of income - which goes to people and with which you are more familiar.

Since the production function has constant returns to scale we can write this in “intensive” or per worker form by dividing by L : y = L = ^F(KL) = ^K" = K^aL^—a = k^a = f (k) where lower case Y K

letters y and k stand for Y and K. So in the above formula:

k = sf (k) — (8 + n) k = sk^a — (8 + n) k

and we can use actual numbers to find k* by setting k = 0 (in equilibrium, things aren't changing, and the dot indicates change so, to find equilibrium, we're looking for when the change is zero). If i

we make up numbers for s, n, a and & we can solve for k) = [8^]戏 ¹ and once you know k*, you know everything. You can get output per worker and savings per worker, consumption per worker (how do you know that?).

As a correction to a flawed assumption by Domar, this was a big advancement. As a model of growth of the economy, though, it had its own limitation. Specifically, it was a model of growth in which there was no growth. Once the economy reached k (and therefore income per worker f (k))), it just stopped except to replace worn out machines and equip more workers at the same level as the original workers but with no more output per worker. So growth in the economy was just n, the rate of growth of the labor force but no one was getting any richer. That didn't seem right either.

So, Solow changed the production function again by adding another term. Now Y = AF(K)L) or, in intensive form y = Af (k). The term "A” was just a term that scales production. "Growth” per worker is now possible if "A” changes and output per capita would grow at the rate of growth of A plus the rate of growth of f(k). But f(k) stops growing altogether when k = k). So the only thing that determines the rate of growth of income is the rate of growth of "A” and the proportionate growth rate of income per capita j? = A and didn't have anything to do with capital at all anymore.

Where did "A” come from? Nowhere. Solow made it up. He kind of thought it represented "technological progress” or increased income that came from new inventions or something, but he was quite candid and honest about it and admitted he had no idea where it came from. He didn't have a theory about how we invent things and called A "the measure of our ignorance”. Since his papers in 1956 and 1957, people have called this "total factor productivity” and talk about it as if it were a real thing. But it's not, really. It's just what's left over after we account for things we call capital. It's fair to say we do believe it reflects technological progress as did Solow. But it's not much of a "theory” of growth, it's just a description of growth. It points out that we can't grow forever just by adding the same old machinery. Real growth has to come from new inventions and innovations either in productive processes or in new and better "things”. This is why you might find this unsatisfying as a theory of development.

Subsequent work on growth has tried to figure out how "A”changes and why. This has lead to a cottage industry of regressions being run (usually across countries) to try to figure out any systematic reason for A or incomes to increase. One thing people seem to agree on is that there are increasing returns to scale which we will discuss in class. Here, I'll discuss two ways of manipulating the ideas and graphs above.

Ways to use the Solow model to explain the world.

As we've seen illustrated in Gapminder, countries seem to cluster around at least two and maybe more general levels of income. (If I didn't convince you of this, that's OK. Humor me for the next few paragraphs). How is this captured in the Solow model? Well, in its original formulation, it isn't. There is one, unique level of k that all economies should be tending toward. Also, poor countries should be approaching it faster than rich because of diminishing returns to capital. But they're not. Also, since the slope of the production function is the marginal return to capital, investments should flow to places where it is high - that is, where the function is steepest; that is, where k is low; that is, in poor countries. But it doesn't. So, the theory has to be modified or else just junked. Let's try modification.

Clustering around a few fixed points brings to mind the concept of "multiple equilibria” or, more than one stable solution to a model. In the Solow framework, this means we need a way to make those two functions cross more than once. In real life, this means that any or all of the behavioral or technological relationships are more complicated than we've assumed so far.

In general, if we can find reasons why the two curves could look like the following, we might have a story to tell that fits the observable facts.

Figure 5: Multiple Equilibria

Here, if curves cross three or more times, the ones in which the(〃 + 8) ray (or curve as illustrated here if we have a story about n or &) cuts the savings function from below (moving left to right), then the intersections are stable. To the left, savings is more than necessary to cover population growth and depreciation and k moves right. To the right of the intersection, savings is insufficient and k moves left. So that's stable. If the two curves cut the other way, (n + 8) cuts from above, then the intersections are unstable. If you happen to be on the intersection, you'll stay there, but if you deviate just a little, the forces in the economy will pull you farther away. To the left, savings are insufficient and you'll be pushed farther left. To the right, savings are more than sufficient and you'll be pushed farther right. That's unstable.

What could make a function behave this way? Well, you can play with s, with f (k), with n (maybe as a function of k or y) or even 8 (as a function of k or y). The O-ring story says that f (k) can have all kinds of wiggles in it depending on what aspect of production may fail at different levels of “development”. Or maybe 8, depreciation, is very low if the type of machinery we're talking about is not terribly sophisticated or that people tend to ignore it at low levels of k and increases rapidly as industrialization begins. At mid-levels, it falls as people catch on that maintenance is crucial. At high levels of capital the rate of depreciation is as low as it can be but it has to maintain high levels of k so it keeps going up. Any of these permutations can induce multiple crossings of the two core functions in the Solow model and therefore the possibility of multiple equilibria. This could help tell a story that rich countries all look the same at some high level of k, and poor countries all get stuck at some lower level reflecting their own, unique way of screwing up.

Growth, foreign aid and investment

Very oddly, at about the same time (1956/7) that Solow had just shown that you couldn't get more growth just by adding more machines (for which he ultimately got the Nobel Prize), development practitioners were taking a giant step back to the Domar model and saying, basically, yes you could. To this day, models used by many development agencies (and other advocates for foreign aid) use the simplest version of the Domar model to figure out how much investment or aid (aid is to augment investment) is “needed” to reach a target rate of growth. So, if the capital output ration is 3 (a common assumption, consistent with various bits and pieces of evidence for the U.S.), depreciation is 5% (who knows why?) and labor force growth is 2%, then in order to reach a per capita growth rate of, say, 6% (a nice, healthy target), you "need” a savings rate determined by .06=s/3 - .02 -.03 or 33% (3 times (.06+.02+.03)). If domestic savings is not as high as 33%, then the country “needs” enough to make up the difference. If it is saving 25% then aid should be 8% of GDP to reach the target. That is essentially what macro-economists in the World Bank, IMF, ADB, IADB, AfDB, USAID, UNDP and a host of other aid organizations covering most permutations of letters of the alphabet, used as their “analytic” foundation for determining aid “requirements” for years and years. Most still do. Like advocates in the Earth Institute at Columbia University. I am not making this up.

From Theory to Empirics: Testing the Solow Model (Mankiw, Romer, Weil 1992)

They wanted to test the predictions of the solow model, so the needed an “equation” that they could estimate using data. They start with a labour augmenting production function. Remember the "A' terms above that improves the efficiency of production. These guys assume instead that productivity improves but only of labour, no "A” term. Output depends only on Labour and Capital:

Y = F (K)L)

But “effective” labor in the economy is:

L_t = E_tP_t

where, Pt is the actual population that grows at rate n. And their efficiency Et grows at rate n. So:

R+i = ^pt (1 + ⁿ⁾

Et+i = Et (1 + 兀)

and the production function is just:

Y = F (K)L) = F (K)EP)

Now, we know that capital next period is savings (= investment) minus depriciation (see equation (3) above):

Kt+i = s" + (1 - 8) Kt

Divide both sides by EtPt and multiply and divide LHS by Et+iPt+i

^{Kt+1 Et+iPt+i} = ― + (i -8

Et+iP+1 EtPt EtPt ' ^J EtPt

This can be rewritten as:

^kt+1 ^{(1 + n)(1 +} 兀)=^syt ^{+(1 — 8}) ^k

Where the hat terms are just in "per unit effective labor”

k = K

EtPt

Yt

yt —

EtPt

9

Assuming nn is a very small number, the above equation equals:

kt+i — k = sy_t — (8 + n + 兀)k

At steady state:

^kt+i ^一如=⁰

syt — (8 + n + 兀)kt = 0

k

y 8 + n + n

Now, let's use a cobb-doglas production fuction:

Y = F (K)EP)= K^a (EP)^1—a —Y ^y= EP -1 = ^k

y =(-

\8 + n + n

)1—a

Take logs

aa

ln s —  ln (8 + n + n)

1—a 1—a
But remember y is output per efficiency unit of labor. To convert it back to per capita note that

ln y

Yt

yt

人=Y = =

^y~P^t~ PtEo (1 + n) — Eo (1 + n)^t

and then equation (7) becomes:

a a

ln yt = ln Eo +1 ln (1 + n) + ln s — ln (8 + n + n)

1a 1a

or,

ln ^t = A + ln Set — ln (8 + n + 兀)

We observe per capita income, savings = I/GDP, population growth rate. Asumme 8 = 0.03, and n = 0.02. Estimating equation:

aa

In% = A ⁺ 奇^lnScl 3^{ln (nc + 0(05) +}

Predictions

Coefficient on savings should be + and of opposite sign than on ln (n + 0.05). This is true (see regression table 1 in the paper). But the implied a is too high as compared to what you get from national accounts. So then?