Quantitative Techniques for Competition and Antitrust Analysis_10 pptx

9.1. Demand System Estimation: Models of Continuous Choice 443 Table 9.2. IV estimation results based on forty-four observations. Robust Regressors Coefficient std. err. tP>jtj [95% Conf. interval] ln.P Sugar / 0.27 0.08 3.41 0.00 [0.43 0.11] Quarter 1 0.10 0.01 9.23 0.00 [0.12 0.08] Quarter 2 0.01 0.01 1.99 0.05 [0.02 0.00] Quarter 3 0.01 0.00 3.71 0.00 [0.01 0.02] Constant 8.69 0.25 34.71 0.00 [8.19 9.20] R 2 D 0:80. The dependent variable in this regression is ln.Q Sugar /. are a cost of producing sugar and will therefore ordinarily affect observed prices according to economic theory (and also farmers!). On the other hand, given that farmers are a small minority of the population and that the increase in their wages is not likely to translate into material increases in sugar consumption, farm wages are unlikely to materially affect the aggregate demand for sugar. The 2SLS estimation proceeds in two stages: 1st-stage regression: ln P t D a b ln W t C  1 q 1t C  2 q 2t C  3 q 3t C " t ; 2nd-stage regression: ln Q t D a b b ln P t C  1 q 1t C  2 q 2t C  3 q 3t C v t ; where W t is the farm wage at time t and b ln P t is the estimated log of price ob- tained from the first-stage regression. Most statistical computer packages are able to perform this procedure and in doing so provide the output from both regressions. 8 The quarterly dummies are also included in the first-stage regression since the requirement for an instrument to be valid is that it is correlated with an endogenous variable conditional on the included exogenous variables. Demand is itself seasonal, so that the quarterly dummies are not correlated with prices conditional on the included exogenous variables and hence are not valid instruments for prices, even if they are valid instruments for themselves, i.e., can be treated as exogenous. The results of the instrumental variable estimation are presented in table 9.2. The resultsshowa lower coefficient for the price variable. The elasticity of demand is now 0:27 and is below the previous OLS estimate. Because of data availability on farm wages some observations had to be dropped so that the data for the two regressions are not exactly the same. Nonetheless, formally a Durbin–Wu–Hausman test could be used to test between the OLS and IV regression specifications (see Greene 2000; Nakamura and Nakamura 1981). The central question is whether the instruments are in fact successfully addressing the endogeneity bias problem that motivated our use of them. Often inexperienced researchers use IV regression results even if the resulting estimate moves the coefficient in the direction opposite to that expected as a result of endogeneity bias. 8 STATA, for example, provides the “ivreg” command. 444 9. Demand System Estimation Results in IV estimations should be carefully scrutinized because they will only be reliable if the instrument chosen for the first-stage regression is a good instrument. We know that for an instrument to be valid it must satisfy the two conditions: (i) EŒ t j .X t ;W t / D 0 and (ii) EŒln.P t / j .X t ;W t / ¤ 0; where in our case X t D .1; q 1 ;q 2 ;q 3 / are the exogenous regressors in the demand equation and W t is the instrument, farm wages. As we described earlier, the first of these conditions is difficult to test; however, one way to evaluate whether it holds is to examine a picture of the estimated residuals against the regressors. We should see no systematic patterns in the graphs—whatever the value of X t or W t the error term on average around those values should be mean zero. Such tests can be formalized (see, for example, the specification tests due to Ramsey (1969)). But there are limits to the extent to which this assumption can be tested since the model will, to a considerable extent, actively impose this assumption on the data in order to best derive the IV estimates obtained. A variety of potential IV results can certainly be tested against each other and against specifications which use more instruments than strictly necessary to achieve identification. But the reality is that the first of these assumptions is ultimately quite difficult to test entirely convincingly and one is likely to ultimately mainly rely on economic theory—at least to the extent that the theory robustly tells us that, for example, a cost driver will generally not affect consumer demand behavior and so will have no reason to be correlated with the unobserved component of demand. The second condition is easier to evaluate and the most popular method is to run a regression of the potentially endogenous variable (here ln.P t /) on all the exogenous explanatory variables in the demand equation and also the instruments, here ln.W t /. To see whether the second condition holds, we examine the results of the following “first-stage” regression: P t D a b ln W t C q 1 C q 2 C q 3 C " t : For the variable farm wage to be a good instrument, we want the coefficient b to be robustly and significantly different from zero in this equation. If the instrument does not have explanatory power in predicting the price, the predicted price used in the second-stage regression will be poorly correlated with the actual price given the other variables already included in the demand equation. In that case, the estimated coefficient of the price variable in the second-stage regression will be imprecisely estimated and indeed may not be distinguishable from zero. Even with “good instru- ments” in the sense that they are conditionally correlated with the variable being instrumented, we will expect the coefficient of an instrumented variable in an IV regression to be lessprecisely estimated (have a higher standard error)than theanalo- gous coefficient estimated using OLS (with the latter a meaningful comparison only if in fact the OLS estimate is a valid one). IV estimation relaxes the assumptions 9.1. Demand System Estimation: Models of Continuous Choice 445 Table 9.3. First-stage regression results. ln.Price/ Coefficient Std. err. tP>jtj [95% conf. interval] Quarter 1 0.05 0.02 2.93 0.01 [0.01 0.08] Quarter 2 0.00 0.01 0.53 0.60 [0.01 0.02] Quarter 3 0.01 0.01 1.43 0.16 [0.02 0.00] ln.Farm wage/ 1.13 0.12 9.71 0.00 [1.36 0.89] Constant 4.94 0.18 27.47 0.00 [4.58 5.30] required to get valid estimates but one must always remember it does so at a price: lower precision. There will, as a result, be cases where the OLS estimates cannot be rejected when compared with the IV estimates and may as a result be preferred. Results for the first-stage regression for our example are shown in table 9.3. First note that, as one would hope, the coefficient for the ln(Farm wage) variable is significant and has a high t-statistic, indicating that it is precisely estimated. However, note that the coefficient reported is rather surprising: its sign is negative! The economic theory motivating our choice of instrument tells us that an increase in costs should translate into higher prices as the supply curve shifts leftward. Indeed, our aim in selecting an instrument is intuitively to use that instrument to allow us to use only the variation in observed prices which we know is due to the variation in the supply curve. When conundrums such as this one arise, one should address them. While for econometric theory purposes “conditional correlation” is all that is required to sup- port the use of the instrument, one should not proceed further without understanding why the data are behaving in an unexpected way. In this case, one may want to inves- tigate, for example, whether farm wages have a trend that negatively correlates with prices and for which we did not control. Figure 9.2 graphs farm wage data over time. In particular, note that there is an upward trend in wages between 1995 and 2006 during the time that sugar prices fell. Clearly, while farm wages may still be an important determinant of sugar prices, they are not likely to be a major factor driving the price of sugar down. We must look elsewhere for an instrument that helps explain the major source of variation in prices conditional on the exogenous variables in the demand equation. To search for a good instrument we must attempt to better understand the factors that are driving sugar prices down over time. One possibility is that other costs in the industry are falling dramatically. Alternatively, there may be an institutional reason such as changes in the amount of subsidy offered to farmers or in the tariff or quota system that governs the supply from imports. There may have been substantial entry during the period. Another possibility is that the price change may be driven by important demand factors that we have omitted thus far from our model. Perhaps the taste for sugary products changed over time? Or perhaps substitutes (e.g., high- fructose corn syrup) appeared and drove down prices? 446 9. Demand System Estimation 1992q1 1995q3 1999q1 2002q3 2006q1 Date P Farm wage 4.2 4.4 4.6 4.8 5.0 5.2 Figure 9.2. Farm wages plotted over time. At this point we need to go back to our industry experts and descriptive analyses of the industry to attempt to find possible explanations for the major variation in prices and in particular the price decline. The data and regression results have provided us with a puzzle which we need to solve by using industry expertise. Only once we have thought hard about what in the industry is generating our data will we generally be able to move forward to generating econometric results we can believe. It is for this reason that we described in the introduction to this section that it is very rare for an econometrician to be able to work in isolation late into the night with her existing data set and generate sensible regression results without going back to think about the nature and drivers of competition in an industry. That said, we generally do not need to understand everything about the price- setting process to obtain reliable demand estimates. In particular, in a demand esti- mation exercise we are not trying to estimate the pricing equations that explain how firms optimally choose their prices. Although the first-stage regression in a 2SLS estimation may closely resemble the reduced form of the pricing equation in a struc- tural model of prices and quantities (see chapter 6), it is not quite the same. We saw in the previous chapters that factors affecting demand are included in the pricing equation and so are cost data. However, the first-stage equation of the 2SLS regres- sion is materially different from a reduced-form pricing equation in that we do not need to have all the cost data: we really only need one good supply-side instrument to identify the price coefficient in a homogeneous product demand equation. 9.1.2 Differentiated Products Demand Systems Most markets do not consist of a single homogeneous product but are rather com- posed of similar but differentiated goods that compete for customers. For instance, in the market for shampoos there is not a single type of generic shampoo. Rather there is a variety of brands and types of shampoo which consumers do not consider 9.1. Demand System Estimation: Models of Continuous Choice 447 absolutely equivalent. We must take such demand characteristics into account when attempting to estimate demand in differentiated product markets. In particular, we need to take account of the fact that consumers are choosing among different prod- ucts for which they have different relative preferences and which will usually have different prices. Differentiated product demand systems are therefore estimated as a system of individual product demand equations, where the demand for a product depends on its own price but also on the price of the other products in the market. 9.1.2.1 Log-Linear Demand Models One popular differentiated product demand system is the log-linear demand system, which is simply a set of log-linear demand functions, one for each product available in the market. We label the products in the market j D 1;:::;J. In each case, the quantity of the good purchased potentially depends on the prices of all the goods in the market and also income y (Deaton and Muellbauer 1980b). Formally, we have the following system of J equations: ln Q 1t D a 1  b 11 ln P 1t C b 12 ln P 2t C :::: C b 1J ln P 1J C  1 ln y t C  1t ; ln Q 2t D a 2  b 21 ln P 1t C b 22 ln P 2t CCb 2J ln P Jt C  2 ln y t C  2t ; : : : ln Q Jt D a 2  b J1 ln P 1t C b J2 ln P 2t CCb JJ ln P Jt C  J ln y t C  Jt : Maximizing utility subject to a budget constraint will generically provide demand equations which depend on the set of all prices and income (see, for example, Pollak and Wales 1992). Clearly, with aggregate data we might use aggregate income as the relevant variable for the demand equations (e.g., GDP). However, since many studies focus on a particular sector of the economy, the consumer’s problem is often recast and considered as a two-stage problem. At the first stage, we posit that consumers decide how much money to spend on a category of goods—for example, beer—and at the second stage we posit that the chosen level of expenditure is allocated across the various products that the consumer must choose between, perhaps the different brands of beer. Under particular assumptions on the shape of the utility function, this two-stage process can be shown to be equivalent to solving a single one-stage utility-maximization problem (see Deaton and Muellbauer 1980b; Gorman 1959; Hausman et al. 1994). Using the two-stage interpretation, “expenditure” may be used instead of income in the demand equations but the demand equations will then be termed “conditional” demand equations as we are conditioning on a given level of expenditure. A well-known example of an such an exercise is Hausman et al. (1994). In fact, those authors estimate a three-level choice model where consumers choose (1) the level of expenditure on beer, (2) how to allocate that expenditure between three broad categories of beer (respectively termed premium beer, popular beer, and light 448 9. Demand System Estimation Table 9.4. Market segment conditional demand in the market for beer. Premium Popular Light Constant 0.501 4.021 1.183 (0.283) (0.560) (0.377) log(Beer exp) 0.978 0.943 1.067 (0.011) (0.022) (0.015) log(P Premium ) 2.671 2.704 0.424 (0.123) (0.244) (0.166) log(P Popular ) 0.510 2.707 0.747 (0.097) (0.193) (0.127) log(P Light ) 0.701 0.518 2.424 (0.070) (0.140) (0.092) Time 0.001 0.000 0.002 (0.000) (0.001) (0.000) log(# of stores) 0.035 0.253 0.176 (0.016) (0.034) (0.023) Number of observations = 101. Source: Table 1, Hausman et al. (1994). beer) which marketing studies had identified as market segments, and (3) how to allocate expenditure between the various brands of beer within each of the segments. At level (3), we could use the observed product level price and quantity data to estimate our differentiated product demand system. However, in fact, since level (3) is modeled as a choice of brands (e.g., Coors, Budweiser, Molsen, etc.), at levels (1), (2), and (3) we would need to use price and quantity indices constructed from underlying product-level data to give measures of price and quantity for each of the brands or segments of the beer industry. For example, we might use a price index with expenditure share weights forthe underlying prices within each segment s to produce a segment-level price index, P st D P j w jt p jt . 9 Similarly, we might choose to use volumes of liquid to help aggregate over the brands to give segment-level quantity indices. 10 Estimates of the second level of their demand system using price and quantity indices are shown in table 9.4. At the second level of the choice tree, the demand system is a conditional demand system because the amount of money to be spent on beer has already been chosen at stage 1. 9 Expenditure shares can be defined as w jt D p jt q jt = P j p jt q jt , where p represents prices and q quantities. 10 Formally, Deaton and Muellbauer (1980b) show that there are “correct” price and quantity indices which can be constructed for this process to preserve the multilevel models’ equivalence to a single utility-maximization problem (under strong assumptions). In practice, the authors do not seem to have settled on a universally best choice of price and quantity indices. 9.1. Demand System Estimation: Models of Continuous Choice 449 Since weare dealingwith alog-linear model,the b jj coefficientsprovide estimates of the own-price elasticity of demand while the b jk (j ¤ k) parameters provide estimates of the cross-price elasticities of demand. If we are using segment-level data, we must be careful to place the correct interpretation on the elasticities. For example, the results from table 9.4 suggest that the own-price elasticity of segment demand is 2:6 for premium beer, 2:7 for popular beer, and 2:4 for light beer. These price elasticities could be used as important evidence toward a formal test of the hypothesis that each beer segment is a market in itself by performing a SSNIP test. That said, generally, the price elasticity relevant for such a test would include the indirect effect of prices through their effect on the total amount of expenditure on beer. If the price of premium beer goes up, some consumption will be reallocated to other beer segments but the total consumption of beer might also fall as people either switch to other products such as wine or reduce consumption altogether. The elasticities we can read off from the equation in this instance are conditional elasticity estimates—they are conditional on the level of expenditure on beer. Thus for market definition, if we use expenditure levels and price indices to perform market definition tests, we must be careful to trace through the effect of a price change back through its effect on total expenditure on beer. To do so, Hausman et al. (1994) also estimates a single top-level equation so that the demand for beer in total is expressed as a function of prices and income. In this case, the equation estimated depended on income (GDP) and also a price index constructed to capture the general price of beer as well as demographics, Z t : ln Q Beer t D ˇ 0 C ˇ 1 ln y GDP t C ˇ 2 ln P Beer t C Z t ı C" t : The choice of instruments in differentiated product demand systems is generically difficult. First, we may need a lot of them. In particular, we need at least one instrument for every product whose price is considered potentially endogenous in a demand function (although sometimes a given instrument may in fact be used to estimate more than one equation). Second, a natural source of instruments involves cost data. However, since products are often produced in a very similar way, and cost data are often recorded less frequently than prices are set, at least in financial or management accounts, we are often unable to find cost variables that are genuinely sufficiently helpful for identification of each of the demand curves. Data such as exchange rates and wages are often useful in homogeneous product demand esti- mation, but fundamentally such data are not product (or here segment) specific and so will face difficulties as instruments in the differentiated product context. The reality is that there are no entirely persuasive solutions to this problem. One potential solution, that Hausman et al. (1994) suggest, is to use prices in other cities as instruments for the prices in a given city. The logic is that if, and it is often a very big “if,” (1) demand shocks are city specific and independent across cities and (2) cost shocks are correlated across markets, then any correlation between the price in this market and the prices in other markets will be due to cost movements. Inthat case, the 450 9. Demand System Estimation prices in other cities will be valid instruments for the price in this city. Obviously, these are strong assumptions. For example, there must not be any effect of, say, national advertising campaigns in the demand shocks since then they would not be independent across cities. Alternatively, another potentially satisfactory instrument would be the price of a good that shares the costs but which is not a substitute or complement. For example, if a product under study had costs that were each heavily influenced by the oil price, then the price of another good also subject to a similar sensitivity might be used. Of course, in such a situation it would be easier to use the oil price so examples where this approach would genuinely be useful are perhaps hard to think of. We will explore another option for constructing instruments once we have discussed models based on product characteristics in a later section. 9.1.2.2 Indirect Utility and Expenditure Shares Models A log-linear demand system is easy to estimate because all the equations are linear in the parameters. However, they also impose considerable assumptions on the nature of consumer preferences. For example, they impose constant own- and cross-price elasticities of demand. In addition, there is a potentially serious internal consistency issue thatwe face whenestimating log-lineardemand functionsusing aggregate data. Namely, the aggregate demand function may well depend on more than aggregate income. If we only include an aggregate income variable, estimates may suffer from “aggregation bias.” 11 Misspecification and aggregation bias is easily demonstrated by taking the log-linear demand equation for an individual, ln Q it D a b ln P t C  ln y it C  t ; transforming it to the level of quantities Q it D exp.a i C  t /.P t / b .y it /  and adding up across individuals, which gives X i Q it D exp.a C t /.P t / b X i .y it /  so that if we take logs again we get ln  X i Q it à D a C t C b ln.P t / C ln  X i .y it /  à : Thus even with this special case, where there is no heterogeneity across individuals other than in their income, estimating a log-linear demand equation using aggregate data will involve estimating a misspecified model. 11 This debate was particularly important for macroeconomists, where it was common practice to estimate a representative agent model using aggregate data. 9.1. Demand System Estimation: Models of Continuous Choice 451 The economics profession searched for models which were internally consistent in the sense that they either only depended on exactly the aggregate analogous data, say P i y it , or in a weaker sensethat they only depended on aggregate data—perhaps the aggregate income but also the variance of income in the population. Doing so was called the study of “aggregability conditions.” The reason to mention this fact is that the study of aggregability provided the motivation for many of the most popular demand system models that are in use today—they satisfy these “aggregability” conditions. One such example is the almost ideal demand system (AIDS) due to Deaton and Muellbauer (1980a). We discuss that model below. 12 Before we do so, however, let us briefly recall the amazingly useful contribution of choice theory to the practical exercise of specifying demand systems. In particular, recall that an indirect utility function V.p;yI#/ is defined as V.p;yI#/ D max q u.qI#/ subject to pq 6 y; so that V.p;yI#/ represents the maximum utility u.qI#/ that can be achieved at a given set of prices and income .p; y/, where p and q may be vectors of prices and quantities, respectively. Choice theory tells us that specifying V.p;yI#/ is entirely equivalent to specifying preferences, provided V.p;yI#/ satisfies some properties. 13 In an amazing contribution, choice theory also tells us that the solution to this constrained optimization problem is described by Roy’s identity: 14 q j .p; yI#/ D @V .p; yI#/ @p j  @V .p; yI#/ @y : On the one hand, this is interesting as a piece of theory. However, it is not just theory—it has an extremely practical implication for anyone who wants to estimate a demand curve. Namely, that we can easily derive parametric demand systems—all we need to do is to write down an indirect utility function and differentiate it. In particular, Roy’s identity allows us to avoid solving the constrained multivariate maximization problem entirely and moreover gives us a very simple method for generating a whole array of differentiated product demand systems. There is a version of Roy’s identity which uses expenditure shares and we shall use this version below. Recall the expenditure share for good 1 is defined as the expenditure on good 1 divided by total expenditure y, w 1 Á p 1 q 1 =y. 12 Historically, there was great focus in the literature on being able to estimate flexible Engle curves from aggregate data. Fairly recently, this tradition has resulted in a number of contributions including the “QuAIDS” model (see Banks et al. 1997; Ryan and Wales 1999). 13 In particular, it must be increasing in y, homogeneous in degree 0 in income and prices, and quasi- concave in income and prices. See your favorite microeconomics textbook, for example, chapter 3 of Varian (1992). 14 This identity is derived by applying the envelope theorem to the Lagrangian expression in the utility- maximization exercise. 452 9. Demand System Estimation In that case, Roy’s identity can be equivalently stated: w j .p; yI#/ Á p j q j .p; yI#/ y D   p j @V .p; yI#/ @p j Ã y @V .p; yI#/ @y à D   @V .p; yI#/ @ ln p j Ã @V .p; yI#/ @ ln y à : Estimating a model using the expenditure share on a good provides exactly the same information as a model of the demand for the good. We can compute own- and cross-price elasticities of demand directly from the expenditure share equation. If the indirect utility function is linear in parameters but involves terms such as ln p j and ln y, then this formulation will tend to provide an algebraically more convenient model for us to work with, as we shall see in the next section. 9.1.2.3 Almost Ideal Demand System AIDS is perhaps the most commonly used differentiated product demand system (Deaton and Muellbauer 1980a). AIDS satisfies a nice aggregability condition. Specifically, if we take a lot of consumers behaving as predicted by an AIDS model and aggregate their demand systems, the result is itself an AIDS demand system. The relevant parameters of an AIDS specification are also quite easy to estimate and the estimation process requires data that are normally available to the analyst, namely prices and expenditure shares. In AIDS, the indirect utility function V.p;yI#/ is assumed to be V.p;yI#/ D ln y  ln a.p/ ln b.p/  ln a.p/ ; where the functions a.p/ and b.p/ are sometimes described as “price indices” since they are (parametric) functions of underlying price data: ln a.p/ D ˛ 0 C J X kD1 ˛ k ln p k C J X kD1 J X j D1  jk ln p k ln p j and ln b.p/ D ln a.p/ C ˇ 0 J Y kD1 p ˇ k k : Applying Roy’s identity for the expenditure share for product j gives w j D   @V .p; yI#/ @ ln p j Ã @V .p; yI#/ @ ln y à D ˛ j C J X kD1  jk ln p k C ˇ j ln  y P à ; [...]... parts and servicing for photocopiers and went to the U.S Supreme Court: Kodak v Image Technical Services, 504 U.S 451 (1992) Not 458 9 Demand System Estimation 9.1.3.2 Homogeneity Choice theory suggests that individual demand functions will be homogeneous of degree 0 in prices and income That restriction implies that if we multiply all prices and income by a constant multiple, the consumer’s demand will... If an individual’s demand depends on their level of income, then this assumption must be violated since that is telling you that income should not drop out and you will probably prefer to work with an alternative functional form For example, Berry et al (1995) believe that the demand for a type of car will depend on a consumer’s level of income and work with the natural logarithm formulation, vj y pj... two firms or “shops” that are respectively located at L1 and L2 and that the firms’ locations and their prices are each for the moment fixed Without loss of generality we assume that L1 6 L2 The situation can be represented in figure 9.3 Consumers “live” in different locations and are therefore differentiated by their closeness to locations L1 and L2 For the purposes of simplicity, we assume that each consumer... Thus, given the uniform density of the consumers, the demand functions for shops 1 and 2 respectively take the form:32 Z x D1 p1 ; p2 I L1 ; L2 / D f Li / dLi D S x 0  à S.p2 p1 / L 1 C L2 D CS ; 2t L2 L1 / 2 Z 1 D2 p1 ; p2 I L1 ; L2 / D f Li / dLi D S.1 x/ x  à p2 p1 L 1 C L2 DS 1 : 2t L2 L1 / 2 The demands depend on the prices of both shops and on the location of both shops For equal prices, firm... Lawrence Klein and Herman Rubin 454 9 Demand System Estimation so that the demand elasticities can be computed as 8 ˆ @ ln qj D @ ln wj 1 ˆ < @ ln p @ ln pk k Áj k D ˆ @ ln qj @ ln wj ˆ : D @ ln pk @ ln pk if j D k; if j ¤ k: Differentiating the AIDS expenditure share equation yields @ ln wj D @ ln pk jk wk ˇj wj and therefore we can see that the own- and cross-price elasticities of demand depend on... the other hand, if prices double and aggregate income doubles but only because a few people increased their income by a very large amount, then aggregate demand may change The people who experienced the income increase will be able to afford more goods than before because their income more than doubled while prices only doubled On the other hand, the rest of the population would be able to afford fewer... composite commodity as an outside good and in fact it is often useful to think of it as the good money We will normalize the price of the outside 23 See Lancaster (1966) and Gorman (1956) There are also, of course, classic individual studies of demand which predate both Lancaster and even Gorman and which use characteristics of products to control for quality differentials For example, Hotelling (1929) uses... akin to the AIDS and Translog style models is provided in Davis (2006b) For a very good introduction, see Pudney (1989) See also a number of classic contributions to the literature in Manski and McFadden (1981) 464 9 Demand System Estimation good to 1, p0 D 1, which we can do without loss of generality since we have the freedom to choose the units of the outside good Formally, for a standard discrete... income and prices in all of the conditional indirect utilities Formally, this means that vj y pj I.j > 0/; wj I Âi / D ˛y C vj pj I.j > 0/; wj I Âi / N for j D 0; 1; : : : ; J: N If so, since maxkD0;:::;J vi k D ˛yCmaxkD0;:::;J vi k , the resulting demand functions for any given individual will be identical whether we solve the problem on the righthand side or the maximization problem on the left-hand... In each case, the people considering whether to switch are different and, moreover, there can be very different numbers of them For each of these reasons, we do not expect to find symmetry in general aggregate demand equations and therefore generally we will have @QVirgin @QCoke ¤ @pCoke @pVirgin and we may need to estimate both b12 and b21 If we impose this restriction on our estimates, we must be . where p represents prices and q quantities. 10 Formally, Deaton and Muellbauer (1980b) show that there are “correct” price and quantity indices which can be constructed for this process to preserve. the elasticities. For example, the results from table 9.4 suggest that the own-price elasticity of segment demand is 2:6 for premium beer, 2:7 for popular beer, and 2:4 for light beer. These. multiply all prices and income by a constant multiple, the consumer’s demand will not change. For instance, if we double all prices and we double the income, the individual demand for all goods remains