# (p.205) Annex 1 : Key Features of Neoclassical Growth

# (p.205) Annex 1 : Key Features of Neoclassical Growth

Box 2.1 in Chapter 2 lists the three main features of neoclassical growth models and I sketch these out algebraically here.^{1} Let us start with a Cobb–Douglas production function for gross domestic product, *Y*, where *K* and *L* are the capital stock and labor force respectively, and $0\le \alpha \le 1$
, that is, there are diminishing marginal returns to capital:

In equation (1), *A* stands for technological progress or total factor productivity (TFP) and is shown as a function of time to capture its exogenous nature in the neoclassical Solow–Swann growth model. Let’s divide both sides of the equation by L to get:

where $y\equiv Y/L$
is output per worker and $k\equiv K/L$
is the capital-to-labor ratio. Assume, to begin with, that technology is constant and that TFP growth is zero (that is, *dA*/*A*=0). In this case, growth in output per worker can be written:

where a “^”or “hat” over a variable denotes a proportional change or growth rate. Assuming a closed economy so that investment equals saving, $\widehat{y}$ can be written:

where *s* is the savings rate, *n* the growth rate of the work force, and δ the rate of depreciation of capital.^{2} Since *y* has a negative exponent, it follows that $\partial \widehat{y}/\partial y<0$
, that is, that the growth rate of output per worker (which equals the growth rate of per capita income if the workforce is a constant share of population) falls as output per worker goes up. In other words, richer countries would tend to grow slower and poorer countries faster, leading to a convergence in per capita income levels. This is the so-called *Convergence Hypothesis*.

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In this simple setup, there is a unique steady state to which all countries will converge. I shall express the right-hand side of (A3’) in terms of the capital–labor ratio, *k,* which will serve as a bridge to two of Robert Solow’s key insights:

By comparing (A3) and ((A3’’)), we see that $\widehat{k}=sA{k}^{\alpha -1}-(n+\delta )$
. The first expression on the right-hand side of this equality falls as *k* increases because $\alpha -1<0$
, while the second, *n* + δ, is constant. Eventually, the two will become equal, so that the growth rate of capital per worker, $\widehat{k}$, goes to zero. In this case, *k* will reach a steady state and so will *y*. ((A3’’)) shows that we need diminishing marginal returns to capital (that is, *α* positive but less than 1) in order to reach a steady-state level for *k* (that is, $\widehat{k}=0$
). This yields the two key results of neoclassical growth: (a) the steady-state rate of per capita output growth will be zero in the absence of exogenous TFP growth, that is, $\widehat{y}=0$
unless $\widehat{A}>0$
. And (b) a rise in the savings rate will lead to a higher *level* of income but not to a permanent increase in the growth rate. That is, if *s* rises, $\widehat{y}$ will increase temporarily (since $\partial \widehat{y}/\partial s>0$
) until k reaches a new, higher steady-state level, but then revert to zero unless $\widehat{A}>0$
.^{3}

# Questioning Convergence

As Paul Romer points out in his 1994 survey, the convergence hypothesis is not supported by the empirical evidence, leading to questions about the validity of the model itself. These questions arise because the model leads to implausible inferences about the level of savings and investment rates in rich countries relative to poorer ones; but I will first provide another equation before I explore this further. The marginal product of capital is given by (you can check that $\partial Y/\partial K=\partial y/\partial k$ ):

from which it is clear that there are diminishing marginal returns to capital, since $\frac{\partial MPK}{\partial k}<0$.

Let us compare an advanced economy and a developing country. In 1960, as Romer (1994) notes, output per worker in the Philippines was about a tenth of that in the USA. Based on equation (A2) with labor’s share in national income (1–*α*) = 0.6, Romer goes on to say that this would imply that the capital–labor ratio in the USA was over 300 times that in the Philippines, suggesting an implausibly high level of savings and investment in the USA.

I am going to focus on the MPK instead in an attempt to tease out some development implications. Putting equations (A2) and (A4) together, these same numbers imply that the MPK in the Philippines would have been 31.6 times that in the USA in 1960! In this case, we would have witnessed an exodus of capital from the USA to the Philippines until the MPKs and hence *k*’s are equalized. This would also imply an equalization of *y*’s—that is, a convergence of income levels (as equation (A2) would tell us).

In trying to resolve these implausible results, there are two parameters up for grabs: 1–*α* and *A*. Since (1–*α*) comes from the national income statistics, let us focus on *A*. The expressions for the ratios of output per worker and MPKs in the two countries are given by:
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where P denotes the Philippines and U, the USA. Now consider the following thought experiment. Assume that the capital–labor ratio in the USA is 10 times that in the Philippines (rather than over 300 times) and further, that, since the USA is technologically more advanced, it is three times as efficient, that is, $\frac{{A}_{P}}{{A}_{U}}=0.33$. Then equations (A5) and (A6) tell us that output per worker in the Philippines is 13 percent that of the USA but its MPK is 31 percent higher ($\frac{MP{K}_{P}}{MP{K}_{U}}=1.31$). These results are more plausible!

Suppose capital flows are liberalized, spurring an outflow from the USA to the Philippines until the MPKs were equalized. According to equation (A6), this would mean that the capital–labor ratio in the Philippines would go up from 10 to 16 percent of that of the USA, while its output per worker would rise from 13 percent to 16 percent of that in the USA. Obviously, the Philippines would grow faster than the USA until this new equilibrium is reached. But convergence in income *levels* does not occur, unlike in the previous case when identical technology (i.e., $\frac{{A}_{P}}{{A}_{U}}=1$) was assumed. Thus, while the liberalization of capital flows will equalize marginal returns to capital, it will not lead to a convergence of income levels unless there is rapid technological catch-up, providing the point of departure for endogenous growth and highlighting the criticality of technology and human capital as discussed in Chapter 2.

Figure A1.1 shows how this might happen. The marginal product of capital for the USA is denoted by curve U, and that for the Philippines, P. The starting points are U_{1} and P_{1}, while the post-capital account liberalization equilibrium is given by U_{2} and P_{2}. While the capital–labor ratio in the Philippines rises, it, and the level of output per worker, remain below those in the USA in the new equilibrium. If the level of TFP in the USA were sufficiently high (corresponding to the curve U’ or above), the capital account liberalization would lead to flows in the “wrong” direction!

# TFP and Growth Accounting

Algebraically, $A=\frac{Y}{{K}^{\alpha}{L}^{1-\alpha}}$
, which is simply output per unit of input measured as a geometric average of *K* and *L*. Hence, TFP is a generalized measure of productivity or efficiency. Its growth rate is computed through a so-called ** growth accounting exercise** using equation (A1), which can be easily manipulated to get:

In (A7)
, g, *dK/K*, and *dL/L* can be obtained from the national income accounts and labor market statistics. Once we assume a value for (1–*α*)—which is simply the share of national income
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Paul Krugman (1994) captures the practical significance of TFP in his polemic questioning whether the rapid East Asian growth from 1960 to 1990 constituted a genuine miracle, that is, was a result of growing productivity or simply a matter of increasing inputs: “Mere increases in inputs, without an increase in the efficiency with which those inputs are used—investing in more machinery and infrastructure—must run into diminishing returns; input-driven growth is inevitably limited” (p. 67). He uses the output of a growth accounting exercise of the sort described earlier in this Annex to conclude that since “the remarkable record of East Asian growth has been matched by input growth so rapid that...[it] ceases to be a mystery” (p. 76).^{4}

## Notes:

(1) . If you really know neoclassical growth, you don’t need this annex. I draw heavily on Paul Romer’s 1994 survey, Robert Solow’s seminal 1956 paper, and a 1990 NBER working paper by Xavier Sala-i-Martin.

(2) . Note that $\widehat{k}=\widehat{K}-\widehat{L}$, with $\widehat{L}=n$. Assuming a closed economy, the net increase in the capital stock $\dot{K}=sY-\delta K$, which gives $\widehat{K}=\frac{sY}{K}-\delta $. Use this together with (A1) and (A3) to derive equation (A3’).

(3)
. See pages 10–12 of Sala-i-Martin (1990) and his Figure 1. NB: There will be no steady-state level either for income or capital per worker if $\widehat{\text{A}}>\text{0}$. However, there will be a steady-state solution for output per *effective* worker defined as $\tilde{y}=\frac{Y}{\tilde{L}}$, where $\tilde{L}\equiv BL$ and $B\equiv {A}^{\frac{1}{\left(1-\alpha \right)}}$. In this case, the
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steady-state rate of growth of output per worker *y* and that of capital per worker *k* will both equal $\frac{\widehat{A}}{\left(1-\alpha \right)}$ along a so-called “balanced growth path” (output and capital grow at the same rate). See also Lucas (1988).

(4) . For a rebuttal, see Bhagwati (1996). And see Chapters 2 and 3 for a policy-oriented discussion of accumulation versus TFP in growth.