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Deciphering the fall and rise in the net capital share: accumulation or scarcity?

IV. Capital Share Theory: A Multisector Model

In this section I expand beyond the one-sector model, constructing a tentative multisector model that allows a more nuanced analysis, including consideration of the housing sector. I then subject this model to four exogenous shocks, namely the required rate of return, the price of equipment investment, the price of residential structures investment, and the quantity of residential land.

IV.A. Design of the Multisector Model

The theory in section III enables a first-pass analysis of how the distribution of income is affected by various forces. It shows that accumulation of capital, all else equal, will likely result in a decline in the net capital share, since the net elasticity of substitution is almost certainly below 1. This counters the central hypothesis of Piketty (2014). It also shows that a decline in the relative price [P.sub.K] of capital, holding the required return r constant, will result in an increase in the net capital share if the gross elasticity of substitution is above 1--a claim that is still hard to reconcile with the bulk of empirical evidence, but for which Karabarbounis and Neiman (2014a) mount a spirited case.

Nevertheless, the one-sector model in section III is in many ways unsatisfactory as a model of the distribution between capital and labor. For instance, sections I and II demonstrated the decisive role of the housing sector in the long-term trajectory of the net capital share--but a one-sector model is by construction unable to account for a shift toward housing. Indeed, Piketty (2015) has recently voiced discomfort with the one-sector interpretation of the rising capital share, arguing that "the right model to think about rising capital-income ratios and capital shares in recent decades is a multisector model of capital accumulation" (p. 81). In this section I will construct a tentative version of such a model.

NESTED FRAMEWORK Given the central role of housing in sections I and II, it is first important to distinguish between nonhousing and housing output. If household preferences are homothetic in these two types of output, the household objective can be written as a monotonic transformation of a constant-returns-to-scale aggregator Z({[Y.sub.nh], [Y.sub.h]) that takes nonhousing and housing services as inputs. We can view Z as the "top-level" production function for the economy.

For the nonhousing sector, it will be useful to model the production process in a way that reflects the different types of capital studied in section II (equipment, structures, and land), so that the results from that disaggregation exercise can be used to inform the model. One natural approach is to assume that structures and land together provide "real estate" services that serve as an input to production, while labor and equipment together provide all other services. This approach enables me to draw upon several empirical literatures, which estimate the relevant elasticities of substitution--for instance, the elasticity of substitution between structures and land in the production of real estate services, or the elasticity of substitution between housing and nonhousing in consumer preferences.

Concretely, let H(N, [K.sub.e]) be a constant-returns-to-scale aggregator combining labor N and equipment [K.sub.e], and let [G.sub.1]([K.sub.s1], [L.sub.1]) be another constant-returns-to-scale aggregator combining nonresidential structures [K.sub.s1] and land [L.sub.1]. Finally, let F be another constant-returns-to-scale aggregator that combines H and [G.sub.1], so that the consolidated production function for the nonhousing sector takes the form

(18) [Y.sub.nh] = F[H(N, [K.sub.e]), [G.sub.1] ([K.sub.s1], [L.sub.1])].

Following section II, 1 assume that gross output in the nonhousing sector is sold at some markup [mu] over marginal cost.

Similarly, suppose that residential structures [K.sub.s1] and land [L.sub.2] are combined by an aggregate [G.sub.2]( [K.sub.s2], [L.sub.2]) to provide housing services, so that the production function for the housing sector takes the form

(19) [Y.sub.h] = [G.sub.2] ([K.sub.s2], [L.sub.2]).

Finally, as already mentioned, Z combines [Y.sub.nh] and [Y.sub.h] into an aggregate that reflects household preferences:

(20) Y = Z([Y.sub.nh], [Y.sub.h]).

This multisector economy captures the distinction between the nonhousing and housing sectors, as well as all five forms of capital analyzed in section II: equipment ([K.sub.e]), nonresidential structures ([K.sub.s1]), nonresidential land ([L.sub.1]), residential structures ([K.sub.s2]), and residential land ([L.sub.2]).

The aggregate, nested structure of production in the economy is depicted in the tree below.


ELASTICITIES OF SUBSTITUTION The response of the multisector model to various shocks is influenced by the local (gross) elasticities of substitution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each of the five constant-returns-to-scale production functions (Z, F, [G.sub.1], [G.sub.2], H) in the model above.

Although there are extensive empirical literatures that study many of these elasticities, a convincing research design is often elusive, and there is rarely strong consensus around a single point estimate. In the absence of such consensus, I will draw upon each literature to obtain plausible ranges for each elasticity, and study the implications of choosing different values within each range. The objective is to see which conclusions, if any, emerge robustly from the multisector model despite allowing for some uncertainty about the as. Another goal is to investigate which [sigma]s matter most to aggregate outcomes, both to clarify thinking and to direct future research toward the most crucial targets.

Surveying the relevant literatures, I find the following.

First, [[sigma].sub.z] equals the elasticity of demand for housing services (as a share of total output) with respect to its price (relative to the aggregate price index for Z). Closely related elasticities of demand for housing have been studied in the literature, which has generally obtained relatively low values. For instance, in a review of the literature, John Ermisch, Jeanette Findlay, and Kenneth Gibb (1996, p. 67) state that "price elasticity estimates are less dispersed than the income elasticity measures, yielding results between 0.5 and 0.8" and they themselves provide an estimate of O.4. (26) I set a range of [[sigma].sub.z] [member of] [0.4, 0.8].

Second, [[sigma].sub.F], the elasticity of substitution between real estate and other services in the nonhousing sector, does not map closely onto any empirically studied elasticity. In the absence of direct evidence, I set a wide range of [[sigma].sub.z] [member of] [0.5, 1.5].

Third, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the elasticities of substitution between structures and land in the nonhousing and housing sectors, respectively. These elasticities play an important role in the urban economics literature, where substitutability between structures and land in the provision of real estate services is of great practical and theoretical interest.

The more voluminous literature is for housing, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with a widely cited early entry by Richard Muth (1971), who estimates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using several approaches. More recently, Paul Thorsnes (1997) surveys the literature and finds that recent estimates have generally been below 1, in the range of [0.5, 1]; but he also argues that some of these estimates may be biased downward due to measurement error and that the true elasticity may not be much below 1. This claim is seconded by Gabriel Ahlfeldt and Daniel McMillen (2014). In light of these findings, I set a range of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [0.5, 1].

The literature for nonhousing real estate, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is more scattered, with a range of elasticity estimates similar to that for housing--generally below one, but with concerns about bias from measurement error. For instance, John Clapp (1979) obtains elasticities from high-rise office data mostly in the range of [0.5, 0.75], but in a tentative attempt to correct for measurement error finds that elasticities closer to 1 may be appropriate. Interpretation is complicated by the fact that nonhousing real estate is much more heterogeneous than housing real estate, spanning everything from high-rise office towers to farmland. Amid this uncertainty, I set the range [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [0.5, 1].

Fourth, [[sigma].sub.H] is the elasticity of substitution between equipment and labor. This is of great speculative interest--there are frequent discussions about the extent to which automation, for instance, can replace existing workers, and [[sigma].sub.H] governs the extent to which the decline in equipment prices documented by Karabarbounis and Neiman (2014a) will lead to substitution away from labor. In his survey, Chirinko (2008) reports a wide range of relevant estimates; the majority are still below 1, but several are above 1 as well, and he suggests that the elasticity for equipment may be higher than the aggregate elasticity. For instance, Cummins and Hassett (1992) obtain implied elasticities of 0.93 for equipment but only 0.28 for structures, and the estimates listed by Chirinko (2008) that use computer investment obtain values as high as 1.58.1 therefore set a range [[sigma].sub.H] [member of] [0.5, 1.5].

IV.B. Response of the Net Capital Share to Exogenous Shocks

I now study the elasticity of the net capital share with respect to various shocks in the multisector model, whose structure was described in the previous section.

GENERAL METHODOLOGY I assume that the quantities (N, [L.sub.1], [L.sub.2]) of labor and both types of land are exogenous. I take final output from Z to be the numeraire, and assume that the prices [P.sub.e], [P.sub.s1], and [Ps.sub.2] of reproducible capital in terms of this numeraire are exogenously fixed by technology. (27) As in equation 2, the user cost of reproducible capital for i

e {e, [s.sub.1], [s.sub.2]) is


where the required return r, the depreciation rate [delta], and the expected real change in prices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are also all assumed to be exogenous. As in section II, r may differ between the nonhousing and housing sectors. The quantities ([K.sub.e], [K.sub.s1], [K.sub.s2]) of reproducible capital are then given endogenously by demand at this user cost.

I will consider exogenous shocks to either the quantities (N, [L.sub.1], [L.sub.2]), the prices ([P.sub.e], [P.sub.s1], [P.sub.s2]), or r, which jointly determine the user costs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The elasticity of factor shares in the model with respect to either of these shocks depends only on the initial gross and net shares and the local elasticities of substitution ([[sigma].sub.Z], [[sigma].sub.F], [[sigma].sub.G1], [[sigma].sub.G2], [[sigma].sub.H]) at each level of production; with these in hand, it can be obtained numerically. (Unfortunately, unlike in Oberfield and Raval (2014), elasticities here cannot be expressed in closed form as a weighted average of the individual elasticities ([[sigma].sub.Z],[[sigma].sub.F], [[sigma].sub.G1], [[sigma].sub.G2], [[sigma].sub.H]). Analytically, this is due to the fact that I assume more than one exogenous quantity.)

I calibrate the initial shares to match the decomposition of the U.S. economy in section II.C for the final year in the sample, 2013. Table 4 displays the resulting gross and net shares of each factor as a fraction of total income, while table 5 shows the gross shares of each factor as a fraction of the parent aggregate.

IMPLEMENTATION AND RESULTS I focus on the elasticity of the net capital share with respect to four specific exogenous shocks:

--A shock to the required rate of return r

--A shock to the price of equipment investment [P.sub.e]

--A shock to the price of residential structures investment [P.sub.s2], and

--A shock to the quantity of residential land [L.sub.2].

As discussed in greater detail below, the first and second correspond to the Piketty (2014) and Karabarbounis and Neiman (2014a) versions of the accumulation view, respectively. The third and fourth shocks, which relate to residential housing, correspond to my proposed alternative of a "scarcity view."

The core results are summarized in tables 6, 7, and 8. For table 6, I calculate the elasticity of the net capital share with respect to each shock over the full range of a, deemed plausible in the previous section


and report the minimum and maximum elasticities of the net capital share obtained for any combination of c, in this range. I also calculate the elasticity of the net capital share at a set of "benchmark" [[sigma].sub.i], which I choose to be the midpoint of this range: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Table 7 provides additional insight into how different assumptions on [[sigma].sub.i], combine to produce an aggregate response to shocks. For each shock, the table shows the sensitivity (partial derivative) of the net capital share elasticity to changes in each of the underlying [[sigma].sub.i] starting from the benchmark values. Essentially, table 7 shows the gradient of the values in the "benchmark" column of table 6 with respect to perturbations in the [[sigma].sub.i].

For instance, in the case of a shock to [P.sub.e], the second row of table 7 shows small sensitivities to all o, except [[sigma].sub.H], for which the sensitivity is -0.29. This means that if [[sigma].sub.H] is increased slightly from its benchmark value--say, from [[sigma].sub.H] = 1.0 to [[sigma].sub.H] = 1.1--the elasticity of the net capital share with respect to [P.sub.e] will decline by 0.029. The intuition in this case is straightforward: when [[sigma].sub.H] is higher, it is easier to replace equipment with labor in response to higher equipment prices, meaning that a rise in [P.sub.e] will result in a smaller increase in (or greater decline in) net capital income.

Finally, table 8 decomposes the elasticity of the net capital share, at the benchmark [[sigma].sub.i], into contributing changes in each source of capital income. Each row of table 8 sums to the elasticity for the corresponding shock in the "benchmark" column of table 6, with one exception: an extra row is included for a shock to [P.sub.e], showing the decomposition in the "high elasticity" case where each elasticity o, is chosen to be at the maximum of the range. (This is because there is virtually no effect from the shock to [P.sub.e] at the benchmark [[sigma].sub.i].) For instance, for a shock to the price [P.sub.s2] of residential structures investment, the contribution of residential structures [K.sub.s2] is 0.09, out of a total elasticity (from table 6) of 0.07; this means that when the cost of residential investment rises, more than 100 percent of the resulting increase in the net capital share is due to a rise in income from residential structures themselves. I now discuss and interpret the results for each shock.

SHOCK TO THE REQUIRED RATE OF RETURN "r" This case tests the Piketty (2014) hypothesis that a rise in savings will push up the net capital share. In general equilibrium, increased savings influences capital income by pushing down the real interest rate; hence, to learn the sign of the effect of savings on the net capital share, it suffices to study the partial equilibrium effect of a change in the real interest rate.

Since the decomposition of the U.S. economy in section II.C allows r in the nonhousing and housing sectors to be different, I define a "shock to r" to be a parallel shift dr in these two rates of return. I then define the elasticity of the net capital share with respect

to this shock to be

[[partial derivative](net capital share)/(net capital share)]/[[partial derivative]r/[r.sup.ave]]

where [r.sup.ave] is the average return on capital across the economy as a whole, including both the nonhousing and housing sectors.

Table 6 shows that for all [[sigma].sub.i] within range (equation 21), the response of the net capital share to r is positive--barely so at minimum (0.04) and strongly so at maximum (0.54). This is inconsistent with the Piketty (2014) hypothesis that a decline in r can produce an increase in the net capital share, and it corroborates the findings from the single-sector model in section III.

Table 7 reveals that the response of the net capital share to r depends primarily on three elasticities, all negatively: [[sigma].sub.Z], [[sigma].sub.F], and [[sigma].sub.H], each with a sensitivity of about -0.20.

Each of these elasticities governs the extent to which an aggregate that includes labor (which is unaffected by r) can be substituted for an aggregate that does not include labor. But even when these elasticities are chosen at the maximum level in the range ([[sigma].sub.Z] = 0.8, [[sigma].sub.F] = 1.5, [[sigma].sub.H] = 1.5), the response of the net capital share to r remains slightly positive.

Table 8 shows that the vast majority of the response to r comes from residential structures: at benchmark [[sigma].sub.i], a contribution of 0.23 out of an overall elasticity of 0.26. This is for two reasons. First, since both housing [G.sub.2] and aggregate consumer demand Z have [[sigma].sub.s] below 1, the direct positive impact of rising r on income from residential structures outweighs the negative effect of substitution--much more so than for nonresidential structures or equipment. Second, since the analysis in section II.C finds a lower r for the housing sector than the nonhousing sector, a parallel shift in these rates has a disproportionate effect on housing. This reinforces the centrality of housing to any assessment of the Piketty (2014) narrative.

Finally, table 8 indicates the importance of a crucial distinction--namely, the distinction between (A) the ratio of housing capital to aggregate income and (B) the share of housing capital income in aggregate income. In response to rising r, (A) falls: higher r pushes down the demand for residential structures relative to aggregate income, (28) and since residential land's share of income remains roughly constant in table 8, higher r will push down the valuation of this land relative to aggregate income. At the same time, as already discussed, (B) rises dramatically. Hence a shock to r pushes (A) and (B) in different directions, making it important to document (A) and (B) separately.

SHOCK TO THE PRICE OF EQUIPMENT INVESTMENT [P.sub.e] This case tests the Karabarbounis and Neiman (2014a) hypothesis that declining investment prices--which have been concentrated in equipment--will push up the net capital share. As table 6 shows, this remains ambiguous for the range of [[sigma].sub.i] specified in equation 21, which are consistent with either a positive or negative relationship between [P.sub.e] and the net capital share.

Table 7 makes clear the source of this ambiguity: the response of the net capital share to [P.sub.e] depends almost entirely on the elasticity of substitution [[sigma].sub.H] between labor and equipment. When [[sigma].sub.H] is near the top of the [0.5, 1.5] range, falling [P.sub.e] leads to arise in the net capital share, consistent with Karabarbounis and Neiman (2014b); when [[sigma].sub.H] is near the bottom, the opposite is true.

The "high elasticity" row in table 8, however, provides cause for skepticism regarding the Karabarbounis and Neiman (2014a) channel. Here, [P.sub.e] has a substantial negative effect on the net capital share. But this effect comes almost exclusively from the net capital income of equipment itself--which, in this partial equilibrium exercise, moves in parallel with the value of the equipment stock--rather than through some less direct channel. Section II found that the value of equipment (which has recently fallen) has followed a path quite distinct from the path of the net capital share (which has recently risen). This is not consistent with a major role for [P.sub.e].

For the [P.sub.e] hypothesis to be consistent with the data, it would be necessary for declining [P.sub.e] to push up the net capital share through some channel other than a rise in the value of the equipment stock. Karabarbounis and Neiman (2014b) sketch one such possibility, where falling [P.sub.e] can lead to an increase in the net capital share despite an actual decline in the aggregate value of equipment, but the multisector model here does not corroborate their mechanism. (29)

SHOCK TO THE PRICE OF RESIDENTIAL STRUCTURES INVESTMENT [P.sub.s2] In table 6, a rise in [P.sub.s2] leads to a rise in the net capital share for benchmark [[sigma].sub.i]; for other choices of [[sigma].sub.i] within range (equation 21), there is at worst roughly no effect. According to table 7, the effect is most sensitive to the elasticity [[sigma].sub.Z] of substitution between housing and nonhousing output; and according to table 8 it works almost entirely through the net capital income of residential structures themselves. The mechanism here is relatively simple: when the ability to substitute away from housing is limited, costlier residential investment leads to a higher-value housing stock and a larger share of income accruing to housing.

SHOCK TO THE QUANTITY OF RESIDENTIAL LAND [L.sub.2] This is similar to the previous case. In table 6, a decline in the quantity of residential land [L.sub.2] leads to a rise in the net capital share for benchmark [[sigma].sub.i]; for other choices of [[sigma].sub.i] within range (equation 21), there is at worst roughly no effect. Again, according to table 7, the effect is most sensitive to [[sigma].sub.Z]; now, however, the effect is smaller and works mainly through the net capital income earned by residential land.

SUMMARY OF RESULTS AND CONCLUSION I have examined the response of the multisector model to four exogenous shocks. The first two shocks correspond to versions of the accumulation view: a shock to r captures the general equilibrium channel through which the rise in savings postulated by Piketty (2014) affects factor shares, while a shock to [P.sub.e] is central to the Karabarbounis and Neiman (2014a) narrative.

In both cases, the results do not support the proposed mechanism. For all choices of {[[sigma].sub.i]} within the range considered, a fall in r leads to a fall in the net capital share, in contrast with Piketty (2014). Meanwhile, although a fall in [P.sub.e] can produce a rise in the net capital share, it only does so by pushing up the net capital income from equipment itself, which is at odds with the evidence from section II.

The latter two shocks both embody some form of the scarcity view, which is more successful in the multisector model. Either a rise in the price [P.sub.s2] of residential investment or a fall in the quantity [L.sub.s2] of residential land leads to a rise in the net capital share, for the vast majority of {[[sigma].sub.i]} in the range (equation 21). In both cases, the mechanism works by increasing the net capital income earned by housing, consistent with the dramatic rise in the contribution of housing documented in section I.

IV.C. Counterfactual Exercise

Building upon the promise of the scarcity view in the previous section, I now use the multisector model to perform a counterfactual exercise, exploring the implications of alternative paths for [P.sub.s2] and [L.sub.s2].

The real price [P.sub.s2] of residential investment has risen in the last several decades in the United States; furthermore, real output has grown substantially, putting pressure on the supply of residential land. I consider a counterfactual where these two forces are not present: where the real price of [P.sub.s2] is instead constant from the beginning (1948) of the sample period onward, and where the quantity of residential land [L.sub.s2] grows in tandem with real output from the beginning of the sample period onward. (30)

In contrast to the exercises in section IV.B, which consider only local shocks to exogenous variables, this counterfactual involves large global changes. It requires additional global assumptions to compute; for this purpose, I will assume that the production functions (Z, F, [G.sub.1] [G.sub.2], H) each have a globally constant elasticity of substitution. I consider two choices of {[[sigma].sub.i]}: first, the benchmark [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; and second, an alternative [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that sets [[sigma].sub.Z], of, and [[sigma].sub.H] (the [sigma]s that govern the response to [P.sub.s2] and [L.sub.2], according to table 7) to the minimum values in the range (equation 21).

Figure 12 displays the results of this exercise, distinguishing between the housing and nonhousing components of the net capital share. Consistent with table 8, there is little effect working through the nonhousing component. Furthermore, the large initial increase and then decline in the housing component, through 1980, is left untouched by the counterfactuals. However, much of the subsequent increase in the housing component is eroded.

This is consistent with a role for rising residential investment costs, along with growing scarcity of residential land, in driving up housing's contribution to the net capital share: when these forces are reversed in a counterfactual, we see less of a rise. At the same time, figure 12 makes clear the limitations of this account. It does not explain the fall and rise in the nonhousing component, nor can it explain all aspects of the housing time series. The scarcity view, therefore, is only a partial replacement for the accumulation view: it achieves better consistency with data and theory, but does not purport to explain more than a fragment of the evolving factor income distribution.

V. Conclusion

The aging Kaldor (1957) facts have retreated in the face of experience. Today, macroeconomists no longer claim that factor shares are constant--but what should replace the old consensus?

It is increasingly commonplace to believe that labor is ceding ground to capital. But a closer look at postwar experience reveals a murkier story, in which steady increase is limited to the gross capital share. The net share, by contrast, has fallen and then recovered; it consists of a large long-term increase in net capital income from housing, and a more volatile contribution from the rest of the economy, with little cumulative movement in either direction.

Even more elusive than these facts is a cohesive explanation of them. The accumulation view, in both its Piketty (2014) and Karabarbounis and Neiman (2014a) variants, falters in multiple respects. It cannot explain the dominant role of housing, nor can it be readily reconciled with the evidence on elasticities of substitution. Outside of housing, there appears to be little correlation between the capital-income ratio and the net capital share.

By contrast, the rise in housing's contribution to the capital share can be explained in part as the result of scarcity. The rising real cost of residential investment and the limited quantity of residential land have conspired to make housing more expensive, and given low elasticities of substitution this has meant a rise in housing's share of income.

With these trends in mind, policymakers concerned about the distribution of income should keep an eye on housing costs. Many urban economists, including Glaeser, Gyourko, and Saks (2005) and Quigley and Raphael (2005), have documented explicitly how restrictions on land use and residential construction inflate the cost of housing. Outside of housing, however, this paper raises more questions than it answers about the evolution of the net capital share: once the accumulation view has been discarded, there is no master narrative at hand that can explain the postwar fall and rise.

If anything, these results suggest that concern about inequality should be shifted away from the overall split between capital and labor and toward other aspects of distribution, such as the within-labor distribution of income. Although the net capital share has at times seen dramatic shifts both up and down, away from housing its long-term movement has been quite small, and there is no compelling reason to suggest that this pattern will change going forward.

No doubt, however, the distribution between capital and labor will continue to be a salient issue: we surely have not seen the last of David Ricardo's "principal problem of Political Economy."

ACKNOWLEDGMENTS For their helpful comments and conversations I thank my discussants Robert Solow and Brad DeLong, the participants at the Spring 2015 Brookings Panel on Economic Activity, Adrien Auclert, Loukas Karabarbounis, Stephen Murphy, Thomas Piketty, Alp Simsek, Ludwig Straub, Ivan Werning, Gabriel Zucman, and the volume editors.


Massachusetts Institute of Technology


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Piketty, Thomas, and Gabriel Zucman. 2014. "Capital Is Back: Wealth-Income Ratios in Rich Countries, 1700-2010." Quarterly Journal of Economics 129, no. 3.

Quigley, John M., and Steven Raphael. 2005. "Regulation and the High Cost of Housing in California." American Economic Review 95, no. 2: 323-28.

Ricardo, David. 1821. On the Principles of Political Economy, and Taxation. London: John Murray.

Rotemberg, Julio J., and Michael Woodford. 1999. "The Cyclical Behavior of Prices and Costs." In Handbook of Macroeconomics, vol. 1, edited by J.B. Taylor and M. Woodford. Amsterdam: North Holland.

Solow, Robert M. 1958. "A Skeptical Note on the Constancy of Relative Shares." American Economic Review 48, no. 4: 618-31.

Thorsnes, Paul. 1997. "Consistent Estimates of the Elasticity of Substitution between Land and Non-Land Inputs in the Production of Housing." Journal of Urban Economics 42, no. 1: 98-108.

Weitzman, Martin L. 1976. "On the Welfare Significance of National Product in a Dynamic Economy." Quarterly Journal of Economics 90, no. 1: 156-62.

(1.) See, for instance, Azmat, Manning, and Van Reenen (2012), who address the role of privatization, and Arpaia, Perez, and Pichelmann (2009), who draw attention to capital-skill complementarity.

(2.) This decomposition potentially applies at many levels of aggregation: for instance, the "sector" may be the entire domestic economy, in which case gross value-added at market prices is called gross domestic product (GDP).

(3.) This invariance can be very useful in analyzing trends--for instance, when high inflation pushes up nominal interest rates, a large share of capital income is often paid to bondholders in the form of nominal interest. As Modigliani and Cohn (1979) memorably observed in the context of late-1970s inflation, this causes recorded profits to dramatically understate true profits, since they do not reflect the gain from real depreciation in nominal liabilities.

(4.) The full set of start dates is 1948 (France, United Kingdom, United States), 1955 (Japan), 1960 (Canada), 1990 (Italy), and 1991 (Germany). Data for France, the United Kingdom, and the United States are available starting even earlier, but I focus on 1948 onward because that is when the necessary data start becoming available for my subsequent, more detailed exercise for the United States in section II. This also keeps the focus on postwar dynamics, detached from the sizable dislocations associated with depression and wartime, and mostly postdates the transition from agricultural self-employment to formal employment that bedeviled older analysts like Johnson (1954).

(5.) When countries have different trends in [s.sub.i,t], there will be an artifactual discontinuity in [[alpha].sub.t] when a country enters the sample, which in principle could deliver a misleading impression of the actual year-to-year changes in [s.sub.i,t]. In practice, this does not seem to be much of an issue here, and alternative approaches--for instance, averaging the first differences Asit across countries in the sample for each year t, then plotting the cumulative average first difference over time--deliver similar results.

(6.) See Feenstra, Inklaar, and Timmer (2013).

(7.) There are two exceptions: the Canadian national accounts already provide a decomposition of mixed income into labor and capital, which I use; and the Japanese national accounts do not fully break out the corporate sector, necessitating some additional imputations.

(8.) For Canada and Japan, the "housing" sector is actually the owner-occupied housing sector due to data limitations. Importantly, Canada and Japan do not drive the trend here: to the contrary, from 1960 (when Canada enters the sample) to 2010, the average contribution of housing to net capital income in Canada and Japan increases by 3 percentage points, while in France, the United Kingdom, and the United States it increases by 4.5 percentage points.

(9.) See, for example, Andrews and Sanchez (2011) for some discussion of trends in homeownership.

(10.) In fact, computations using the multisector model in section IV.B will show that this is backward: the fall in r induced by savings should lead to a concentrated decline in the housing component of the net capital share.

(11.) As described in footnote 7, different imputations are used for Canada and Japan. Furthermore, since separate data for the corporate sector are not available in Japan, figure 4 displays the overall capital share for Japan instead.

(12.) Explicitly, [((1 - ,177)/(1 - .264)).sup.(1/25)] - 1 [approximately equal to] .45% and [((1 - .236)/ (1 - .177)).sup.(1/13) - 1 [approximately equal to] -.57%.

(13.) Explicitly, for unweighted: [((1 - .214)/[(1 - .229)).sup.(1/62)] - 1 [approximately equal to] .03%. For weighted: [((1 - .231)/(1 - .245)).sup.(1/62)] - 1 [approximately equal to] .03%.

(14.) To be clear, the long-run impact in individual countries can be larger. Perhaps the most extreme example is Japan, which table 1 shows to have experienced a decline in the average nonhousing share of aggregate capital income from 31 percent in the 1960s to 20 percent in the 2000s, implying an annualized contribution to wage growth of roughly three-tenths of a percentage point. But table 2 does not suggest any long-run tendency for corporate capital shares in different countries to diverge from each other; the distinct paths across countries are therefore probably best interpreted as mean-reverting variations around an apparently trendless average.

(15.) Since F is constant-returns-to-scale, marginal and average costs are equal, so I will refer to them both as "cost."

(16.) Ideally, this exercise would extend to all seven of the G7 countries covered in section I, but the additional data required make this difficult.

(17.) Since the flow of funds provides end-of-year values for capital, I average the adjacent end-of-year values to obtain the effective capital stock used in production during each year.

(18.) Although this seems high for a real return, note that it is a pretax return, the return before taxes are applied either to corporate profits or distributions of interest or dividends. Interestingly, it is slightly lower than the constant return in figure 7 estimated using my alternative approach, which is roughly 12.8 percent. As explained later in this paper, this return is higher because according to the flow of funds, the total market value of the corporate sector in the United States has actually been lower than the book value, on average, in the postwar era--suggesting that pure profits are, if anything, negative, and that the assumption that pure profits are zero on average is not misattributing these profits to an exaggerated return r on capital.

(19.) The equipment investment deflator rose relative to the GDP deflator at an annualized rate of 1.5 percent during the sample period, as opposed to a 1.1 percent average rise in the deflator for nonresidential structures.

(20.) For simplicity, I will call the total value of the firm's fixed assets its "book value," even though this is not necessarily book value in the usual sense: 1 will define it to exclude financial assets--these are instead subtracted from the market value, which includes net financial liabilities--and to use values from the flow of funds for real estate and equipment, which are updated to reflect changes in price.

(21.) Online appendixes for papers in this volume may be found at the Brookings Papers web page,, under "Past Editions."

(22.) This causes some anomalies in the early postwar years, when the corporate sector was left with large cash balances and relatively little debt, making net liabilities negative while equity valuations were already quite low, and leading to an extremely low market relative to book value. To avoid undue influence from this period. I exclude data from prior to 1955 in the benchmark results displayed here; otherwise, there is an even more dramatic estimated downward trend in r(t).

(23.) Note that this imputation, which uses data on the value of fixed assets in the non-corporate sector, is different from the imputation in section I.B, where these data were not available for the full sample and the net capital share of income--rather than the return r--in the nonhousing, noncorporate sector was assumed to be the same as in the corporate sector.

(24.) For a few sources that list a range of elasticities, I take the midpoint. This has minimal effect on the distribution.

(25.) There is some conflict between the assumption of exogenous s for all income and the emphasis on r - g. If only this fraction s of capital income r is saved, then existing fortunes will grow at the rate s x r - g, not r - g; and for plausible values of s as a share of all income, s x r - g is likely to be quite negative, implying the rapid erosion of existing wealth.

(26.) [[sigma].sub.z] < 1 is strongly supported by casual observation as well. For instance, as the real price of housing services has risen in the United States over the last several decades, its share of consumption has increased slightly; there is also a well-known tendency for consumers to spend a larger share of their budgets on housing in areas where housing is expensive.

(27.) Since housing is probably not an input to the production of equipment or structures, it would be slightly more natural to assume that these prices are fixed relative to the price of nonhousing output F; I assume they are fixed relative to Z for convenience, and in general the relative prices of F and Z do not change enough that this has a sizable impact on the results.

(28.) This occurs in the model but is not directly visible in tables 6 through 8.

(29.) Karabarbounis and Neiman (2014b) devise a model where two types of capital, high-depreciation (which can be interpreted as equipment) and low-depreciation (which can be interpreted as structures) combine to form a capital aggregate; the elasticity of substitution between these types of capital is less than 1, while the elasticity of substitution between the capital aggregate and labor is greater than 1. A decline in the price of equipment lowers the price of the capital aggregate, which induces substitution from labor to the capital aggregate; but since the elasticity of substitution between equipment and structures is less than 1, this also causes a decline in equipment relative to structures. With the right parameters, it is possible for a decline in the price of equipment to increase the net capital share while net capital income from equipment itself actually declines.

(30.) To make this modification, I assume that the quantity of land [L.sub.s2] was in reality constant, and then expand it in each year by a fraction equal to cumulative real GDP growth since 1948. Depending on the interpretation of [L.sub.s2], the assumption that it has been constant may or may not be appropriate.



J. BRADFORD DELONG Let me begin by thanking Matthew Rognlie for his serious and thoughtful digging into this set of factor-payments data. That digging leaves me in an ideal position as a discussant. There are interesting and important numbers here, numbers that have not been put together in this way before. The author is wise enough to know he has not nailed to the floor what these numbers mean, leaving me in an excellent position, if not to add intellectual value, at least to claim a lavish intellectual-rent share of Rognlie's product.

I was weaned on the education-deficit explanation of recent trends in U.S. inequality, perhaps best set out by the very sharp Claudia Goldin and Lawrence Katz (2009) in The Race Between Education and Technology. In their view, the bulk of U.S. inequality trends since the 1980s were driven by education's losing this race. In the era that began in 1636, the United States (and its founding colonies) made increasing the educational level of the population a priority. But that era came to an end in the 1970s, while skill-biased technological change continued. That meant that the return to education-based skills began to rise. And it was that rise that was the principal driver of rising income inequality.

But recently, reality has not been agreeing with what had once seemed to me to be a satisfactory explanation. First, to get large swings in the income distribution out of small changes in the relative supply of educated workers requires relatively low substitutability between college-taught skills and other factors of production. As inequality has risen, the degree of substitutability required to fit the data has dropped to what now feels to me an unreasonably low magnitude. Second, while it is true that we have seen higher experience-skill premiums and sharply higher education-skill premiums, the most rapid growth in inequality appears now to be unduly concentrated in the upper tail. The distribution of the rise in inequality does not seem to match the distribution of technology-complementary skills at all.

I can illustrate this by looking simply at my own family history. My Grandfather Bill's income reached not just the top 1 percent or the 0.1 percent but the 0.01 percent back in 1968, in the days before the rise in inequality, by selling his construction company to a conglomerate. A good many of us who are his grandchildren have been very successful- consider my cousin Phil Lord's The LEGO Movie, and the other franchises for which he gets "director" credit. But even should any of us be as lucky as my Grandfather Bill was in terms of our peak income and wealth as a multiple of median earnings, we would still be a multiple of his rank further down in the percentile income distribution.

Today, in the United States you need roughly 3.5 times the wealth you needed in 1968--and eight times the wealth worldwide you needed in 1968--to achieve the same percentile rank in the distributions (see Atkinson, Piketty, and Saez 2011, Piketty 2014, and Saez and Zucman 2014). I find it simply impossible to conceive that such an extreme concentration is in any way a return to a factor of production obtained as the product of "hours spent studying" times "brainpower," even when I also multiply by a factor of "luck" and a factor of "winner-take-all economy."

What, then, is going on to drive this sharp rise in inequality, if it is not some interaction between our education policy on the one hand and the continued progress of technology on the other? Piketty (2014) offers a guess: the real explanation, he writes, is that the period 1914-80 is the anomaly. Without great political disturbances, wealth accumulates, concentrates, and dominates. The inequality trends we have seen over the past generation are simply a return to the normal pattern of income distribution in an industrialized market economy in which productivity growth is not unusually fast, and political, depression, and military shocks are not unusually large and prevalent.

What about what John Maynard Keynes (1936) called the "euthanasia of the rentier"? Eighty years ago, Keynes guessed that, as accumulation proceeded and the capital-output ratio rose, the relative rate of profit would decline and it would decline by more. Thus more and more concentrated wealth would mean a smaller and smaller share of income received by pure rentiers--as opposed to entrepreneurs and risk-bearers. Keynes strongly believed that the returns to investment at the margin were likely to drop rapidly enough to make this "euthanization of the rentier" the most likely possibility. Rognlie agrees. And, indeed, it is difficult to see how, if investment takes the form of the accumulation of useful physical capital, it could be otherwise.

In an anticipatory response to this part of the Rognlie critique, Piketty (2014) points to a remarkable constancy in the rate of profit. His data show it to have been stuck between 4 and 5 percent per year across centuries with very different capital-output ratios. Piketty, however, appears agnostic as to whether the cause is easy capital-labor substitution, rent-seeking through control of the government by the rich, or social structures that set 4-5 percent a year as the "fair" rate of profit. This question is left hanging by Piketty (2014), which is why it is truly excellent that Matt Rognlie has written this paper, bringing well-ordered and insightfully organized data to these questions.

Rognlie's paper focuses on the net rather than the gross capital share. That focus is surely right. I never understood why, in the Solow model, gross savings was supposed to be a function of gross output anyway.

I have the usual big worries about the data. (1) But let me skip over those: I cannot resolve them here, and it will take a great deal of additional thought and work before one can even begin to think of resolving them. Let me focus, instead, on the big news. As Rognlie stresses, the big news in the post-World War II net capital share is the surge in housing, rising from 3 percent to 8 percent of private domestic value-added. How much of this is a real increase in housing intensity? How much reflects congestion driven by exhaustion of the low-hanging superhighways? How much is rent-extraction via NIMBYism? And how much trust do we place in these "imputed rent" imputations--and what do they mean, anyway?

There has always been a problem with using our GDP estimates as social accounts. In GDP, we measure each unit's contribution to production at the final unit's marginal cost and each unit's contribution to societal well-being at the final unit's money-metric marginal utility. In the presence of anything like near-satiation in consumption, or of near-exhaustion in productive capacity, this does not convey a true picture. These are very hard questions, and we do not have any very good answers to them.

The fact that the big news since World War II is a rise in housing as a share of value-added is striking. It raises the question of whether this surge--the rise of valuable urban housing now--is the only such shift in value-added shares. Was there another significant shift a century and more ago? With the coming of the railroad, the iron-hulled steamship, and the first era of globalization, the value of European farmland and European mine installations crashed under competition from what were then developing resource-abundant economies. How significant was that crash for aggregate wealth and income distributions? Was it large enough to drive a significant share of the great equalization Piketty sees occurring between 1900 and 1930? We do not know.

Moreover, to the extent that shifts in land values are the drivers of shifts in the capital-output ratio, is this really a problem? It is a problem, or rather a reflection of and a consequence of a problem, to the extent that it is driven by NIMBYism. But is it a problem otherwise?

As Rognlie has also rightly stressed, a secondary piece of big news in his numbers is the pre-1990 fall in the net capital share. This fall is driven by a rise in calculated depreciation rates. That depreciation is real, as our capital stock today contains many more machines that are rapidly made obsolete by Moore's Law and so are not built for durability.

However, this too raises a puzzle. The pre-1990 fall in the net capital share is not matched by a decline in the relative capitalization of the corporate sector. Rognlie points out a steady rise in capitalization up to the late 1960s, followed in the 1970s by a "negative bubble"--earnings yields on equities that were truly absurdly high--that lasted well into the 1980s. Since the start of the 1990s, we have seen a bubbly rise in the relative capitalization of the corporate sector, a rise that persists in spite of sub-par business-cycle performance. There is thus a severe dissonance between what the production-function and depreciation logic say should be the value of claims to capital ownership and what financial markets say the value is. Once again, these are very hard questions, and we do not have any very good answers to them.

In conclusion, I strongly endorse what I take to be Matthew Rognlie's bottom line. The post-World War II variation in the observed net capital share cannot be explained by returns on the underlying assets. Instead, the decomposition in the paper attributes most of the variation in the factor distribution of income to shifts in markups and pure profits, with accumulation and returns outside of housing playing a distinctly secondary role, if any role at all.

It is equally hard to find any role for the race between education and technology. There should be such a role, for we do think the factors of production are labor, education-skills, machines, and buildings (including residences). Variations in factor supplies should show themselves in factor returns. Likewise, variation in income inequality is hard to attribute to wealth ownership, or human capital investment or to differential shifts in rewards to factors like raw labor, experience-skills, education-skills, and machines. Rognlie thus concludes that "concern about inequality should be shifted away from the overall split between capital and labor and toward other aspects of distribution, such as the within-labor distribution of income." The only dissent I wish to make is this: Rognlie is correct, today, but if Piketty is right he may no longer be correct in 50 years.

Matthew Rognlie's conclusion is bad news for us economists. It leaves us in the same position as those trying to explain an earlier large puzzle in the production function, the twentieth-century retardation of the British economy. It was Robert Solow (1970) who said: "Every discussion among economists of the relatively slow growth of the British economy compared with the Continental economies ends up in a blaze of amateur sociology" (pp. 102-3). But this time, I really would like us to be able to do better than we did then.


Atkinson, Anthony B., Thomas Piketty, and Emmanuel Saez. 2011. "Top Incomes in the Long Run of History." Journal of Economic Literature 49, no. 1: 3-71.

Goldin, Claudia, and Lawrence F. Katz. 2009. The Race Between Education and Technology. Cambridge: Belknap Press.

Keynes, John Maynard. 1936. The General Theory of Employment, Interest and Money. London: Macmillan.

Piketty, Thomas. 2014. Capital in the Twenty-First Century. Cambridge: Belknap Press.

Saez, Emmanuel, and Gabriel Zucman. 2014. "Wealth Inequality in the United States Since 1913: Evidence from Capitalized Income Tax Data." Working Paper no. 20625. Cambridge, Mass.: National Bureau of Economic Research (revised version forthcoming in Quarterly Journal of Economics).

Solow, Robert M. 1970. "Science and Ideology in Economics." Public Interest no. 21: 94-107.

(1.) These big worries concern (i) depreciation allowances in the accounts (which I perhaps worry about more than most do); (ii) how much of the value that comes from installing capital comes from (local) learning about how to handle the technology, something that does not depreciate from the point of view of the individual firm that is not captured; (iii) from the societal point of view, how much of the value that comes from installing capital comes from global learning about how to handle the technology; and (iv) the perennial questions about what in high-end labor incomes are really incomes earned by raw labor and human capital, and what are rent-extraction and thus sharing in the returns to capital.


ROBERT SOLOW Matthew Rognlie's excellent paper circles around a fundamental question in medium-run macroeconomics: how strongly, if at all, does the rate of return on capital fall as capital intensity increases? I describe it as fundamental because it lies at the heart of at least two important and contentious current issues. Capital intensity may be increasing for some time in developed economies if only because the growth of population will slow with no commensurate reduction in saving. Then the behavior of the return on investment will certainly affect the demand for investment and thus the plausibility of secular stagnation. In addition, the response of the rate of return will affect the functional distribution of income between compensation and profits and thus, eventually, the degree of income inequality, which is already a political issue, at least rhetorically. (It is interesting, although not directly relevant, that another imponderable, the likely future of total factor productivity, connects both these issues: rapid technological progress could sustain the return on investment as it has in the past, but that may not happen again.)

This question of diminishing returns to capital intensity has preoccupied economists for a long time, from Ricardo and Mill to Keynes and Schumpeter. As an indication of how little was ever settled, it is not so long ago that growth theory was littered with so-called "AK models" that were founded on little more than the assumed absence of diminishing returns to capital intensity. Those models are not so fashionable now. So at last I find myself with the delightful task of discussing a paper--by someone younger than several of my grandchildren--that makes a serious and intelligent effort to see what we know or what we might be able to find out about diminishing returns to capital intensity. No doubt this effort was stimulated by the Piketty phenomenon, but it is of more general interest.

CAPITAL SHARE AND RETURNS TO CAPITAL The paper does a useful service by documenting in some detail that a substantial fraction of recent real capital accumulation in the United States took the form of land and buildings, including housing. How should we think about this fact? For some purposes we can say (and do say) that houses just represent a very capital-intensive form of production: they produce housing services, measured by market and imputed rents, with very little labor input. That is okay for national income and product accounting, but it misses the deeper point: we are really interested in the intensity of diminishing returns to capital.

For estimating an economywide elasticity of substitution, it would be better to eliminate the housing stock and associated land on the capital-input side and the rents on the output side, recognizing that the motives underlying behavior are slightly different from those whose effects we are trying to isolate. I would also favor eliminating some other sectors: financial services, because it is so unclear what one means by output; unincorporated enterprises, because it is impossible to separate labor income from return to capital; and general government, because the accounting conventions make no sense. The usual calculations of the elasticity of substitution should probably be confined to the inputs into and the value-added produced by nonfinancial corporations, just under half of gross domestic product. The paper does, very sensibly, omit unincorporated enterprises and general government, but it includes financial corporations along with nonfinancial. I would recommend excluding them as well. In the 1960s and 70s, the profits of financial corporations were about 15 percent of all corporate profits; just before the financial crisis they were up to nearly 40 percent of the total (and are rather less now). I cannot believe that this has anything to do with the marginal product of capital, as we understand that notion, or with the substitutability of capital for labor.

The paper spends more time and effort than I would have done on the consequences of the growth of housing for the economy wide share of capital. This is not to say that the accumulation of capital in the form of housing is not important for the understanding of capital accumulation and the functional distribution of income. But one has to recognize that much of that capital is acquired as a store of value (and perhaps a vehicle for speculation) rather than as a productive input. This is certainly true of the 20-million-dollar condominiums bought by crooks from Russia, Latin America, and elsewhere, and their offspring. It probably also played a substantial part in the housing boom and bubble of the previous decade, although that may of course change. Whether willingness to invest in housing can provide an offset to otherwise excess saving and might thus be a factor in warding off secular stagnation is a possibility; but it seems like a weak reed. If diminishing returns should drive down the return on industrial capital, would that increase the demand for housing? There is not much evidence.

This question of the relation between housing and the relative share of capital reminds me of a complaint that I have been nursing. It is directed not at this excellent paper but at the literature. The 19th century German mathematician Leopold Kronecker--he of the Kronecker delta--is supposed to have said: "God created the integers; everything else is the work of man." There is a strong implication that God knew what She was doing, but mankind has made a mess of the rest. If Kronecker had been an economist he might have said that God created prices and quantities, and all the rest is a manmade mess. The real subject of Rognlie's paper is the effect of increasing capital intensity on the rate of return. To put it in terms of a relative share--a ratio of prices times a ratio of quantities--is to add unnecessary complication to an already complicated question. Rognlie does a nice, clearheaded job, and he has some very interesting things to say. It is the literature that creates a detour.

I do not want to spend much time on the net-gross distinction. Once one focuses on the rate of return, it becomes obvious that the return net of depreciation is what matters, both for distribution and with respect to investment demand (and hence secular stagnation). Nevertheless, it is worth remembering that the only reason research has devoted so much effort to gross concepts was the sense that measured depreciation might verge on the meaningless because it reflected accounting conventions and tax incentives that had little or nothing to do with the changing productive capacity of existing plants and equipment. The conceptual basis of the data might be much better nowadays. One further reminder: modelers now universally assume, without comment, that depreciation is proportional to the stock of capital. This is an overwhelmingly convenient assumption: it is the only assumption that makes depreciation independent of the history of gross investment. Convenience may be its only advantage.

Back in the early years of my research, when I used to see an occasional survival table for some class of capital goods, what I saw did not look much like declining exponentials. Maybe this does not matter, but how do we know?

ROGNLIE'S "PURE PROFIT"--AND ITS IMPLAUSIBLE VALUE The most exciting result in Rognlie's paper is his finding that, during the postwar period, most of the action in the distribution of corporate value-added (after taxes on production) comes not in the compensation of labor nor in the market return to capital but in a residual. He calls it "pure profit," but I like to think of it as monopoly rent, broadly conceived. This is a big deal, because it can help to explain many things, but it is also a big annoyance because it makes for very difficult analytical-empirical problems.

The easy way to solve them is to just assume that value-added is divided between labor and capital roughly in accord with marginal products; this is of course the competitive allocation. But I suspect we do not believe it is true. A corporation facing a demand curve with elasticity [epsilon] (a sort of "as if' elasticity reflecting many things) will choose inputs and output so that each real factor price is ([epsilon] - 1)/[epsilon] times its marginal product. The result will be a monopoly rent equal to a fraction 1/[epsilon] of value-added. Looked at differently, 1/[epsilon] is equivalent to (price - marginal cost)/price. It is what Abba Lerner long ago defined as "the degree of monopoly" for that firm or for the representative firm. According to Rognlie's calculations, that is what has been rising for U.S. corporations since about 1980. So, how big is it?

According to Rognlie's calculations, 1/e averages to about zero, and it manages to grow only by going from negative to positive. This strikes me as wholly implausible. It worries Rognlie, too. He appeals to the idea of Chamberlinian large-group monopolistic competition: free entry overcrowds the market and drives pure profit to zero. But this way out seems just as implausible: it is precisely barriers to entry, of which there are many, that create monopoly rents in the first place. The full calculation leads to the further conclusion that the market return on capital was about 13 percent a year between 1950 and 2010, if it is assumed to have been constant, and to have fallen from above 16 percent in 1950 to below 12 percent in 2010 if it is allowed to have a linear trend. A quadratic trend does no better in the author's figure 7. It is hard to believe that the discount rate was this high from 1950 to 2010. (Household saving was available at an interest cost of 4 to 5 percent; one would have expected more investment to have taken place.) If the market rate of return were assigned a lower value, presumably the estimated monopoly rent would be a larger fraction of value-added.

All of this provokes an interesting question, to which I do not have an answer: Why do Rognlie's sensible calculations conclude that pure profit or monopoly rent was negative nearly all the time between 1950 and 2010? (or, almost equivalently, Why was his version of Tobin-Brainard's q less than one most of the time?) Equation 5 in the paper looks very busy, but the basic idea is simple and smart: the difference between the stock market value of a corporation and the "book value" of its assets is interpreted as the present discounted value of the anticipated stream of rents. Maybe the version of book value that he uses, in which physical capital appears not as reproduction cost but at historical value (or something else), is peculiar, especially when there is inflation. Maybe stock market valuations are equally garbage-ridden. Rognlie needs to use the difference between these numbers, which must certainly have a lot of noise, and not necessarily white noise.

The best suggestions I can manage are a couple of almost-constructive suggestions for further work. First, I think it is essential to get the financial services industry out of the calculation. The profits of financial firms, mostly from trading and mostly from asymmetric information, are not to the point here. Second, a clearer picture would allow for the fact that recorded wages include a certain amount of monopoly rent. This is obviously true of executive compensation, but even garden-variety compensation has a nontrivial rent component.

THE PRICE-TO-MARGINAL-COST RATIO The real issue here is the ratio of price to marginal cost in American industry (or nonfinancial industry, as I would prefer). There is a large literature on average mark-ups of price over cost, mostly concerned with cyclical behavior. Much of it is summarized and discussed in the article by Julio Rotemberg and Michael Woodford (1999), cited in Rognlie's paper. But I am more interested in work that aims explicitly at the ratio of price to marginal cost ([epsilon]/([epsilon] - 1) in that notation). Robert Hall (1986) estimates that ratio to be between 2 and 3, which would imply that monopoly rents amount to between 1/2 and 2/3 of value-added. That seems shockingly high. Mark Bils (1989) has an ingenious method that puts rent at about 30 percent of value-added. Both of those papers go back to the 1980s; if Rognlie is correct, as I think he is, the right number, whatever it is, would be higher now.

At the BPEA conference where Rognlie presented this paper, Robert Hall remarked that his current estimate of the ratio of price to marginal cost is about 1.2, which would make rent about 16 to 17 percent of value-added. He suggested that this might just about cover fixed costs, leaving net rent at zero. My conclusion is that the degree of monopoly in U.S. industry remains an open question and needs more research, both microeconomic and macroeconomic. The matter of fixed costs strikes me as more complicated. In the short run, one imagines fixed costs to be mainly capital costs. In the medium to long run, as in Rognlie's paper, capital costs are modeled explicitly and treated as variable. Remaining fixed costs are a little hazy.

All of this work makes a tacit assumption which, as I have already suggested, may be in error, namely that all of the rent accrues to the capital-income part of value-added. It seems likely that, at least

in many industries, the reported compensation of labor includes some rent, either in the form of wages or benefits or working conditions. I have always taken it for granted that the division of rent was what collective bargaining was all about, back when there actually was collective bargaining. Even without formal bargaining, I would imagine that accepted business practices, social norms, and even public opinion, all have an influence on the division of rents within a firm and thus in the aggregate. It may not be mere coincidence that the share of rents accruing to the capital side began to rise about when Ronald Reagan was elected president.

Imagination is one thing; measuring what has happened will be very difficult. I would like to see Rognlie stay with this aspect of the problem. It has both analytical and policy implications. For instance, when it comes to estimating the elasticity of substitution, the presence of a significant amount of rent means that reported input prices (and relative shares) are a bad basis for inference. Unless factor prices can be purified of the rent element, the best (or only) bet would seem to be estimating production functions directly from data on inputs and output.

FINAL THOUGHTS This brings me to a final comment. Rognlie makes a valuable contribution by organizing a multisector model as a vehicle for some inferences about what matters most for movements in relative shares. There he simply assigns values of the elasticity of substitution to different sectors in accordance with the literature. That is a useful step. I want to suggest that a further extension in the direction of general equilibrium might even change the picture.

The fundamental question of interest is this: How far would the rate of return have to fall for the economy to absorb a likely increase in capital intensity? One way the economy does that is by substituting capital for labor in the production of final output. That is why that elusive elasticity of substitution enters the story. But there is another route by which the economy can absorb capital. When the return on capital falls, capital-intensive goods should become cheaper relative to labor-intensive goods. (Housing is one example, of course.) If these cost changes are passed into prices, consumers may shift toward more capital-intensive goods. The same process may affect producers' choices among alternative intermediate inputs.

The economy can become more capital-intensive even apart from shifts within production processes. I have no idea about the likely quantitative importance of this kind of adjustment, but there is no theoretical reason why it should be negligible.


Bils, Mark. 1988. "The Cyclical Behavior of Marginal Cost and Price." American Economic Review, December.

Hall, Robert E. 1988. "The Relation between Price and Marginal Cost in U.S. Industry." Journal of Political Economy 96 (October), no. 5: 921-47.

Rotemberg, Julio J., and Michael Woodford. 1999. "The Cyclical Behavior of Prices and Costs." In Handbook of Macroeconomics, vol. 1, edited by J. B. Taylor and M. Woodford. Amsterdam: North Holland.

GENERAL DISCUSSION Robert Hall opened the discussion by observing that much of the literature, including Thomas Piketty's work, treats capital as a primary factor, whereas in his view capital is an intermediate factor. Following an Arrow-Debreu view of intertemporal economics, he said, people who own capital can be understood as having chosen to defer consumption. Agreeing with a point discussant Robert Solow had made in his comment, he said the purchase of land is an exception and must be considered a primary factor. Hall believes there is still a need to understand what has happened in the stock market. It is a mystery what is being capitalized there, although it is certainly not realized future cash flows, and in his view finance economists have not made progress in clarifying this. In 1980, the valuation of the stock market was about half of any reasonable measure of intrinsic value, yet he thinks that today it is closer to 1.5 times such a value.

The work he had done in 1986 to measure the residual elasticity of demand, mentioned by Solow, has had to be heavily revised since then, and he now considers that residual elasticity to be 6, resulting in a markup ratio of 1.2. But Hall also thought almost all of that markup was absorbed, as in a Chamberlinian equilibrium, by the fixed cost of entry. Measured profit might therefore not include any monopoly rent, but instead be roughly what one would expect in a competitive setting. Investors battle to get into these markets, they pay the fixed cost, they get a markup of 1.2, and it all comes out even. In sum, Hall believes, one should not treat capital as a primary factor. None of this should matter for the basic issue of the rapidly growing inequality in the power to consume, Hall added, because most of that increase comes from earnings. As Solow pointed out, that may be partly due to monopoly rents. What one ought to focus on is the fact that an extraordinarily high level of actual earnings is paid out as cash, and this accounts for most of the rise in inequality.

Steven Braun added to Hall's comment on the role of income in inequality. A 2011 CBO study of the increase in income inequality in the United States from 1979 to 2007 concluded that 79 percent of the rise in inequality is accounted for by the rise in inequality within income categories, and that only 21 percent is accounted for by the redistribution between income sources (such as labor and capital income). This would mean, he noted, that most of what this panel has been arguing about is limited to that 21 percent.

Ilyana Kuziemko thought that applying Piketty's thinking to U.S. data might be inappropriate, since Piketty makes it clear that in the United States the inequality issue is more about labor than capital income. Piketty's thinking applies much better to Europe, and it may be unfair to critique his work using U.S. examples. It is not surprising to find some tension when trying to explain the U.S. pattern with his analysis.

John Haltiwanger said he was nervous about relying heavily on delta K in any of the standard measures. Concerning the national product accounts, he found it embarrassing how little was actually known. While labor compensation is understood and measured relatively well, as is nominal GDP, which is measured by adding up the nominal value of final goods and ser vices, much of the rest is not understood well. He emphasized that in the United States capital is not measured from the production side but from the supply side--adding the production of the capital goods industries, subtracting exports, and then adding imports. Estimating nominal investment flows therefore depends on modeling capital stock through inventories. Haltiwanger also agreed with Solow that analysts' ability to model depreciation rates has not made much progress. Depreciation is an area that he and many of his colleagues are worried about.

Haltiwanger was also struck by the possibility that something significant has changed in the United States over the last 30 years, and wondered if it has to do with barriers to business entry. His research into the substantial decline in entrepreneurship over the last 30 years, during which the concentration of wealth in large national and multinational firms has sharply increased, leads him to think that this may be part of the story.

Joe Beaulieu disagreed with Haltiwanger's view that the official measurements of GDP were nearly futile. He considered it remarkable that in the United States the national product is measured two different ways, from very different data sets, which nevertheless produce remarkably similar results. The statistical debt discrepancy leaves some uncertainty, but both accounts capture the same story and match what one sees in the macro-economy. He also disagreed with Haltiwanger concerning depreciation: the Bureau of Economic Analysis does not assume depreciation but simply follows an exponential decay function, albeit one that is estimated from outdated studies. The Bureau of Labor Statistics and those who produce its multifactor productivity data set have a more complex view on appreciation and decay, and one could use their studies--with the same assumptions and models--and just calculate the dual for the income side. It is a complicated issue but, he believes, ultimately it can be solved as a programming problem.

Finally, Beaulieu endorsed the idea that one must look at the net product. Like Hall, he found it interesting that in the history of the national accounts new things are continually being capitalized, most recently intellectual property research and development, which was once treated as an intermediate good. This suggested to him that economists should have been looking at net product all along. He acknowledged that he did not know whether the various national statistical agencies have a consistent accounting standard for that, although the OECD tries to make net product measures consistent, notwithstanding time delays that can cause them to be off.

Martin Baily mentioned that he has been studying trends in corporate profits from an international perspective and has found that although examining production functions in the United States requires looking at factors within the country, to understand income distribution one must also consider foreign profits and earnings. He found it striking that over the last 10 or 15 years corporate profits have risen sharply in the United States and in Europe but actually declined in China and other emerging markets. Asian companies have invested heavily to serve rapidly expanding domestic markets and exports, but at the cost of profitability. In the United States, he added, rising profits in the nonfinancial sector are concentrated in three sectors: technology, pharmaceuticals, and oil and gas. Profitability in the financial sector also rose very strongly until the crisis. Whether those profits are returns to entrepreneurship or stem from rents, they are concentrated and not spread equally across the whole economy.

David Romer said he would like to know the authors' thoughts about Solow's final comment, which raised something Romer had not thought about before: whether substitution in final demand could possibly be large enough to rescue things by giving us a large elasticity of substitution between capital and labor.

Matthew Rognlie responded, first by addressing what he considered Hall's well justified complaint that capital is really an intermediate factor. When one talks about the capital share, he said, one is dealing with a mongrel creature, one that consists of elements from replaceable capital, which is intermediate, and elements from the buying and selling of land, and possibly other rents, which are not intermediate. On the whole he agreed with Hall, noting that the convention has been to look at capital shares as a consumption decision--investing in capital for tomorrow rather than today--like any other Arrow-Debreu consumption decision. Even if it is just one of many consumption decisions, it is an intertemporal one and has special significance because of its correlation with so many other issues. He also agreed with Hall's point about the Chamberlinian equilibrium. He treated the average rate of pure profits over time as zero to reflect such an equilibrium, since in the long run free entry causes pure profits to be zero, though in the shorter run there are increases or decreases.

Concerning inequality, he said, in the United States most of it comes from labor rather than capital. He agreed that Piketty's focus on capital made his work much more relevant to Europe, although new evidence--such as Gabriel Zucman's job market studies--suggests that a larger share than people realize is coming from capital income. It is not a completely settled issue. He agreed with Haltiwanger that the measurement of depreciation rates may be sketchy, but noted that rising depreciation seems to be a consistent and long-term trend across many countries, including the United States.

Regarding Solow and Romer's idea that substitution in final demand might be another source of substitution, Rognlie said he has looked into this. He mentioned a recent paper by Ezra Oberfield and Devesh Raval, who showed how elasticities of substitution at a lower level, such as within a product between capital and labor, aggregate with elasticities of substitution between products, such as in final demand, to yield a total elasticity of substitution. It turns out that adding yet adding another layer of substitution does not necessarily raise the elasticity. Instead, the aggregate elasticity is an average of all the elasticities. This was a little surprising, because the intuition is that having another margin of substitution would expand the ability to substitute. Instead, there is an offsetting aggregation bias. When products with dispersed labor capital shares are at the lowest level they are less able to have a big shift, since they are starting from an uneven position and have more uneven distributions holding the elasticity constant. That aggregation bias cancels out the top-level ability to substitute.

At the end of the day, Rognlie said, what one finds is that aggregate elasticity of substitution is a weighted average of the different elasticities. He noted that while his study into this yielded a clean conclusion, Oberfield and Raval had already done the same work and reached the same finding. Nevertheless, in his view more focus is still needed on this higher level of elasticity of substitution.

Table 1. Decadal Averages for the Net Capital Share of Private
Domestic Value-Added, G7 Countries, 1950s to 2000s (Percent)

                           1950s    1960s    1970s    1980s

United States    Housing     5.3      6.5      5.7      7.2
                 Other      22.0     21.7     18.6     18.4
                 Total      27.3     28.2     24.2     25.6
Japan            Housing              4.2      3.6      4.1
                 Other               31.2     26.9     25.7
                 Total               35.4     30.5     29.8
Germany          Housing
France           Housing     3.6      5.1      5.9      7.1
                 Other      21.3     19.8     17.9     16.6
                 Total      24.9     24.9     23.8     23.7
United Kingdom   Housing     1.2      2.1      3.8      4.6
                 Other      27.2     23.9     18.3     21.6
                 Total      28.4     26.0     22.1     26.2
Italy            Housing
Canada           Housing              6.6      6.6      8.1
                 Other               22.5     24.0     25.8
                 Total               29.1     30.6     33.8

                           1990s    2000s

United States    Housing     8.4      8.2
                 Other      19.2     19.4
                 Total      27.5     27.6
Japan            Housing     5.2      7.0
                 Other      21.6     20.1
                 Total      26.9     27.1
Germany          Housing     2.9      3.4
                 Other      23.5     28.0
                 Total      26.4     31.4
France           Housing     9.8     10.8
                 Other      19.9     18.0
                 Total      29.7     28.8
United Kingdom   Housing     5.8      7.3
                 Other      23.2     23.4
                 Total      29.0     30.7
Italy            Housing     4.3      6.4
                 Other      33.9     32.5
                 Total      38.2     38.9
Canada           Housing    10.4      8.6
                 Other      21.2     27.2
                 Total      31.6     35.8

Source: National accounts; Piketty and Zucman (2014).

Table 2. Decadal Averages for the Net Capital Share of Value-Added in
the Domestic Corporate Sector, G7 Countries, 1950s to 2000s (Percent)

                 1950s    1960s    1970s    1980s    1990s    2000s

United States     23.2     23.2     19.7     19.8     20.9     21.1
Germany                                               24.2     29.0
France            22.1     20.9     19.0     17.9     22.1     20.1
United Kingdom    27.6     24.4     19.0     22.7     24.7     25.3
Italy                                                 35.4     34.6
Canada                     24.5     26.1     28.5     24.3     30.1

Source: National accounts; Piketty and Zucman (2014).

Table 3. Distribution of Chirinko Elasticity Estimates in Gross and
Net Terms *

                           [0,0.5)   [0.5,1)   [1,1.5)

Frequency of gross sigma     14        12         3
Frequency of net sigma       21         8         1

                           [1-5,2)    [2,4)

Frequency of gross sigma      1         1
Frequency of net sigma        0         1

(a.) Based on estimates compiled by Chirinko (2008); gross terms as
originally stated, and net terms converted using equation 11.

Table 4. Cross and Net Shares of Factors and Higher-Level Aggregates
Used to Calibrate the Multisector Model (a) (Percent)

                                                   Gross        Net
                                                 aggregate   aggregate
                                                   share       share

Labor (N)                                           60          68
Equipment ([K.sub.e])                               12           7
Nonresidential structures ([K.sub.s1])              12          11
Nonresidential land ([L.sub.1],)                     3           3
Residential structures ([K.sub.s2])                 10           8
Residential land ([L.sub.2])                         1           1
Pure capital profits ([pi])                          1           2
Nonhousing production, other (H)                    72          76
Nonhousing production, real estate ([G.sub.1])      15          14
Housing production, real estate ([G.sub.2])         11           9
Nonhousing production, total (F)                    88          90

Source: Author's calculations; see text for details.

(a.) Factors and aggregates taken from the 2013 decomposition
presented in section II.C of this paper.

Table 5. Gross Shares of Production Within Each Higher-Level Aggregate
Used to Calibrate the Multisector Model (a) (Percent)


Nonhousing production,      Labor (N)                              83
  other (H)                 Equipment ([K.sub.e])                  17
Nonhousing production,      Nonresidential structures              82
  real estate ([G.sub.1])     ([K.sub.s1])
                            Nonresidential land ([L.sub.1])        18
Housing production, real    Residential structures ([K.sub.s2])    90
  estate ([G.sub.2])        Residential land ([L.sub.2])           10
Nonhousing production,      Nonhousing production, other (H)       83
  total (F)                 Nonhousing production, real estate     17
Total production (Z)        Nonhousing production, total (F)       89
                            Housing production, real estate        11

Source: Author's calculations; see text for details,

(a.) Based on shares presented in table 4.

Table 6. Minimum and Maximum Elasticities of Net Capital Share
with Respect to Shocks (a)

Shock                                        Min     Max   Benchmark

Real interest rate (r)                       0.04   0.54        0.26
Price of equipment investment ([P.sub.e])   -0.18   0.15        0.00
Price of residential structures investment   0.00   0.16        0.07
Quantity of residential land ([L.sub.2])    -0.04   0.00       -0.01

Source: Author's calculations; see text for details.

(a.) Elasticities for choices of G, within range.

Table 7. Sensitivity of the Elasticity of Net Capital Share to Changes
in Each of, Starting at Benchmark Values


Shock                       sigma_z   sigma_F   sigma_Gl

Real interest rate (r)      -0.21     -0.21     0.03
Price of equipment           0.02      0.01     0.00
  investment ([P.sub.e])    -0.17     -0.03     0.00
Price of residential
  investment ([P.sub.s2])    0.03      0.00     0.00
Quantity of residential
  land ([L.sub.2])


Shock                       sigma_G2   sigma_H

Real interest rate (r)      -0.01      -0.19
Price of equipment           0.01      -0.29
  investment ([P.sub.e])     0.01      -0.04
Price of residential
  investment ([P.sub.s2])    0.01       0.00
Quantity of residential
  land ([L.sub.2])

Source: Author's calculations; see text for details.

Table 8. Contribution to the Aggregate Elasticity of the Net
Capital Share for Benchmark [[sigma].sub.i]


Shock                            Equipment    Nonresidential
                                ([K.sub.e])     structures

Real interest rate (r)              0.04           0.03
Price of equipment investment       0.00           0.00
Price of equipment investment      -0.12           0.00
  ([P.sub.e]) (high
  elasticity case)
Price of residential               -0.01          -0.01
  structures investment
Quantity of residential land        0.00           0.00


Shock                            Nonresidential    Residential
                                land ([L.sub.1])    structures

Real interest rate (r)               -0.04             0.23
Price of equipment investment         0.00             0.00
Price of equipment investment         0.00            -0.01
  ([P.sub.e]) (high
  elasticity case)
Price of residential                 -0.01             0.09
  structures investment
Quantity of residential land          0.00             0.00


Shock                             Quantity of      Profit
                                land ([L.sub.2])

Real interest rate (r)                0.00         -0.01
Price of equipment investment         0.00          0.00
Price of equipment investment         0.00          0.00
  ([P.sub.e]) (high
  elasticity case)
Price of residential                  0.01          0.00
  structures investment
Quantity of residential land         -0.01          0.00
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Title Annotation:p. 37-69
Author:Rognlie, Matthew
Publication:Brookings Papers on Economic Activity
Article Type:Report
Geographic Code:1USA
Date:Mar 22, 2015
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