The illusions of calculating total factor productivity and testing growth models: from Cobb-Douglas to Solow and Romer

Abstract

Growth models in the tradition of Solow and Romer are framed in terms of production functions. Consequently, they are equally subject to a criticism developed by, among others, Phelps Brown, Simon, and Samuelson. These authors argued that production function estimations are flawed exercises because output, labor and capital stock, are definitionally related through an accounting identity. The identity argument helps demystify two illusions in the literature: (i) finding the Holy Grail: total factor productivity is, by construction, a weighted average of dollars per worker and a pure number (the rate of profit or the rental rate of capital); and (ii) the possibility of testing: if estimated properly, production function regressions will yield: (a) a very high fit, potentially an R² of unity; and (b) estimated factor elasticities equal to the factor shares, hence they must always add up to 1. We illustrate these points through a series of well-known growth accounting exercises and models directly derived from production functions. They are futile exercises. We conclude that we know substantially less than we think about growth and that many of the discussions in the neoclassical growth literature are Kuhnian puzzles that only make sense within this paradigm.

Keywords:

• Accounting identity
• production function
• Romer
• Solow
• total factor productivity

JEL CLASSIFICATIONS:

• E22
• E23
• E25
• O11
• O33
• O47

Notes

Notes

1 We acknowledge the large empirical literature of growth regressions that claim that the number of robust regressors “related” to growth is significant (Sala-i-Martin 1997).

2 Much less in the light of the damaging conclusions of the Cambridge Capital Controversies and the aggregation problem. On these, see Felipe and Fisher (2003).

3 Some of the points we make in this paper draw on Felipe and McCombie (2013).

4 Reder (1943), Bronfenbrenner (1944), and Marschak and Andrews (1944) had already made references to the problem in discussions of Cobb and Douglas’ (1928) results, although not in the clearest way.

5 Solow’s (1957, 312) warning that “it takes something more than the usual willing suspension of disbelief to talk seriously about aggregate production function” did not have any impact on the profession. Solow himself did not follow it.

6 Actually, Shaikh’s (1974) paper was criticized by Solow (1974). It is important to read Shaikh’s (1980) rebuttal.

7 Phelps Brown, Simon, and Samuelson, expressed the problem in slightly different ways. However, once the issue at hand is understood, it is clear that their expositions amount to the same.

8 Growth accounting exercises are pervasive in the growth literature despite that most authors who undertake them are somewhat skeptical about what they do. This is partly the result of the problems measuring capital properly (Pritchett 2000). Moreover, it is generally accepted that the exercises do not prove causality (Aghion and Howitt 2007). Nevertheless, most growth theorists consider that performing a growth accounting exercise is a useful first step in looking at the data. Our view is that the concerns about TFP expressed by many authors are, at best, second-order issues.

9 The specifics of how national account statisticians construct equation (1a)(1a) $$Y_t\equiv W_t+S_t$$(1a) in practice are not important for us.

10 We just note that while it is self-evident that the wage bill ($W_t$) is split into the product of a price ($w_t$ is measured in $/worker) times a quantity ($L_t$ is measured in number of workers), it is much less obvious that this is also the case of the operating surplus ($S_t$). This is because the units of $r_t$ and $K_t$ are a percentage and dollars of a base year, respectively. This does not mean that writing $S_t=r_t{K}_t$ is incorrect as the product still yields dollars. Also, it should be obvious that $w_t$ and $r_t$ may or may not be the marginal products of labor and capital, respectively, in the sense of being derived from a production function, even though this is what equation (1b)(1b) $$Y_t\equiv w_t{L}_t+r_t{K}_t$$(1b) will always indicate, that is,$$(\partial Y/\partial L)\equiv w$$ and $$(\partial Y/\partial K)\equiv r.$$

11 Van Beveren (2012) provides an up-to-date survey of the methods to estimate TFP, especially of those that take account of the endogeneity problem that results from the fact that productivity and input choices are likely to be correlated; and of the selection bias that emerges if no allowance is made for entry and exit. There is another class of methods based on index number theory that we do not discuss.

12 There is an important point to be made, which refers to the factor shares used in equation (3c)(3c) $${\widehat{\mathit{TFP}}}_t={\widehat{Y}}_t-a_t{\widehat{L}}_t-\left(1-a_t\right){\widehat{K}}_t$$(3c) . The derivation in the text assumes that technical progress is Hicks-neutral. In this case, technical progress does not affect the factor shares. If technical progress were biased, one should not use the observed factor shares in the National Accounts, to the extent that these incorporate the effect of technical progress. See Ferguson (1968).

13 Note that while the left-hand side of equation (2c)(2c) $${\widehat{\mathit{TFP}}}_t^1\equiv {\widehat{Y}}_t-a_t{\widehat{L}}_t-(1-a_t){\widehat{K}}_t\equiv a_t{\widehat{w}}_t+(1-a_t){\widehat{r}}_t\equiv {\widehat{\mathit{TFP}}}_t^{D1}$$(2c) is identical to (3c)(3c) $${\widehat{\mathit{TFP}}}_t={\widehat{Y}}_t-a_t{\widehat{L}}_t-\left(1-a_t\right){\widehat{K}}_t$$(3c) , the right-hand side of equation (2c)(2c) $${\widehat{\mathit{TFP}}}_t^1\equiv {\widehat{Y}}_t-a_t{\widehat{L}}_t-(1-a_t){\widehat{K}}_t\equiv a_t{\widehat{w}}_t+(1-a_t){\widehat{r}}_t\equiv {\widehat{\mathit{TFP}}}_t^{D1}$$(2c) is not identical to equation (5)(5) $${\widehat{\mathit{TFP}}}_t^D={\vartheta }_{Lt}{\widehat{w}}_t+{\vartheta }_{Kt}{\widehat{\rho }}_t$$(5) . This difference will be discussed in the section “Recent growth accounting controversies: The illusion of calculating total factor productivity”.

14 GDP in most countries is generated through the demand side of the NIPA. This does not invalidate the claim that output $Y$ comes from an identity, not from $$Y=F(K,L,t).$$

15 This seems to be the view of, for example, Jorgenson and Griliches (1967, 252–253) or Prescott (1998, 532). From: $$Y=F(K,L,t)$$ one can write $$Y=F_{K}K+F_{L}L$$(Euler’s theorem), and from the first-order conditions, $$F_K=r$$ and $$F_L=w.$$ Hence $$Y=wL+rK,$$ taken to be identity (1b). That is, the neoclassical framework considers that the production function through Euler’s theorem implies the identity. While this derivation is mathematically correct, it does not mean that the production function provides a theory of the accounting identity. See also Hulten (2009), who traces the history of growth accounting during the 1930s–1950s, with the identity as starting point. This formulation was “atheoretical” (Hulten 2009, 4). Solow’s (1957) contribution was to provide the economic structure that the approach lacked.

16 See Lucas (1990) and Romer (1994), who argued that, under conventional assumptions about the extent of diminishing returns, the observed differences of over 30 times in labor productivity across countries cannot be explained by differences in capital intensity. Using the identity, it is straightforward to show that what accounts for most of the ratio of labor productivities between two countries is the ratio of wage rates, an insight resulting from circular reasoning that does not require a production function.

17 Some readers may think that the assumption that factor shares are constant implies a Cobb-Douglas production function. This is incorrect. See Fisher (1971).

18 Strictly speaking, perfect multicollinearity will mean that the coefficients cannot be estimated.

19 Expected value of the elasticity of capital:

$$E({\widehat{\beta }}_{\mathit{OLS}})=(1-a)+E[\frac{\mathit{Cov}({\widehat{K}}_t,{\lambda }_t^{*})\mathit{Var}({\widehat{L}}_t)-\mathit{Cov}({\widehat{L}}_t,{\lambda }_t^{*})\mathit{Cov}({\widehat{L}}_t,{\widehat{K}}_t)}{\mathit{Var}({\widehat{L}}_t)\mathit{Var}({\widehat{K}}_t)-{[\mathit{Cov}({\widehat{L}}_t,{\widehat{K}}_t)]}^2}].$$ The second part of this expression will be zero if: (i) ${\lambda }_t^{*}$ is a constant, then $$\mathit{Cov}({\widehat{K}}_t,{\lambda }_t^{*})=\mathit{Cov}({\widehat{L}}_t,{\lambda }_t^{*})=\mathrm{0;}$$ or if (ii) by coincidence the whole expression in the numerator is zero, then $$\mathit{Cov}({\widehat{K}}_t,{\lambda }_t^{*})\mathit{Var}({\widehat{L}}_t)=\mathit{Cov}({\widehat{L}}_t,{\lambda }_t^{*})\mathit{Cov}({\widehat{L}}_t,{\widehat{K}}_t).$$

20 See also Shaikh (1987, 2005).

21 Solow actually used a slightly different approach for calculating growth rates, but it makes no significant difference to the argument.

22 Hogan’s (1958) comment on Solow (1957), the exchange between Shaikh (1974) and Solow (1974), as well as Shaikh’s (1980) reply, are summarized and discussed in Felipe and McCombie (2013, 167–176).

23 Solow (1957) was followed by a series of papers that tried to investigate what was behind such large residual, for example, Denison (1962, 1967); Jorgenson and Griliches (1967). These set the pace for the TFP growth research program that has lasted until today.

24 Solow (1958) is a reply to Hogan (1958), who pointed out the nature of Solow’s tautological procedure. Hogan noted that it is obvious that the coefficient $b$ in the regression $\text{ln}\hspace{0.17em}q=c+b\hspace{0.17em}\hspace{0.17em}\text{ln}\hspace{0.17em}k$ will be the capital share. Solow claimed that if the capital share had been exactly constant, then indeed the procedure would have been a bad tautology. However, insofar as the capital share showed some variation, he argued that it was a good tautology. We think this was not a particularly convincing argument.

25 Fernald’s analysis covered until 2011. Real wage growth in the US declined from an annual average of 1.5% during 2000–2002, to 0.2% during 2008–2009, and then slightly recovered to 0.4% during 2010–2011, and to 0.6% during 2010–2017. The labor share declined by 3 percentage points between 2000 and 2017. This implies that $a_t{\widehat{w}}_t$ declined significantly between 2000–2007 (0.82%) and 2008–2009 (0.13%), and then increased in 2010–2011 (0.23%) and during 2010–2017 (0.33%). Data for the profit rate indicate that it actually increased after the financial crisis. This, together with the increase in the share of capital, implies that $(1-a_t){\widehat{r}}_t$ increased and compensated the decline in $a_t{\widehat{w}}_t,$ though the sum is still lower than during 2000–2007.

26 Equation (3c)(3c) $${\widehat{\mathit{TFP}}}_t={\widehat{Y}}_t-a_t{\widehat{L}}_t-\left(1-a_t\right){\widehat{K}}_t$$(3c) is often derived by assuming a translog production function and the Tornqvist approximation (discrete approximation to a continuous Divisia index). This requires the use of the average factor shares at the start and end periods, that is, $${\overline{a}}_t=(a_0+a_T)/2.$$ This is what Young (1992, 1995) used.

27 This seems to be Harberger’s (1998) approach, who wrote the identity in growth rates but gave it a completely neoclassical interpretation by assuming that each factor is rewarded according to its marginal product. He interpreted the residual as the rate of “real cost reduction.”

28 It is worth quoting them on this. Barro (1999, 123) noted that: “the dual approach can be derived readily from the equality between output and factor income.” He continued: “it is important to recognize that the derivation of equation (8)(8) $${\overline{Y}}_t\equiv w_t{L}_t+{\rho }_t{K}_t$$(8) [the growth accounting equation in Barro’s paper] uses only the condition $$Y_t\equiv w_t{L}_t+r_t{K}_t.$$ No assumptions were made about the relations of factor prices to social marginal products or about the form of the production function” (Barro 1999, 123; emphasis added). And: “If the condition $$Y_t\equiv w_t{L}_t+r_t{K}_t$$ holds, then the primal and dual estimates of TFP growth inevitably coincide […] If the condition $$Y_t\equiv w_t{L}_t+r_t{K}_t$$ holds, then the discrepancy between the primal and dual estimates of TFP has to reflect the use of different data in the two calculations” (Barro 1999, 123–24). To show this, he writes the income accounting identity, differentiates it, and expresses it in terms of growth rates (Barro 1999, equations [7](7) $$({\widehat{Y}}_i-{\widehat{L}}_i)=c+\eta ({\widehat{K}}_i-{\widehat{L}}_i)+u_i$$(7) and [8](8) $${\overline{Y}}_t\equiv w_t{L}_t+{\rho }_t{K}_t$$(8) , 123). It is noticeable from the above quotation that Barro assumed that there exists an underlying production function, in spite of any impression to the contrary. We think that Barro is using Euler’s theorem to connect the production function to the identity. We already disputed this in the section “Statement of the problem”. Hsieh (2002, 505), likewise, concurred that “with only the condition that output equals factor incomes, we have the result that the primal and dual measures of the Solow residual are equal. No other assumptions are needed for this result: we do not need any assumption about the form of the production function, bias of technological change, or relationship between factor prices and their social marginal products” (emphasis added).

29 It should be noted that in the textbook definition, “nominal” $\rho$ (let us call it ${\rho }^n$) is actually a price, whose unit is $ per unit of physical capital (e.g., “leets”). Consequently, in “real” terms $$\rho =({\rho }^n/p),$$ where $p$ is the dollar price of a unit of output (e.g., dollars per widget). It is measured in units of physical output (i.e., widgets) per unit of physical capital (e.g., per square meter of office, per computer).

30 Empirically, it might well be that one finds that $$\overline{Y}>Y.$$

31 Note that equation (10)(10) $$Y_t\equiv C_t+Z_t\equiv w_t{L}_t+{\rho }_t{K}_t+Z_t$$(10) may be written as:

$$Y_t\equiv w_t{L}_t+{\rho }_t{K}_t+Z_t\equiv w_t{L}_t+{\rho }_t{K}_t+{\rho }_t^{*}{K}_t\equiv w_t{L}_t+({\rho }_t+{\rho }_t^{*})K_t=w_t{L}_t+r_t{K}_t,$$ where ${\rho }^{*}$ can be defined as the “monopolistic” rate of profit.

32 See Solow (1994) and Romer (2001) for critiques of Mankiw ET AL. (1992).

33 They concluded that the production function consistent with their results is $$Y=A{K}^{1/3}{H}^{1/3}{L}^{1/3},$$ where $H$ denotes human capital. As can be seen, the elasticity of physical capital is not different from its share in income and there are no externalities to the accumulation of physical capital.

34 Assuming that factor shares are constant, and that output and capital grow at the same rates, and “imposing” them on the identity does not make the latter a model. This simply means that MRW’s model will work when both assumptions are approximately true in the data; and it will not work otherwise.

35 Surely a regression of the growth rate of GDP on initial income per capita tells us something about absolute convergence. This is not the MRW hypothesis and test.

36 See also Hall and Jones (1999).

37 It is straightforward to see that the introduction of human capital makes no difference whatsoever because it disappears if we substitute instead $$A_i={(y_i/k_i)}^{(1-\alpha )/\alpha }(y_i/h_i)$$ into the ratio of steady-state level of productivity with respect to that of the US. We obtain equation (16d)(16d) $$\frac{y_i^{*}}{y_{US}^{*}}=\frac{{\left(\frac{s_i}{n_i+g+\delta }\right)}^{(1-\alpha )/\alpha }}{{\left(\frac{s_{US}}{n_{US}+g+\delta }\right)}^{(1-\alpha )/\alpha }}\frac{A_i}{A_{US}}$$(16d) again.

38 Combining identity (14d) and the definition $$s\equiv \widehat{K}(K/Y),$$ it follows that $${\widehat{K}}_t\equiv (n+\delta +{\tilde{g}}_t).$$

39 For estimation purposes, Hall ran the inverse of this regression, that is, $$a_t{\widehat{k}}_t^{*}=c+d({\widehat{Y}}_t-{\widehat{K}}_t)+u_t,$$ where now the estimate of the markup is $$\widehat{\mu }=(1/\widehat{d}).$$

40 The estimated $\mu$ can be interpreted as a biased estimate of ${\mu }^{*}=1$ due to the misspecification of ${\lambda }_t.$ The expected value of $\mu$ will be 1 plus a term that depends on the covariance of the omitted and included term and the variance of the latter: $$E[{\widehat{\mu }}_{\mathit{OLS}}]=1+\frac{\mathit{Cov}({\lambda }_t,a_t{\widehat{k}}_t^{*})}{\mathit{Var}(a_t{\widehat{k}}_t^{*})}.$$$\mu =1$ if $\mathit{Cov}({\lambda }_t,a_t{\widehat{k}}_t^{*})=0,$ which in the context of the identity does not imply that markets are perfectly competitive; and $\mu >1$ if $$\mathit{Cov}({\lambda }_t,a_t{\widehat{k}}_t^{*})>0.$$

41 Their calculations of ${\widehat{\mathit{TFP}}}_t$ and ${\widehat{\mathit{TFP}}}_t^D$ involve two additional terms each, corresponding to energy and materials, because their measure of industry output is gross output.

Additional information

Notes on contributors

Jesus Felipe

Jesus Felipe is at Asian Development Bank (ADB), Manila, Philippines.

John McCombie

John McCombie is at University of Cambridge, Cambridge, United Kingdom.