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"Is the "residual factor" a measure of the contribution of knowledge or is it simply a measure of our ignorance of the causes of economic growth?"

J. Vaizey,

The Residual Factor and Economic Growth, 1964: p.5

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Contents

(1) Adding Technical Progress

(2) Empirical Implications

A property of the Solow-Swan growth model which is a bit disturbing is the fact that, at steady-state, all ratios -- the capital-labor ratio, output per person and consumption per person -- remain constant. This is a bit of a disappointment for it implies that standards of living do not improve in steady-state growth. This is not only despiriting, it is also empirically dubious: it contradicts at least two of the "stylized facts" of industrialized economies laid out in Kaldor (1961) -- namely, that the capital-labor and output-labor ratios have been rising over time and that the real wage has been rising.

Of course, just because industrialized countries, and others besides, have experienced ever-increasing per capita consumption and output over the past three centuries does not, by itself, "contradict" the Solow-Swan model. After all, *out* of steady-state, we can easily have changing ratios. So, one possible explanation for the "stylized facts" that is consistent with the Solow-Swan model is simply that industrialized nations are still in the process of adjusting and just have not reached their steady-state equilibrium yet. And why not? It is not unreasonable to assume that adjustment to steady state might take a very long time (cf. Sato, 1963; Atkinson, 1969).

But economists are a rather impatient sort. They like to believe that economies tend to be at or around their steady-states most of the time (a noble exception is Meade (1961)). As a consequence, in order to reconcile the Solow-Swan model with the stylized facts, it is tempting to argue that there has been some sort of "technical progress" in the interim that keeps pushing the steady-state ratios outwards.

(1) __Adding Technical Progress__

Recall that when we write our production function as Y = F(K, L), we are expressing output as a function of capital, labor *and* the production function's form itself, F(ｷ). If output is growing, then this can be due to labor growth (changes in L), capital growth (changes in K) *and* productivity growth/technical progress (changes in F(ｷ)). We have thus far ignored this last component. It is now time to consider it.

Technical progress swings the production function outwards. In a sense, all we need to do is simply add "time" into the production function so that:

Y = F(K, L, t).

or, in intensive form:

y = ¦ (k, t)

The impact of technical progress on steady-state growth is depicted in Figure 1, where the production function ¦
(ｷ, t) swings outwards from ¦
(ｷ, 1) to ¦
(ｷ, 2) to ¦
(ｷ, 3) and so on, taking the steady-state capital ratio with it from k_{1}* to k_{2}* and then k_{3}* respectively.. So, at t =1, ¦
(ｷ, 1) rules, so that beginning at k_{0}, the capital-labor ratio will rise, approaching the steady-state ratio k_{1}*. When technical progress happens at t = 2, then the production function swings to ¦
(ｷ, 2), so the capital-labor ratio will continue increasing, this time towards k_{2}*. At t =3, the third production function ¦
(ｷ, 3) comes into force and thus k rises towards k_{3}*, etc. So, if technical progress is happens repeatedly over time, the capital-labor ratio will never actually settle down. It will continue to rise, implying all the while that that the growth rates of level variables (i.e. capital, output, etc.) are higher than the growth of population for a rather long period of time.

Fig. 1- Technical Progress

Before proceeding, the first thing that must be decided is whether this is a "punctuated" or "smooth" movement. Is technical progress a "sudden" thing that happens only intermittently (i.e. we swing the production function out brusquely and drastically and then let it rest), or is it something that is happening all the time (and so we swing the production function outwards slowly and steadily, without pause). Joseph Schumpeter (1912) certainly favored the exciting "punctuated" form of technical progress, but modern growth theorists have adhered almost exclusively to its boring, "smooth" version. In other words, most economists believe that ｦ(ｷ, t) varies continuously and smoothly with t.

The simple method of modeling production by merely adding time to the production function may not be very informative as it reveals very little about the *nature* and *character* of technical progress. Now, as discussed elsewhere, there are various types of "technical progress" in a production function. The one we shall consider here is *Harrod-neutral* or *labor-augmenting* technical progress. In fact, as Hirofumi
Uzawa (1961) demonstrated,
Harrod-neutral technical progress is the *only* type of technical progress consistent with a stable steady-state ratio k*. This is because, as we prove elsewhere, only
Harrod-neutral technical progress keeps the capital-output ratio, v, constant over time.

Formally, the easiest way to incorporate smooth Harrod-neutral technical progress is to add an "augmenting" factor to labor, explicitly:

Y = F(K, A(t)ｷL)

where A(t) is a shift factor which depends on time, where A > 0 and dA/dt > 0.

To simplify our exposition, we can actually think of A(t)ｷL as the amount of *effective labor* (i.e. labor units L multiplied by the technical shift factor A(t)). So, output grows due not only to increases in capital and labor units (K and L), but also by increasing the "effectiveness" of each labor unit (A). This is the simplest way of adding Harrod-neutral technical progress into our production function. Notice also what the real rate of return on capital and labor become: as Y = F(K, A(t)ｷL) then the rate of return on capital remains r = F_{K}, but the real wage is now w = A(t)ｷ(¶
F/¶
(A(t)ｷL)] = AｷF_{AL}.

Modifying the Solow-Swan model to account for smooth Harrod-neutral technical progress is a simple matter of converting the system into "per effective labor unit" terms, i.e. whenever L was present in the previous model, replace it now with effective labor, A(t)ｷL (henceforth shortened to AL). So, for instance, the new production function, divided by AL becomes:

Y/AL = F(K/AL, 1)

so, in intensive form:

y

^{e}= ¦ (k^{e})

where y^{e} and k^{e} are the output-effective labor ratio and capital-effective labor ratio respectively. Notice that as F(K, AL) = ALｷ¦
(k^{e}), then by marginal productivity pricing, the rate of return on capital is:

r = F

_{K}= ¶ (ALｷ¦ (k^{e}))/¶ K

But as ¦
(k^{e}) = ¦
(K/AL), then ¶
(ALｷ¦
(k^{e}))/¶
K = ALｷ¦
¢
(k^{e})ｷ(¶
k^{e}/K), and since ¶
k^{e}/¶
K = 1/AL, then ¶
(ALｷ¦
(k^{e}))/¶
K = ¦
¢
(k^{e}), i.e.

r = ¦ ¢ (k

^{e})

the slope of the intensive production function in per effective units terms is still the marginal product of capital.

What about the real wage? Well, continuing to let the marginal productivity theory rule, then notice that:

w = ¶ (F(K, AL)/dL

= ¶ (ALｷ¦ (k

^{e})) /dL= Aｷ¦ (k

^{e}) + ALｷ¦ ¢ (k^{e})ｷ(¶ k^{e}/dL)

as dk^{e}/dL = -AK/(AL)^{2} = -K/AL^{2} = -k^{e}/L then:

w = Aｷ¦ (k

^{e}) - ALｷ¦ ¢ (k^{e})ｷk^{e}/L

or simply:

w = A[¦ (k

^{e}) - ¦ ¢ (k^{e})ｷk^{e}]

The macroeconomic equilibrium condition I = sY, becomes:

I/AL = s(Y/AL)

or:

i

^{e}= sy^{e}= s¦ (k^{e})

where i^{e} is the investment-effective labor ratio.

Now, suppose the physical labor units, L, grow at the population growth rate n (i.e. g_{L} = n) and labor-augmenting technical shift factor A grows at the rate q
(i.e. g_{A} = q
), then effective labor grows at rate q
+ n, i.e.:

g

_{AL}= g_{A}+ g_{L}= q + n

Now, for steady-state growth, capital must grow at the same rate as *effective *labor grows, i.e. for k^{e} to be constant, then in steady state g_{K} = q
+ n, or:

I

^{r}= dK/dt = (q +n)K

is the required investment level. Dividing through by AL:

i

^{re}= (q +n)k^{e}

where i^{re} is the required rate of investment per unit of effective labor.

The resulting fundamental differential equation is:

dk

^{e}/dt = i^{e}- i^{re}

or:

dk

^{e}/dt = s¦ (k^{e}) - (n+q )k^{e}

which is virtually identical with the one we had before. The resulting diagram (Figure 2) will also be the same as the conventional one. The significant difference is that now the growth of the technical shift parameter, q , is included into the required investment line and all ratios are expressed in terms of effective labor units.

Consequently, at steady state, dk^{e}/dt = 0, and we can define a steady-state capital-effective labor ratio k^{e}* which is constant and stable. All level terms -- output, Y, consumption, C, and capital, K -- grow at the rate n+q
.

Fig. 2- Growth with Harrod-Neutral Technical Progress

If the end-result is virtually identical to before, what is the gain in adding Harrod-neutral technical progress? This should be obvious. While all the steady-state ratios -- output per effective capita, y^{e}*, consumption per effective capita, c^{e}*, and capital per effective capita, k^{e}* -- are constant, this is not informative of the welfare of the economy. It is *people* -- and not *effective people* -- that receive the income and consume. In other words, to assess the welfare of the economy, we want to look at output and consumption per *physical* labor unit.

Now, the physical population L is *only* growing at the rate n, but output and consumption are growing at rate n + q
. Consequently, output *per person*, y = Y/L, and consumption *per person*, c = C/L, are *not* constant; they are growing at the steady rate q
, the rate of technical progress. Thus, although steady-state growth has effective ratios constant, actual ratios are increasing: actual people *are* getting richer and richer and consuming more and more even when the economy is experiencing steady-state growth.

(2) __Empirical Implications__

How valid is this empirically? From the outset, notice that the Solow-Swan model with technical progress accounts for all of the Kaldorian stylized facts. Namely, at steady-state (we are dropping the asterisk):

(1) the investment-output ratio, I/Y = i

^{e}/y^{e}= s¦ (k^{e})/¦ (k^{e}) = s is constant,(2) the capital-output ratio K/Y = k

^{e}/¦ (k^{e}) is constant,(3) the capital-labor ratio k = K/L and the output-labor ratio y = Y/L are growing at rate q ;

(4) the rate of return on capital, r = F

_{K}= ¦ ¢ (k^{e}) is constant;(5) the real wage is w = F

_{L}= A[y^{e}- k^{e}ｷ¦ ¢ (k^{e})] is growing at rate q(6) the relative share of capital rK/Y = ¦ ¢ (k

^{e})ｷk^{e}/y^{e}is constant, and the relative share of labor wL/Y = w/Ay^{e}= A[y^{e}- k^{e}ｷ¦ ¢ (k^{e})]/Ay^{e}= 1 - k^{e}ｷ¦ ¢ (k^{e})/y^{e}is constant.

The main implication of all this is that the Solow-Swan growth model can only explain steadily-increasing standards of living (growing y and c) via technical progress.

There is an entire body of empirical literature, known as "growth accounting", which attempts to address the empirical validity of this modified Solow-Swan model. Unlike the model just described, they usually assume that the technical progress factor A(t) is *outside* the production function, i.e.:

Y = A(t)¦ (K, L).

where A > 0 and dA/dt > 0, is the technical progress parameter (in this context, A is referred to as the "Total Factor Productivity" or "TFP" parameter). Thus, unlike the model above, growth accounting literature assumes that technical progress is *Hicks-neutral* or *TFP-augmenting* rather than Harrod-neutral/labor-augmenting.

[Note: Hello, does this not contradict Uzawa's (1961) proof? Not quite. It is possible for technical progress to be *both* Hicks-neutral and Harrod-neutral *if* the production function has constant unit elasticity of substitution, i.e. s
= 1. As we prove elsewhere, the Cobb-Douglas form of the production function is the *only* functional form that fulfills this. And you were wondering why it was so popular?]

The growth accounting literature asks the simple question: given the history of output growth, how much of it was due to growth of capital inputs (g_{K}), growth labor inputs (g_{L}) and technical progress (g_{A})? Output growth, labor growth and capital growth are observable, but technical progress is not. How do we estimate it?

Empirical growth accounting began with the famous studies of Moses Abramovitz (1956, 1962) and Robert
Solow (1957). Their procedure in calculating g_{A} was to deduct the growth rates of capital and labor (multiplied by their respective factor prices) and ascribing the "residual" to technical progress. For example, if we assume Cobb-Douglas form, so that the production function is:

Y = AK

^{a}L^{(1-a )}

where 0 £ a £ 1, the growth accounting literature is interested in the relationship:

g

_{Y}= g_{A}+ a g_{K}+ (1-a )g_{L}

where, as g_{Y}, g_{K}, g_{L} and a
are more or less observable, then g_{A} can be imputed residually. In fact, total factor productivity growth, g_{A}, is often referred to simply as the *Solow residual*.

The striking feature of the early investigations of growth accounting was the *size* of the Solow residual.
Solow (1957), for instance, calculates that only 12.5% of growth in output per capita in the 1909-1949 period in the United States was due to factor accumulation -- leaving 87.5% to be explained by technical progress! This is a bit dispiriting as it implies that the overwhelming majority of the growth that is empirically observed is "outside" the explanatory power of the Solow-Swan growth model!

In a series of famous studies, Edward Denison (1962), Zvi Griliches (1963) and Dale W. Jorgenson and Zvi Griliches (1967) argued that the there were errors in measurement in the early growth accounting work. For instance, if we remind ourselves that technical progress usually arrives "embodied" in new capital goods, then a lot more of growth can be ascribed to the "qualitative growth" of capital inputs. Thus, the importance of the Solow residual -- the growth in "total factor productivity" -- was argued to be substantially less than that estimated by earlier researchers. We will turn to technical progress again when examining endogenous growth theory.

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