2. Resource allocation and productivity

Economic theory suggests that more productive companies should have a greater incentive to, and be more able to, attract inputs, be it capital or labour, relative to companies that are less efficient. Over time, less productive companies are forced to become more efficient or go out of business. This process brings about capital and labour reallocation, which will show up in measured aggregate Total Factor Productivity (TFP), and which will foster productivity growth across the economy as a whole.

There is a substantial body of literature demonstrating the importance of resource allocation in driving productivity. Reference Disney, Haskel and HedenDisney et al. (2003) find that during the 1980s and 1990s recessions external restructuring (measured as the exit, entry or changing market share of firms) could explain around 50 per cent of UK labour productivity growth within the manufacturing sector. More recently, Reference Barnett, Barriel, Chiu and FranklinBarnett et al. (2014) find that labour reallocation across firms explained 48 per cent of labour productivity growth for most UK sectors^{Footnote 3} in the 5 years prior to 2007. They also find, however, that since 2007, the contribution of reallocation to aggregate productive growth fell to nearly zero, indicating that the process of efficient resource allocation may have been impaired. Their work focuses on labour reallocation across firms and sectors. This paper builds on that work but focusses specifically on the capital reallocation channel. It contributes to the growing body of literature linking resource misallocation to aggregate productivity (Reference Bartelsman, Haltiwanger and ScarpettaBartelsman et al., 2013; Reference Restuccia and RogersonRestuccia and Rogerson, 2013). And it also relates to the body of work, dating back to Schumpeter, that considers the role of resource restructuring in crises (Reference CaballeroCaballero, 2007).^{Footnote 4}

Misallocation of resources can arise if there are impediments to the movement of factors between heterogeneous firms and sectors (Reference BroadbentBroadbent, 2012). This can give rise to persistent rates of return differentials across firms, and reduce aggregate TFP and labour productivity growth.^{Footnote 5} Impediments to the efficient allocation of capital can take various forms. In the context of the recent financial crisis, it is plausible that these impediments have intensified – notably in the form of financial market frictions and weak and uncertain demand conditions. At the same time the shocks associated with the crisis may have increased the necessity for some reallocation of resources. Since only a part of capital reallocation occurs as a result of relatively slow depreciation of less productive capital assets, any restraint on investment by more productive firms can have a significant impact in slowing the process of reallocation.

The 2008 financial crisis was associated with weak demand conditions and a large increase in uncertainty, both of which can distort the market signals that drive incentives to reallocate and depress investment.^{Footnote 6} Weak demand can mean that current resources are underutilised, leading firms (including those with high productivity) to delay expansion plans. Uncertainty can lead firms to delay investment decisions because capital choices are (at least partially) irreversible (Reference Dixit and PindyckDixit and Pindyck, 1994; Reference Caballero and PindyckCaballero and Pindyck, 1996). Indeed, it is the costs associated with adjusting capital that help explain the ‘lumpy’ nature of firm investment: firms’ capital stocks do not tend to evolve in a continuous and linear way but include discrete steps (Reference Doms and DunneDoms and Dunne, 1998; Reference NickellNickell, 1978). Delaying investment decisions until more information is available creates an option value that can be shown to be increasing in the level of uncertainty (Abel et al., 1996; Reference Abel and EberlyAbel and Eberly, 1996). Reference Bloom, Bond and Van ReenenBloom et al. (2007) show that higher uncertainty reduces firms’ responsiveness to demand shocks.

Financial market frictions, in particular, are commonly cited as affecting the allocation of capital.^{Footnote 7} They do this by increasing the cost of capital or producing capital constraints for some firms in ways that are unrelated to fundamental characteristics and thereby distorting investment decisions. Reference Gilchrist, Sim and ZakrajsekGilchrist et al. (2013) relate financial frictions, measured as an increased dispersion in borrowing costs, to capital misallocation for a subset of US manufacturing firms over the period 1985–2010.^{Footnote 8} The 2008 financial crisis was associated with a sharp fall in liquidity and an increase in borrowing costs (both in the banking sector and on capital markets). Lenders may also have become more risk averse, such that some firms (e.g small firms) or some projects (e.g. those involving assets that cannot be collateralised) found it harder to finance new investments. In addition, accommodative monetary policy, forbearance by banks and tax forbearance by HMRC may have contributed to preventing unproductive firms from exiting the market, thereby slowing one part of the reallocative process (Reference Arrowsmith, Franklin, Gregory, Griffiths, Wohlmann and YoungArrowsmith et al., 2013).

Reference Hsieh and KlenowHshieh and Klenow (2009) develop a framework with many industries, heterogeneous firms and monopolistic competition, to show that firm-level distortions to capital choices lead firms to make inefficient input choices. This in turn prevents the rates of return to capital from equalising across firms and leads to lower aggregate TFP.^{Footnote 9} In such a framework, in response to relative demand shocks for example, distortions will prevent capital from fully adjusting across firms, which will lead to persistent dispersion in prices and rates of return across firms.

The framework underlying our analysis draws on the spirit of Reference Hsieh and KlenowHsieh and Klenow (2009). We consider the financial crisis as having generated incentives for capital and labour to re-allocate because there were large relative shocks – including the large exchange rate depreciation in 2007 and commodity price shocks. At the same time, uncertainty and financial market frictions potentially created higher barriers to capital reallocation. These uneven shocks may have reduced prices for some products and increased them for others, resulting in changes in relative profitability. In the longer run we would expect resources to move towards the more profitable firms. If, however, there are distortions to capital reallocation, we would expect to see this manifested in a persistently higher dispersion of prices, and indeed marginal products of capital, across firms and sectors.

We consider evidence relating to dispersion in prices in section 3 and across rates of return in section 4. In both cases we attempt to measure the scale of the loss of TFP resulting from allocation frictions.

3. Price dispersion and allocation

Figures 1 and 2 provide some evidence that UK firms have faced large relative shocks since the 2008 crisis. Figure 1 plots the mean (black) and two standard deviations around the mean (the red and dashed red lines) of the distribution of deviations in output from long-run industry trends across 17 industrial sectors. Figure 2 plots the comparable dispersion in output prices.^{Footnote 10} In both cases dispersion increased markedly since the crisis suggesting that resources may have faced incentives to move. If that was the case we would expect a move in resources towards firms and sectors where returns were highest, which in time may act to reduce the dispersion in price. The persistence of the dispersion in prices suggests that relatively little reallocation of resources has taken place to date.

Figure 1. Dispersion in gross value added across sectors

Figure 2. Dispersion in output price deflators across sectors

Reference Hsieh and KlenowHsieh and Klenow (2009) show that frictions to labour and capital choices lead to persistent dispersions in prices and output. In turn, under some assumptions, these dispersions can be mapped into an estimate of the aggregate loss to TFP. However, this approach relies on a particular functional form for firms’ production functions, and a specific way of aggregating over firm and sector level production functions.

In what follows, we employ a model used by Reference BroadbentBroadbent (2013) in a policy speech. This model uses a general static perfect competition framework that does not require a specific production function to map dispersion of sectoral prices following a demand shock into labour productivity changes. And although this relies on a perfect competition assumption, the implications for productivity are similar if one relaxes this assumption.

Specifically, all firms, indexed i, use inputs K _i and L _i to produce y _i = f _i(K _i, L _i). Firms operate in a perfectly competitive market, where the price of a firm's output, denoted p _i, is equal to marginal cost.^{Footnote 11} The question that we want to ask is what the loss in output and productivity is when firms face uneven demand shocks and labour can freely adjust, but capital cannot. More specifically, suppose we increase a firm's employment by ΔL _i, holding fixed K _i. Then the (base-weighted) value of its output will change by

(1)$p_i^0\Delta y_i=p_i^0{\int }_{L_i^0}^{L_i^0+\Delta L_i}{f}_{iL}^{\prime }\left(K,L\right)dL$

where a zero superscript indicates the starting value and $f_{iL}^{\prime }$ the marginal product of labour. We want to think about what happens to the change in aggregate output $\Delta Y={\Sigma }_i{p}_i^0\Delta y_i$ when shifts in relative demand are met by changes in labour alone. In doing so we assume there is a fixed supply of labour in aggregate (call it L) and that the labour market clears.

The intuition behind this model is relatively straightforward. A sector with a positive demand shock will grow by increasing its labour input. The marginal cost of production will increase. Since capital is fixed, the marginal cost, and therefore the price, will be higher than if capital could fully adjust. Output will be lower and so will aggregate labour productivity.

More generally, the resulting percentage change in output is proportional to the cross-sectoral covariance between inflation and size-weighted employment growth (see appendix for details on how to derive equation 2) as in the expression below:

(2)$\begin{array}{cc}\frac{\Delta Y}{Y}& \approx \frac{w^{0}L}{Y}{\Sigma }_i\left[1+\frac{1}{2}\left(\frac{\Delta w}{w^0}-\frac{\Delta p_i}{p_i^0}\right)\right]\frac{\Delta L_i}{L}\\ & =-\frac{1}{2}\alpha {\mathrm{cov}}_i\left(\frac{\Delta p_i}{p_i^0},{\lambda }_i\frac{\Delta L_i}{L_i^0}\right)\end{array}$

where α is the share of wages in national income (wL/Y) and λ_i is employment in sector i relative to the average, namely $\frac{L_i}{L/N}$.

The share of wages in GDP is roughly two-thirds. So this relationship says the loss in productivity is (to a first-order approximation) one third the cross-sectoral covariance between inflation and size-weighted employment growth. Further approximating the relationship between price and employment growth from (2), and using Σ as the elasticity of substitution between capital and labour in sector i, and α_i as the share of labour income in that sector, one can re-express this in terms of prices alone:

(3)$\begin{array}{cc}\frac{\Delta Y}{Y}& \approx \frac{1}{2}\alpha {\mathrm{cov}}_i\left(\frac{{\lambda }_i{\sigma }_i}{1-{\alpha }_i}\frac{\Delta p_i}{p_i^0},\frac{\Delta p_i}{p_i^0}\right)\\ & -\frac{1}{2}\alpha \mu {\mathrm{var}}_i\left(\sqrt{\frac{{\mu }_i}{\mu }}\frac{\Delta p_i}{p_i^0}\right)\end{array}$

where ${\mu }_i=\frac{{\lambda }_i{\sigma }_i}{1-{\alpha }_i}$ and $\mu =\frac{\sigma }{1-\alpha }$ is the same quantity for the economy as a whole. Intuitively, equation (3) shows that relative demand shocks won't affect aggregate productivity if prices remain unchanged. Prices remain unchanged as long as productive resources, and in our case capital, move seamlessly in response to demand shocks. When there are frictions to the movement of capital across firms or sectors, relative marginal costs and therefore relative prices will change. In reality, there will be some frictions to capital mobility and relative prices will fluctuate in the short term as the economy reacts to shocks. However, large and persistent increases in price dispersion would be indicative of capital misallocation and more binding constraints on capital movement.

Figure 3 plots the cross-sectoral variance derived in equation (3). The scale of the variance since 2008 suggests that slow reallocation might have knocked 3–4 per cent off aggregate labour productivity compared with the pre-crisis period.^{Footnote 12}

Figure 3. Significant increase in price dispersion in the UK(a)

Aggregate labour productivity can be thought of as a combination of capital per worker and Total Factor Productivity (TFP), the latter embodying a range of factors such as technology. There is a large degree of uncertainty around aggregate estimates of the UK capital stock. However, latest estimates suggest that the change in the capital to labour ratio since the crisis can only account for a small part of the shortfall in productivity relative to its pre-crisis trend. Therefore, it is likely that much of the fall in measured labour productivity is accounted for by a fall in TFP.^{Footnote 13} As such, we make the inference that the loss in labour productivity identified in this model will largely reflect a loss in measured aggregate TFP due to the misallocation of capital across sectors.

These results are very general. All they require is that the labour demand curves are well defined,^{Footnote 14} but they do not need restrictions on production functions: these don't need to be the same in all sectors, or to exhibit constant returns to scale (CRS).^{Footnote 15} In the extreme case that there are constant returns, and if capital too is fully mobile, then relative prices don't change in response to demand shocks (relative supply curves are flat) and productivity is invariant to shifts in relative demand.^{Footnote 16}

We have presented a model in which competitive markets mean that prices are equal to marginal costs. However, the broad point that higher price dispersion is indicative of capital misallocation requires only that a demand shock creates or increases differences in marginal costs across firms that feed through into prices if capital does not move freely to where returns are highest.

4. Firm level evidence of lack of capital reallocation

In a neoclassical model of firm factor demand the optimal capital stock is chosen to maximise the value of the firm. In a static setting this requires that the marginal product of capital be equated with the user cost of capital in each period. However, firms face adjustment costs when changing the capital stock, such that they may not respond instantly to a demand shock. The presence of adjustment costs also means that firms face a dynamic decision in which investment today is based in part on expected future demand conditions. Commonly, investment in any given year is thought of as advancing the firm towards an optimal capital choice. For a discussion of models of firm investment and empirical counterparts see Reference Bloom, Bond and Van ReenenBloom et al. (2007).

The recession was characterised by a series of large shocks. For example, as highlighted in section 3, we expect firms that experience a positive (negative) demand shock to see an increase (fall) in the marginal product of capital while firms respond to this shock. Initially, this will be seen as an increase in the dispersion in the marginal product of capital across firms. But as capital moves towards firms that produce a higher return on capital, dispersion in returns should decrease. Any frictions to the process of allocation (including those that arise in normal times, and any additional frictions caused by the financial crisis) will impair this allocation of resources, such that the dispersion in rates of return across firms will be persistent.

In this section we are interested in estimating an empirical relationship between rates of return to capital and subsequent investment. We test whether the expected positive relationship has changed since the 2008 financial crisis, more specifically, whether firms have become less responsive to investment incentives.

Although we cannot directly measure the marginal product of capital or observe the demand shocks that firms face, we can calculate the average rate of return to capital at the firm level each year in a rich firm-level dataset. The following three subsections describe the firm level data, our empirical approach and then the results.

4.1 Data description

We use a data set of around 8,000 UK firms per year over the period 2000–2011. This sample uses data from the ONS Annual Business Survey (ABS) and its predecessor.^{Footnote 17} The ABS is an annual survey of around 60,000 businesses from most sectors of the UK economy^{Footnote 18} and covers around two-thirds of the economy in terms of Gross Value Added (GVA). As well as GVA, the data include information on employment, wages and capital expenditure. For each firm, real GVA is calculated using deflators at the 2 digit Standard Industry Classification (SIC) level.

Following Reference GilhoolyGilhooly (2009),^{Footnote 19} for each reporting unit, a perpetual inventory method is used to create a measure of three different capital assets namely plant and machinery, buildings and vehicles. We use annual depreciation rates for each asset, as set out in ONS (2007), of 6 per cent, 2 per cent and 20 per cent respectively to calculate an overall net measure of firm-level capital. Rates of return are calculated as Gross Operating Surplus – that part of Gross Value Added that is not used to remunerate labour but retained by firms as a form of profit – divided by the net capital stock. For further details about the dataset please see Reference Barnett, Barriel, Chiu and FranklinBarnett et al. (2014).

Our sample size is significantly smaller than the ABS full sample mainly because there are many firms for which we do not observe capital expenditure data. In addition, since we use lags in our empirical specification (described next), the number of firms for which data is available in consecutive years is even smaller. Table 1 summarises the key variables.

Figure 4 plots the standard deviation (second column of table 1) for the rates of return and the change in the capital stock over time. This shows an increase in the variation in rates of return across firms since 2007 (red line) and a flat variance of changes in capital stock (black line).

Figure 4. Standard deviation of firm rates of returns has increased since the crisis

4.2 Empirical specification

As described in the previous section, our aim is to test the hypothesis that firms have become less responsive to investment incentives since the crisis. To do this, we estimate the reduced form model below:

(4)$\begin{array}{cc}\Delta K_{ijt}& =\mu +{\Sigma }_s\left({\alpha }_s+{\beta }_{s}crisis\right)RO{R}_{ijt-s}\\ & +{\Sigma }_1\left({\alpha }_1+{\beta }_{1}crisis\right)Deman{d}_{kt+1}\\ & +{\beta }_{5}crisis+{\gamma }_i+{\gamma }_{j}9+{\epsilon }_{ijt}\end{array}$

where ΔK_ijt is the change in the capital stock measured as investment (capital expenditure) divided by the previous years’ capital stock for firm i, in industry j, in period t. ROR_ijt–s is the rate of return on capital of firm i, in industry j, in period t–s. We include the first and second lag in recognition that there may be a delay to firms’ capital adjustment. This is captured by the s subscript. Crisis is a dummy variable that takes the value of 0 up to and including 2007 and 1 after.

One possible concern is that the rate of return does not adequately capture investment incentives. To address this we present results of a specification that also includes an independent measure of output demand,^{Footnote 20} as one would expect firms that face higher demand (now and in the future) to face a larger incentive to increase investment. The variable Demand is constructed following Reference Bailey, Bartelsman and HaltiwangerBailey et al. (2001), Reference Bartelsman, Caballero and LyonsBartelsman et al. (1994) and Reference SheaShea (1993), and is a downstream demand indicator specific to the two-digit industry, k, that each firm belongs to. This sectoral indicator is a weighted average of changes in real GVA of the downstream industries with weights equal to their share of purchases of output from that specific upstream industry. Furthermore, we also apply a Sheaexogeneity test in order to rule out the endogenous effect that the upstream industry's demand may have, via intermediate goods prices, on the downstream industry's activity.^{Footnote 21} More specifically, we calculate this using the input-output matrices and two-digit real output indices from the ONS and exclude from our indicator those downstream industries whose purchases of the upstream industry are larger than 5 per cent of their expenditure on intermediate inputs.

Our specification also uses γ_j as industry fixed effects, defined at the two-digit industry level. These should capture systematic differences in investment rates across firms in different industries. The firm fixed effects, γ_i, should capture systematic firm differences that affect investment choices, such as differences in production technology and time invariant differences in the cost of investment. ε_ijt is an i.i.d error, assumed to be normally distributed.

This empirical specification should be seen as a reduced form model, in which the change in the capital stock is expressed as a function of factors that affect the choice of the optimal capital stock, as these factors may signal an incentive to increase investment. We have not specified the precise relationship between investment and the optimal capital stock (which will be a function of the expected costs and benefits of capital investment) or the precise functional form of any adjustment process. The coefficients on ROR and Demand will be some function of the structural parameters relating firms’ expectation generating process to the adjustment technology. As such, this approach will not allow us to identify the underlying structural parameters of the firms’ problem. However, that is not our goal here. Instead, our interest is in considering whether the relationship between the signal to invest and investment has changed since the recession, controlling for other factors that affect firms’ decisions. The idea is to test whether or not there appear to be barriers or constraints to the efficient movement of capital across firms.

We interpret the α coefficients as the average or ‘normal’ process of adjustment of capital to investment incentives. We interpret the β coefficient as representing distortions to the adjustment of firms’ capital stocks since 2008. This may encompass various sources of adjustment friction that arose as the result of (or at least at the same time as) the financial crisis. For example, it may capture the effect of higher uncertainty, which has been shown to make firms less responsive to changes in demand (Reference Bloom, Bond and Van ReenenBloom et al., 2007), or the effects of credit constraints.^{Footnote 22}

The validity of this approach requires two sets of assumptions. First, we assume that firms have not changed the way they calculate their optimal capital stock and therefore that rates of return and demand conditions still provide as much information with regard to firms’ incentives to invest as they did before the crisis. In fact we find that the correlation between investment incentives today and in the future (i.e. the correlation between ROR_t and ROR_t+1) did not change after the crisis.

Second, we assume that ROR and Demand can be treated as exogenous after controlling for various factors. One further possible concern is that we do not directly include a measure of the user cost of capital (which is a function of price of capital goods relative to output), how this is expected to change, the firm's required rate of return or the rate of depreciation of capital (see Jorgensen, 1963).

We do not observe firm-specific interest rates or investment prices and therefore cannot include these in our specification. The industry fixed effects will capture systematic (time-invariant) differences across firms in different industries. In addition, the firm fixed effects will capture systematic differences in the cost of investment faced by firms. While there may be firm-year specific shocks to the user cost of capital, we argue that these are unlikely to be correlated with rates of return.

4.4 The effect on productivity

We use the estimated pre-crisis relationship between firm-level capital growth and rates of return (table 2) to construct a back-of-the-envelope counterfactual measure of firms’ capital stock. The idea here is to estimate what aggregate labour productivity would have been if capital had moved in response to changes in rates of return. In this counterfactual experiment, firms with higher relative rates of return would have invested more, increasing the size of their capital stock. Based on this estimate of the capital stock, and holding the level of each firm's Total Factor Productivity (TFP) and employment constant at 2008 levels, we estimate what the potential level of output could have been using a simple growth accounting identity shown below.

$Y_{ijt}^{\ast }=L_{ijt}^{\alpha }{K}_{ijt}^{\ast 1-\alpha }$

For this exercise, we assume that the labour share, α, is 2/3. ∗ indicates the estimated counterfactual measures. The difference between the observed and counterfactual capital levels gives us an indication of the degree of capital misallocation. And the difference between observed and counterfactual output levels gives us an indication of the degree of the associated output loss, which we can use to derive an aggregate TFP loss.

However, there are a number of limitations to this type of counterfactual exercise. First, it ignores any beneficial spillover effects to individual firm level TFP through new investment. Second, we do not capture firm formation and bankruptcies that might have taken place had capital been allocated differently, channels that we think are important for the capital allocation process. Third, the estimation strategy only considers within-firm effects since we use a fixed effects panel specification. There may be between-firm effects and within-sector dynamics that are important to capture. Fourth, any counterfactual exercise is highly endogenous, as future rates of return will be affected by changes in the capital stock in the current period. These effects are hard to capture accurately in this simple exercise. Last, companies may respond to incentives differently in recessions compared to normal times. And our data set does not include any previous recessions. Indeed, as one might expect given these caveats, this counterfactual experiment suggests that the aggregate TFP loss for this sample of firms is relatively small and a bit less than 1 percentage point. However, we think that this may be a distinct lower bound when aggregating to the rest of the UK economy, since we are unable to capture any of the effects described above. More importantly, given the uncertainties with this counterfactual exercise, the key finding to highlight is that the relationship between rates of return and subsequent investment growth appears to have materially weakened since the crisis. This is suggestive that this channel may be one contributing factor to the impaired allocation of capital across firms.

5.2 Next steps

There are important questions left to be explored in further work. In this paper we do not distinguish between the sources of impaired capital allocation. However, understanding the source of the frictions, including how persistent these are, is important for considering the appropriate policy response. We highlight uncertainty and credit market imperfections as two important candidates. And there is evidence that both have been defining features of the UK experience. Since 2008 there has been uncertainty over demand conditions, fiscal policy and conditions in the Eurozone. Furthermore, indicators of uncertainty were particularly high in 2008 and 2011 and have only recently returned to more normal levels.

These factors have clearly been persistent, but we think it unlikely that either uncertainty or credit market frictions represent permanent changes. As such, we would expect to see a process of capital reallocation contributing positively to TFP as the economy recovers.