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Performance measurement in an inputoutput framework
Journal of Economic Structures volume 1, Article number: 2 (2012)
Abstract
This paper fruitfully combines two complementary theories: performance measurement and inputoutput analysis. Our point of departure is the theory of the consumer, who maximizes utility subject to a budget constraint. His wellbeing can be measured by the change in the consumption bundle, valued at constant prices. Inputoutput analysis is invoked to impute the change in this bundle to technical change, a termsoftrade effect and two types of efficiency change. The analysis is extended to environmental economics.
JEL Classification:C67, O47, Q56.
1 Introduction
In this paper I bring to fruition the neoclassical measurement of performance change in an inputoutput framework. Some pitfalls are avoided. First, market prices need not be perfectly competitive and, more generally, the observed allocation of resources may be inefficient. Solow’s ascription of TFPgrowth to technical change holds in an idyllic neoclassical world where resources are rewarded according to their marginal productivities, but not beyond. Market power, managerial slacks and other sources of inefficiency abound and we must factor in their variations to properly account for performance changes. Since these conditions differ between industries, an inputoutput framework seems promising.
The strategy of this paper is the following. I find it reasonable to assume that households are price takers. This point of departure is quite powerful, because the first order condition of the problem of the consumer, which equates the marginal rates of substitution to the price ratios, reveal that prices measure marginal utilities (up to an uninteresting multiplicative constant, the marginal utility of money) even when those prices are distorted. So I measure changes in the level of wellbeing of consumers by tracking final consumption. Then I use inputoutput analysis to decompose that bill of final goods and services. It may increase due to reductions in input coefficients, Solow’s technical change, but obviously for other reasons as well. This paper sorts it out.
2 Utility based performance measurement
In general equilibrium analysis  of which inputoutput is an important instance  consumers have initial endowments, including their labor skills. Some consumers are better off than others, i.e. have bigger endowments, and this translates into greater budget sets, hence opportunities. If, however, we assume away income effects in their utility functions  as is the case for constant elasticity of substitution (including CobbDouglas and Leontief) utility functions, then the relationship between the percentage rate of change in utility and in consumption is independent of the level of well being.
Formally, a consumer maximizes $U(x)$ subject to $px\le p\omega $, where x is a consumption vector, ω the initial endowment, p the price (row) vector, and U the utility function. I assume U is linearly homogeneous (no income effects). Exogenous to the consumer are p and ω (price taking behavior). How does the consumer respond to change? Well, that’s determined by the envelope theorem, according to which the change in the objective value is given by the partial derivatives of the objective function with respect to the exogenous parameters. (The point of the theorem is that the change in the endogenous parameter, consumption, may be ignored.) Since the envelope theorem is formulated for free maximization problems, we must first handle the constraint. This is done in the usual way by setting up the Lagrangian function, $U(x)+\lambda (p\omega px)$, and maximizing that.
Change the parameters: dp and dω. By the envelope theorem dU is determined by the partial derivatives with respect p and ω. The first effect is $\lambda (\omega x)\phantom{\rule{0.2em}{0ex}}dp$; the consumer is better of his resources are priced higher and worse off if his consumption bundle is priced higher. The second effect is $\lambda p\phantom{\rule{0.2em}{0ex}}d\omega $; the consumer is better off if he is better endowed. In total $dU=\lambda \phantom{\rule{0.2em}{0ex}}dp(\omega x)+\lambda p\phantom{\rule{0.2em}{0ex}}d\omega $. This can be simplified. (For simplicity I assume, quite realistically, that the budget constraint is binding, but otherwise the simplification still holds by the phenomenon of complimentary slackness, $\lambda =0$.) Differentiating the budget equation, $p(\omega x)=0$, we obtain $dp(\omega x)+p\phantom{\rule{0.2em}{0ex}}d\omega p\phantom{\rule{0.2em}{0ex}}dx=0$. Substituting, $dU=\lambda p\phantom{\rule{0.2em}{0ex}}dx$. The wellbeing of the consumer is assessed by tracking consumption. An immediate and well known consequence is that the vector of marginal utilities is ${U}^{\prime}=\lambda p$.
Here the Lagrange multiplier is the marginal utility of income, but the noincome effects assumption gets rid of it. Formally, linear homogeneity means $U(\theta x)=\theta U(x)$. Differentiating with respect to (positive) θ, using the chain rule, ${U}^{\prime}(x)x=U(x)$. (This is Euler’s theorem.) The (percentage) rate of change of utility becomes now $dU/U=\lambda p\phantom{\rule{0.2em}{0ex}}dx/{U}^{\prime}(x)x=\lambda p\phantom{\rule{0.2em}{0ex}}dx/\lambda px=p\phantom{\rule{0.2em}{0ex}}dx/px$. If we have two consumers, then their rates of change are $p\phantom{\rule{0.2em}{0ex}}d{x}^{1}/p{x}^{1}$ and $p\phantom{\rule{0.2em}{0ex}}d{x}^{2}/p{x}^{2}$. Defining x as the total consumption bundle, the weighted average is $(p{x}^{1}/px)p\phantom{\rule{0.2em}{0ex}}d{x}^{1}/p{x}^{1}+(p{x}^{2}/px)p\phantom{\rule{0.2em}{0ex}}d{x}^{2}/p{x}^{2}=p\phantom{\rule{0.2em}{0ex}}dx/px$. In other words, $p\phantom{\rule{0.2em}{0ex}}dx/px$ is a robust measure for tracking utility.
Performance, however, is the latter per unit of endowment, hence as a growth rate performance is measured by subtracting the change in endowment from the change in consumption: $(p\phantom{\rule{0.2em}{0ex}}dxp\phantom{\rule{0.2em}{0ex}}d\omega )/px$.
3 Embedding performance measurement in the inputoutput framework
I assume that utility is a function of the consumption of the produced commodities. This neglects the utility of leisure time, but is not bad if labor is supplied inelastically (e.g. when overall utility is a CobbDouglas function of commodity consumption utility and leisure). I also identify household consumption with domestic final demand, i.e. incorporating investment. This has been given a theoretical foundation by Weitzman ([1976]). Net output $y=x+z+s$, where z is net exports and s is slack (output not allocated to intermediate demand, domestic final demand or exports, i.e. inventory investment). The initial endowment does not enter this equation. I assume that the initial endowment consists of (currently) nonproduced factors, capital K and labor L. This dichotomy between produced and nonproduced commodities is handled by reserving the last two dimensions for capital and labor, with prices r and w. With a slight abuse of notation I reserve p for the price (row) vector of the produced goods and servicesthe price vector of Section 2 becoming ($p\phantom{\rule{0.5em}{0ex}}w\phantom{\rule{0.5em}{0ex}}r$). The performance measure becomes $(p\phantom{\rule{0.2em}{0ex}}dxr\phantom{\rule{0.2em}{0ex}}dKw\phantom{\rule{0.2em}{0ex}}dL)/px$.
If there is no slack ($s=0$) and balance of payments ($pz=0$) then consumption x may be replaced by net output y in the performance measure and we recognize the Solow residual. I will not make these neoclassical assumptions. I will use, however, the balance equations, for gross output $q=Aq+y$, and for the factor inputs $K=kq+\kappa $ and $L=lq+\lambda $. Inputoutput coefficients populate matrix A (intermediate inputs) and row vectors k and l (factor inputs). κ and λ represent idle capital and labor. (Product slack s was already accounted for in net output y.) Substitution turns performance $(p\phantom{\rule{0.2em}{0ex}}dxr\phantom{\rule{0.2em}{0ex}}dKw\phantom{\rule{0.2em}{0ex}}dL)/px$ into $[p\phantom{\rule{0.2em}{0ex}}d(qAqzs)r\phantom{\rule{0.2em}{0ex}}d(kq+\kappa )w\phantom{\rule{0.2em}{0ex}}d(lq+\lambda )]/px$. Applying the product rule to Aq, kq and lq, and rearranging, the numerator of this performance measure is decomposed into four parts:

(i)
$$(p\phantom{\rule{0.2em}{0ex}}dA+r\phantom{\rule{0.2em}{0ex}}dk+w\phantom{\rule{0.2em}{0ex}}dl)q$$

(ii)
$$p\phantom{\rule{0.2em}{0ex}}dz$$

(iii)
$$(ppArkwl)\phantom{\rule{0.2em}{0ex}}dq$$

(iv)
$$(p\phantom{\rule{0.2em}{0ex}}ds+r\phantom{\rule{0.2em}{0ex}}d\kappa +w\phantom{\rule{0.2em}{0ex}}d\lambda )$$
All have to be divided by px. The first term measures the reductions in the input coefficients, i.e. technical change. Because we multiply with gross output components and divide by the main component of net output, this is a weighted average of industrial Solow residuals, with the weight summing to the gross/net output ratio of the economy (Domar aggregation, ten Raa [2006]). The second term measures the value in the change in the net imports, −z, i.e. the termsoftradeeffect. It is straightforward to understand this. If the balance of payments does not change, pz constant, then $p\phantom{\rule{0.2em}{0ex}}dz=(dp)z$, which measures the improvement in the prices of the net exports. The third term measures the reallocation of output in terms of profitability, i.e. allocative efficiency change. The fourth term measures the reduction of slack in the utilization of goods and services, capital and labor, i.e. Xefficiency change (Comanor and Leibenstein [1969]).
A perfectly competitive economy shows only the first two effects, technical change and the termsoftrade effect. Indeed, the trade sector can be considered an industry where the exports are the inputs and the imports are the outputs. An improvement in the terms of trade is a reduction of the exports required per unit of imports, just like a reduction of the input coefficients in the regular industries. Solow ([1957]) analyzed a closed perfectly competitive economy and discovered that the residual (now called Solow residual) measures technical change. The role of the termoftrade has been in the air for long and was articulated by ten Raa and Mohnen ([2002]). They also relate the broad decomposition into technical change and efficiency change to the operations research/productivity literature. The further decomposition into Xefficiency and allocative efficiency has been studied in the same literature, but in micro contexts. This paper encompasses all, using an inputoutput framework of the economy.
I have not assumed that the prices are equilibrium prices. Even when prices are distorted, they point the way to the greatest increase of utility, by the first order condition of the problem of the consumer, and that’s all we needed. Things get different when one wants to assess the level of efficiency. This is matter of determining how much better off the consumers could be. There are several ways to find the Pareto frontier. A natural one  particularly when assuming linearly homogeneous utility functions  is to find alternative allocations that maximize the utility levels with a common multiplicative factor, subject to the balances. Then the prices are the shadow prices to the latter constraints. This methodology is particularly helpful when markets are incomplete, prices are missing.
4 The environmental extension
The classical example of missing prices is that for externalities, such as pollution. Elsewhere I ([2008]) have argued that a conservative way to estimate the inefficiency involved is to assume Leontief preferences. I have also shown that, then, the problem to determine inefficiency is solved by maximizing the level of consumption subject to the balance constraints. Denote that level by u, where the observed level corresponds to $u=1$. Hence u is the expansion factor for consumption. For example, if u can be 1.25, then consumption could be 25% higher and, therefore, it is only at 80% of its potential. We say that the economy is 20% inefficient. The constraints are $Aq+xu+z\le q$$kq\le K,lq\le L$$pz\le p{z}^{\circ}$. Here the variables are q u z. $p{z}^{\circ}$ is the observed trade balance, at world prices. Maximizing u, the shadow prices to the produced balance constraint can be seen to be proportional to p (ten Raa [2006]). The shadow prices to the next two constraints are denoted r and w. Bring in emission coefficients, organized in row vector m, and a policy level M of maximum emission. That is equivalent to adding a constraint $mq\le M$. Denote the shadow price by t, then the dual constraint becomes $p\le pA+rk+wl+tm$, with slack indicating that the output of a product must be zero. These shadow prices are competitive prices, sustaining the efficient allocation, and reflect the cost of pollution, which is equal to the Pigovian tax.
Performance growth accounting à la Section 3 is straightforward, because formally pollution is equivalent to the addition of a third resource.
5 Conclusion
The reconciliation of performance measurement and inputoutput analysis is mutually beneficial. The inputoutput framework facilitates economywide performance measurement, including a quantification of the termsoftrade effect and a decomposition of efficiency change in allocative and Xefficiency changes, even down to the level of industry. Conversely, performance measurement makes inputoutput analysis richer. The doctor not only knows how the parts of the body interact, but also how healthy the patient is.
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ten Raa, T. Performance measurement in an inputoutput framework. Economic Structures 1, 2 (2012) doi:10.1186/2193240912
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Keywords
 performance measurement
 growth accounting
 terms of trade
 environmental constraint