# Can We Measure Elasticity of Demand From Time-Series Data on Prices and Quantities?

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Deepak K. Sinha (1994) ,"Can We Measure Elasticity of Demand From Time-Series Data on Prices and Quantities?", in AP - Asia Pacific Advances in Consumer Research Volume 1, eds. Joseph A. Cote and Siew Meng Leong, Provo, UT : Association for Consumer Research, Pages: 213-219.

The constant elasticity specification has been used in several empirical studies of consumer demand in the past which interpreted the slopes of time-series plots of price-current volume on log-log paper as (the negative of) industry price elasticities. According to this paper, however, time series data on prices and quantities may not contain any information on elasticities. Prices and quantities appear to be functions of time and the slope of relative price-current volume plot of an industry indicates the ratio of the rate of growth of industry productivity relative to the economy and the rate of growth of industry output.

Consumer behavior ultimately manifests itself in the form of demand for products. Hence the importance of studying consumer demand. One of the most commonly used specifications of systems of demand has been the constant elasticity specification, also called the double-log or the log-linear demand:

Log q_{jt} = a_{j} + b_{j} Log P_{jt} (1)

q_{jt} is the quantity demanded of industry j at time t; P_{jt} is industry price relative to the price index of the economy; and b_{j} [__<__0] and a_{j} are constants. Equation (1) has been used extensively in research in the areas of marketing [Bass, 1980; also see Tellis(1988) for more references] and economics (Fogel and Engerman, 1971; Williamson, 1967; Wold and Jureen, 1953; Stone, 1956; Passell and Schmundt, 1971) for reasons of "goodness of fit, ease of estimation, and immediacy of interpretation" (Houthakker, 1965, p. 278).

The coefficient b_{j} in logarithmic regression in (1) is believed to represent elasticity. Researchers in the past therefore considered (1) a good description of demand in an industry if the R^{2} was high and the estimate of b_{j} was negative. A statistically insignificant R^{2} was taken to imply that the demand elasticity for the industry was not constant, and a positive value of b_{j} was considered inconsistent with a downward sloping demand curve.

The elasticity interpretation of b_{j} has led to the use of equation (1) by practitioners for making pricing decisions, and by economists for making policy recommendations such as those relating to sales taxes. Trade analysts have fitted equation (1) to data on import prices and quantities for making trade policy recommendations and marketing scholars have used variants of (1) to estimate advertising and promotion elasticities for marketing strategy.

In this paper we suggest that time series data may not contain any information on elasticities. In cross-sectional data at one point of time the slope of a log-linear plot of prices and quantities of different firms in an industry j will indicate the price elasticity of demand for that industry, e_{j}. e_{j} is defined as - (dq_{j}/dp_{j})/(q_{j}/p_{j}), which upon integration yields:

Log q_{j} = A_{j} - e_{j} Log P_{j} (2)

where A_{j} is a constant of integration. Note the absence of subscript t in equation (2). In time series data, as we will show in this paper, relative prices and quantities appear as functions of time rather than of each other:

Log P_{jt} = z_{j} + W_{j}t (3a)

Log q_{jt} = Z_{j} + w_{j}t (3b)

Thus, equation (1) represents the reduced form of (3a)-(3b) describing temporal movements of industry relative price and volume and b_{j} [= w_{j}/W_{j}] does not indicate the price elasticity of demand but the ratio of the rates of movement of industry output and industry price relative to the economy.

An interesting implication of this is that statistically significant and insignificant R^{2}s for equation (1) as well as positive and negative values of b_{j} are all consistent with the constant elasticity specification of demand. We demonstrate this by analyzing shipment data from the Bureau of Economic Analysis. During 1972-1988 relative prices of industries such as newspaper publishing and printing went up. Relative price-current volume plots for these industries will slope upward and b_{j}'s will be positive without implying that these are Giffen goods. Relative prices of industries such as bakery goods and concrete, gypsum, and plaster remained constant during the period. Their relative price-current volume plots are horizontal resulting in statistically insignificant R^{2}s for equation (1), which, however, do not imply that the elasticity of demand in these industries is not constant.

THE MODEL

The Demand

We assume an economy with N industries (j = 1, ..., N), each comprising n_{j} (i = 1, ..., n_{j}) differentiated products manufactured by monopolistically competitive firms (Dixit and Stiglitz, 1977). N and n_{j}'s are large. The demand for the outputs of firm i, industry j, and the economy are:

q^{i}_{j} = (Mb_{j}/P_{j})a^{i}_{j}(P^{i}_{j}/P_{j})^{-ej} (4a)

q_{j} = Mb_{j}/P_{j} (4b)

q = M/P (4c)

where M [= M_{0}e^{ut}] is the dollar GNP; P^{i}_{j}[=p^{i}_{j}e^{ut}], P_{j}={Sia^{i}_{j}(P^{i}_{j})^{-(ej-1)}}^{-1/ej-1)}, and P=P_{j}(P_{j})^{bj} are the nominal prices for firm i, industry j and the economy. P^{i}_{j} is firm i's real price. bi_{j} represents industry j's share of dollar GNP, and aij represents the non-price influences on firm i's market share. b_{j} and a^{i}_{j} are constant. Since n_{j} is large, a small change in p^{i}_{j} has little impact on p_{j}. e_{j}, firm i's own price elasticity of demand, therefore is constant.

At constant M, q^{i}_{j} in (4a) is homogeneous of degree -1 in real prices. If all real prices decreased by some factor then q^{i}_{j} will increase by the same factor. Homogeneity also implies that all real prices in an industry must move together. To demonstrate this we differentiate q^{i}_{j} [=a^{i}_{j}(p^{i}_{j}, ..., p^{h}_{j})] totally:

j = J if firms i and h are in the same industry. Dividing (5) by quantities:

where q^{i}_{j} = -(p^{.i}_{j}/p^{i}_{j}) and Q^{i}_{j} = (q^{.i}_{j}/q^{i}_{j}). e^{ih} [__<__ 0] is firm i's cross price elasticity. For j = J, e^{ih} = 0. Equation (6) comprises N x n_{j} linearly independent non-homogeneous equations and has therefore a unique solution. The own and cross price elasticities also obey the following relationships:

Dividing (6) throughout by the right hand side and comparing terms with (7) I find that either e^{ih} = 0 or q^{h}_{j} = q^{i}_{j} = Q^{i}_{j} = q_{j}. That is, either firms i and h are in different industries or they move their prices together.

This result does not require the elasticities to be constant but follows directly from the homogeneity of demand. If the competitors of a firm facing homogeneous demand reduce price and the firm fails to reduce price at the same rate then it will lose all its market share instantaneously. By incorporating in this model search costs, or switching costs, or lags in the transmission of price information, or by formulating it as a stochastic differential game with autoregressive errors it is possible to analyze the more realistic case in which the loss of market share will be gradual rather than instantaneous. For simplicity, in this paper we will ignore these refinements.

The above result says nothing about the direction of movement of prices. In the differential game that we will describe below we will show that in the non-cooperative closed loop Nash-equilibrium all firms in an industry find it optimal to invest in productivity at a rate that depends on the interest rate, on their own price elasticity of demand, and on how expensive it is to improve productivity in their industry. If these parameters are constant over time then firms invest in productivity at a constant rate over time. Failure to improve productivity with the industry is not profitable since the lost market share will be extremely expensive to regain later. Nor is improving productivity faster than the optimal rate profitable because other firms will then also invest and improve productivity faster to protect their market shares. Investment in productivity in this model does not lead to higher profits but is required for survival.

The Firms

Firms are asymmetric and maximize present value of future profits over an infinite horizon. The production technology exhibits constant return. At constant M, and with other prices constant, firm i's profit will go up if it reduces either its price or its marginal cost by a small amount. We assume that firms can reduce their real marginal cost through investment. Firm i's nominal marginal cost at t [=c^{i}_{j}e^{mt}] therefore will reflect movements in its real marginal cost [=c^{i}_{j}__>__0] and M. Changing price, however, is costless. At every instant, therefore, each firm chooses price that maximizes its instantaneous profit given its current cost resulting in price-cost margin equal to the reciprocal of demand elasticity, that is, (p^{i}_{j}-c^{i}_{j})/p^{i}_{j}=1/e_{j}

Productivity Enhancement Technology

At constant M, firm i's profit will go up if it reduces its marginal cost by a small amount while its competitors do not. In reality, competitors will also decrease cost and price. To determine the optimal paths of costs and prices we use the concept of non-cooperative Nash equilibrium in which during a small time period dt at t all firms in industry j choose levels of costs and prices at constant M without expecting to influence the average industry cost or price. Firm i's profit at time t will therefore be a function of its own price at constant M, p^{i}_{j}, and cost, c^{i}_{j}, and the costs of its competitors. Thus, c^{i}_{j} represents the state variable for firm i. During a small time period dt at t let firm i make cost reducing investment and reduce c^{i}_{j} by x^{i}_{j} [=c^{.i}_{j} = -(dc^{i}_{j}/dt)__>__0]. Firms use a nominal rate [= r + m; r > 0] for discounting profits. Ignoring the fixed cost of entry, we can write firm i's payoff as:

F^{i}_{j}dt is the investment required to reduce c^{i}_{j} by dc^{i}_{j} during the period. Using firms' static profit [p^{i}_{j}=q^{i}_{j}c^{i}_{j}e^{mt}/ej-1)] as an indicator of its relative size in industry, I assume that dF^{i}_{j}/dp^{i}_{j} > 0 and dF^{i}_{j}/d(x^{i}_{j}/c^{i}_{j}) > 0. The first assumption implies that for any percentage reduction in cost, a larger firm will have to invest more. A larger firm has more employees and larger facilities, and any cost reducing measures such as training or changes in equipments may have to be implemented at a larger scale. The second assumption implies that a firm will have to spend more to achieve a larger percentage reduction in cost during the period. Thus, at constant M [= M_{0}]:

whereF_{j} is a constant that indicates how expensive it is to improve productivity in an industry. To insure existence and uniqueness of the solution to the differential game I will assume that:

F_{j} < r / (e_{j} - 1) (10)

I can now write firm i's objective function as:

Optimal Time Path for Productivity

To be a Nash-equilibrium a strategy n-tuple (x_{j}^{1*},...,x_{j}^{nj*}) must satisfy, for each i = 1, ..., n_{j}, a system of n_{j} Bellman equations:

where V^{ii} = dV^{i}_{j}/dc^{i}_{j} and V^{ih} = dV^{i}_{j}/dc^{h}_{j}; h /= 1: h = 1, ..., nj. Maximizing (12):

This suggests the following value functions as solutions for i = 1, ..., n_{j}:

We can evaluate SV^{ih}x^{h}_{j} from the homogeneity condition for the demand function:

At constant M, q^{i}_{j} is a function of costs of all firms. Differentiating q^{i}_{j}:

Since c^{.}_{j}/c_{j}=c^{.h}_{j}/c^{h}_{j}=-q^{.i}_{j}/q^{i}_{j}=-F_{j} due to homogeneity, (16) yields:

Substituting from (14-17) into (12) and simplifying I get:

From (18):

Thus firm i's marginal cost at time t is:

where c^{i}_{j0} is firm i's marginal cost at t = 0.

THE EMPIRICAL STUDY

Results from the Model and Hypotheses

Time and industry average relative price. Economy's output grows at q [= S_{j}b_{j}q_{j}] which is a weighted average of all rates of growth of productivity. Since dollar GNP grows at m, the price indexes of the industry and the economy grow at (m - q_{j}) and (m - q). Industry relative price will then be:

Defining W_{j} = -(q_{j} - q), and ignoring super- and sub-scripts, we obtain our first hypothesis which is non-directional since EQUATION according as EQUATION :

H1. Log P = z + Wt (22)

Time and industry current volume. From (4b), we obtain the current volume of industry j as:

qj = qj0e^{q}jt (23)

Taking logarithm, ignoring super- and sub-scripts, and defining w = q_{j}, we obtain our second hypothesis which is directional since w __>__ 0:

H2. Log q = Z + wt (24)

Price and current volume. From (21) and (23):

k_{j} is constant. Writing b_{j} for -(q_{j}-q)/q_{j} [= W/w]:

H3. Log q = a + b Log P (26)

Since EQUATION according as EQUATION , H3 is non-directional.

The Data

We analyzed current- and constant-dollar shipments for 1972-1988 for all manufacturing industries at the 3-digit level of the Standard Industrial Classification available from the Bureau of Economic Analysis. To compute relative prices we used the ratio of GNP at current and constant dollars as deflator. We report results for six industries with increasing, decreasing, and constant relative prices. Results for other industries are available from us. Durbin-Watson statistic indicated presence of autocorrelation. For brevity, we report only the OLS estimates; GLS estimates correcting for autocorrelation are available from us. Accounting for autocorrelation did not affect the pattern of our results. Results of Dickey-Fuller (1979; 1981) tests for unit roots and cointegration (Granger, 1986) are also available from us.

Results

H1. Table 1 shows OLS estimates from fitting equation (22). Relative prices of office, computing, and accounting machinery (office equipment) and optical instruments and lenses (optical instruments) declined during the period 1972-1988, while those of cigarettes and newspaper publishing and printing (newspapers) went up. Relative prices of bread and bakery products (bakery products) and concrete, gypsum and plaster (concrete), on an average, neither increased nor decreased during the period.

H2. Table 2 shows results from fitting equation (24). Current volumes of three of the six industries, namely, office equipment, optical instruments, and newspapers show significant growth over time, while those of the other three, namely, cigarettes, bakery products, and concrete exhibit no growth.

LINEAR REGRESSION BETWEEN TIME (T) AND THE LOGARITHM OF INDUSTRY RELATIVE PRICE (P)

LINEAR REGRESSION BETWEEN TIME (T) AND THE LOGARITHM OF CURRENT VOLUME (q)

LINEAR REGRESSION BETWEEN LOGARITHMS OF CURRENT VOLUME (q) AND RELATIVE PRICE (P)

H3. Table 3 shows OLS estimates for (26). The value of b is positive for newspapers whose relative price and volume increased and negative for office equipment, optical instruments, and cigarettes whose relative price and volume moved in opposite directions. The relative price-current volume plots for the other two industries, namely, bakery products and concrete, are horizontal.

IMPLICATIONS FOR RESEARCH AND PRACTICE

Studying Systems of Demand

Several empirical studies in the past have estimated the elasticity of demand by the slope of log-linear plots of price and quantity over time. We have shown, however, that it may not be possible to estimate demand elasticities from time series data on prices and quantities. A demand function with constant elasticity is homogeneous of degree zero in all prices and income. This, in turn, requires that all real prices in an industry move together. As a result, in time series data, all prices and quantities become functions of time rather than of each other. A statistically significant fit for a log-linear specification of demand will be consistent with constant elasticity hypothesis, but the slope of the plot will not have an elasticity interpretation. And a statistically non-significant log-linear relationship between relative price and current quantity of a product will not signify lack of fit of the data to the constant elasticity form of the demand function but that the price of the product has been falling at a rate similar to that of the price index for the economy. According to Consumer Reports (1992), the relative price of chicken has not changed over the past decade. Even if the elasticity of demand for chicken is constant and finite, its relative price-current volume plot will be horizontal and a log-linear regression of relative price and quantity will yield a statistically non-significant R^{2}. This, however, will tell us nothing about the elasticity of demand for chicken.

These results have implications beyond the measurement of elasticity of demand. International economists in the past have estimated elasticity of trade from time series data on import prices and quantities. These estimates form important inputs to decisions relating to tariffs and trade policies. Our results suggest that estimates of trade elasticity based on time-series data may be misleading. Similarly, advertising executives who have in the past measured elasticity of sales to advertising and promotion from time series data for determining advertising budgets and for formulating promotion strategies must be careful in interpreting results from time series data.

Beyond Dynamic Demand and Dynamic Pricing

Studies of demand and pricing in recent years have focused on dynamic considerations (Robinson and Lakhani, 1975; Kalish, 1983; Wernerfelt, 1986; Dolan and Jeuland, 1981) that represented a major advance over the state of art of pricing that had been documented by Monroe and Del Bitta (1978). The addition of a dynamic involving price and cost (Bass, 1980; Dolan and Jeuland, 1981; Wernerfelt, 1985) yielded important insights in dynamic demand models that had earlier been based largely upon the concepts of diffusion of innovation and product life cycle. Most of these models, however, were based on analysis of either a monopoly (Dolan and Jeuland, 1981) or a single oligopolistic industry (Wernerfelt, 1985).

Our model, however, suggests that all prices and quantities in the economy are functions of all other prices and are together determined by a process of equilibrium whose solution is given by a system of equations such as (20) that results in all prices and quantities becoming functions of time. We are not suggesting that time causes changes in prices and quantities. In the non-cooperative equilibrium firms find it optimal to invest in productivity. Other things being equal, the optimal percentage rate at which firms in an industry improve productivity is higher the lower the interest rate, the higher the own price elasticity of demand, the lower the cost of capital, and/or the less expensive cost reduction is in the industry. Thus our model provides a mechanism that links characteristics of the industry and the economy to firm's pricing, output, and productivity enhancement decisions.

Dixit and Stiglitz (1977) and Sattinger (1982) have earlier described how demand functions with constant elasticity may arise naturally as a result of utility maximization by consumers. Dixit and Stiglitz and Sattinger have also shown that monopolistic competition in which firms face demand whose elasticity is constant is a very general model of competition in a differentiated goods industry; perfect competition and monopoly are its special cases. And, this paper has shown that a profit maximizing firm facing such a demand will reduce price exponentially over time. Thus the strong statistical evidence in favor of a log-linear relationship between price and quantity should be considered evidence for the existence of rational economic agentsCutility maximizing consumers and profit maximizing firms.

Implications for Managerial Practice and Public Policy

Forecasting cost, price, and volume. Since the rate of movement of prices and quantities depends on the interest rate, on the elasticity of demand, and on how expensive it is to reduce cost, our model allows managers to incorporate important characteristics of the economy and the industry into future estimates and enhances their ability to forecast turning points.

Measurement of elasticity. Although the elasticity of demand can not be measured directly from time series data on prices and quantities, indirect measurements can be obtained in a variety of different ways. Estimates of price-cost margins in an industry could be expected to be equal to the reciprocal of the elasticity of demand in that industry. Cross-sectional data on prices and output of firms in an industry at any point of time could also yield an estimate of the price elasticity of demand.

Implications for antitrust. Investment in productivity and output growth in this model are the results of competition. The collusive outcome results in no growth. Ignoring cost of entry, the value of a firm in this model is:

In case of collusion q_{j} will be zero and V^{i}_{j} will be higher. Thus, the rate of improvement in productivity could help identify collusion.

Capital budgeting for productivity: The model in this paper has two important implications for practice in the areas of accounting and finance. First, our costing systems and accounting methods and techniques should take into account the fact that the productivity and the output of a firm must increase along an optimal path and its real cost and price move exponentially over time. This realization should help control all operating expenses, improve profits, and reduce surprises in financial reports.

And second, our capital budgeting procedures may need an "equilibrium" orientation to assess investment proposals for productivity projects. Most firms employ a net present value, or an internal rate of return, or a payback period, or a hurdle rate, or some similar criteria to assess the feasibility of investment projects. Such procedures appear to be suitable for assessing the desirability of projects which produce proprietary benefits to the firm such as those relating to a patent. This model points out that for assessing investment proposals for productivity projects we may need criteria that are different from those used for other projects. Investment in productivity by a firm lowers its cost. But other firms also invest to reduce their costs at the same rate, so that there is no increase in any firm's market share. When all firms invest in productivity then industry real profit goes down and so does the profit of each firm. But such investment is necessary for survival.

Economic growth. Since the studies by Solow (1957) economists have known that growth in capital and labor could not fully account for the observed increases in the output of economies. By incorporating the capital sector in the model it can be shown that all economic growth is the result of investment in productivity. Investment in productivity by producers of consumption goods accounts for all of the growth in consumption. Investment in productivity by producers of capital goods leads to lower prices for capital goods, which in turn results in capital investment by new entrants. Such investment reduces firms' incentive to invest in productivity as their future benefit from such investment goes down. All capital investment by new businesses therefore has an adverse impact on the growth of output, although such investment could lead to increased consumers' utility if consumers' attach sufficient value to variety. Hence the need for policy makers to know consumers' preferences.

CONCLUSION

We would like to close this paper with a story which illustrates the point we made earlier about the need for different criteria for assessing investment in productivity. During the latter half of the eighties there was a multi-million dollar research project at Massachusetts Institute of Technology funded by International Business Machines, Xerox Corporation, Digital Equipment Corporation, and other leading manufacturers of information technology equipment. The purpose of the project was to document the economic benefit of the investment companies had made in information technology equipment. The project had begun with great fanfare and with all the enthusiasm in the world. But as the data began trickling in, a pall of gloom descended on the researchers, which had nothing to do with the downturn in the economic fortunes of Massachusetts. Most responding companies said that they could not justify the investment they had made in information technology equipment on economic criteria. This paper perhaps provides an answer to the puzzle. If investment in information technology equipment improves productivity but does not yield proprietary benefits then firms in an industry will make such investment and none will be able to show profits for it, and yet the investment would have been economically justified.

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