Aggregation with a fixed distribution of income

Two issues that have become increasingly important while estimating the parameters of aggregate demand functions to study firm behavior are the endogeneity of marketing activities (typically, price) and heterogeneity across consumers in the market under consideration. Ignoring these issues in the estimation of the demand function parameters can lead to biased and inconsistent estimates for the effects of marketing activities. Endogeneity and heterogeneity have achieved prominence in large measure because of the increasing popularity of logit models to characterize demand functions using aggregate data. The logit model accounts for purchase incidence and brand choice by including a “no-purchase” alternative in the consumer's choice set. This allows for category sales to change as a function of the marketing activities of brands in the category.There are three issues with using the logit model with the no-purchase option to characterize demand when studying competitive interactions among firms. (1) The marketing literature dealing with brand choice behavior at the consumer level has found that the IIA restriction is not appropriate, as each brand in the choice set is more similar to some brands than it is to others. (2) Studies have found that the purchase incidence decision is distinct from the brand choice decision. Hence, it may not be appropriate to model the no-purchase decision as just another alternative in the choice set with the IIA restriction holding across all brands and the no-purchase option. (3) Even if the distinction between the purchase incidence and brand choice decisions is accounted for via, for example, a nested logit specification, accounting for the purchase incidence decision with aggregate data requires assumptions for computing the share of the no-purchase alternative which is otherwise unobserved.In this paper, we propose a probit model as an alternative to the logit model to specify the aggregate demand functions of firms competing in oligopoly markets. The probit model avoids the IIA property that affects the logit model at the individual consumer level. Furthermore, the probit model can naturally account for the distinction between the purchase incidence and brand choice decisions due to the general covariance structure assumed for the utilities of the alternatives. We demonstrate how the parameters of the proposed model can be estimated using aggregate time series data from a product market. In the estimation, we account for the endogeneity of marketing variables as well as for heterogeneity across consumers.Our results indicate that both endogeneity as well as heterogeneity need to be accounted for even after allowing for a non-IIA specification at the individual consumer level. Specific to our data, we also find that ignoring endogeneity has a bigger impact on the estimated price elasticities than ignoring the effects of heterogeneity. A comparison of the elasticities obtained from the probit model with those from the corresponding logit specification indicates that the range of elasticities obtained from the probit model across brands is larger than that obtained from the logit. The results have implications for issues such as firm-level pricing.In addition to specifying a probit model and providing comparisons with the logit model, the paper also addresses the third issue raised above. We propose a simple alternative to the purchase incidence/brand choice specification by decomposing the demand for a brand into a category demand equation and a conditional brand choice share equation. We provide a comparison of results from this specification to those from the specification that includes the no-purchase alternative and find that estimated elasticities are sensitive to the specification used. We also estimate the demand function parameters using a traditional specification such as the double-logarithmic model. Here, we find that the estimated elasticities could be signed in such a manner as to be not useful for firm-level pricing decisions.One of the key limitations of the proposed model is that while it accounts for the purchase incidence and brand choice decisions of households, it does not account for differences across consumers in their purchase quantities. The model and analysis are best suited for product categories in which consumers typically make single-unit purchases. Another limitation is more practical in nature. While recent advances have been made in computing probit probabilities, it could nevertheless be a challenge to do so when the number of alternatives is large.

We are now in a position to return to our main theme, from which we broke off at the end of Book I in order to deal with certain general problems of method and definition. The ultimate object of our analysis is to discover what determines the volume of employment. So far we have established the preliminary conclusion that the volume of employment is determined by the point of intersection of the aggregate supply function with the aggregate demand function. The aggregate supply function, however, which depends in the main on the physical conditions of supply, involves few considerations which are not already familiar. The form may be unfamiliar but the underlying factors are not new. We shall return to the aggregate supply function in chapter 20, where we discuss its inverse under the name of the employment function. But, in the main, it is the part played by the aggregate demand function which has been overlooked; and it is to the aggregate demand function that we shall devote Books III and IV.

This paper studies the monotonicity of individual and market demand with the aid of the indirect utility function. We identify sufficient (and in a sense, necessary) conditions on an agent's indirect utility which will guarantee that he has a monotonic demand function. Our conditions also point to a natural way of extending the result of Hildenbrand (1983). Hildenbrand showed that market demand is monontonic if the income distribution has a downward sloping density, even though individual agents' demand function might violate monotonicity. Using the indirect utility function, we introduce a measure of violations of individual monotonicity that allows us to identify a larger class of density functions that will generate a monotonic market demand.

This empirical investigation is based on consumers' reports about their weekly purchases of everyday commodities. From the data we have corresponding price and consumption vectors of a number of individual demand functions and one mean demand function. By comparison with simulated demand functions, the correspondence between the price and consumption vectors is shown to be systematic and in agreement with the axioms of revealed preference.

1. To evaluate the effects of any economic policy upon a single individual, it is necessary to be able to write some well-defined cardinal index of that individual's ordinal utility solely in terms of observable information: prices pi, quantities consumed Xi, money expenditure Y and the parameters of his demand function. Consumer's surplus measures and index numbers all represent attempts at achieving this aim, but all suffer from wellknown weaknesses which render their general application rather suspect.' Many economists have argued that since no other techniques are available, such measures, principally consumer's surplus, must nevertheless be resorted to if any practical policy recommendations are to be made at all. However, an alternative does exist. In this paper, we show how an exact measure may be derived, free of all undesirable restrictions, capable of defining, for the individual consumer, a precise ordering of social states, however close in the ordering any alternative states may be. A numerical example illustrates the superiority of the new approach over the more traditional welfare indicators. 2. In essence the problem is as follows. Suppose that the individual consumer possesses a preference function U(X1, ..., XJ), that we do not know its functional form, but that we do know the commodity demand functions associated with it. Utilizing the first-order maximizing conditions, we can derive expressions for local changes in consumer satisfaction, dU = A)pidXi, or equivalently dU = A(dY-ZXidpi) where A is the marginal utility of money, a non-observable. Since A is always positive, for small changes we can simply use as a welfare index, YpidXi or dY-ZXidpi. Either version can be interpreted as measuring an infinitesimal change in consumer's surplus. Alternatively, the former expression may be called a quantity whereas the latter, assuming that expenditure is unchanged, may be labelled a price variation . For discrete changes in the variables, however, such measures are no longer applicable. Quantity or price variations written in Laspeyres and Paasche form are subject to regions of ignorance . The addition of the well-known consumer's surplus triangle in an attempt to construct an approximate discrete welfare indicator suffers from the same drawback.2 3. We could, however, do the following. Substitute into the above differential equations, the observed inverse and ordinary demand functions for pi and Xi respectively. Then the problem becomes one of finding some integrating factor A, equal to the marginal utility of money, such that the differential equations are integrable.3 In other words, if we can find such a A, we can retrieve the underlying preference function from our knowledge of the consumer's demand functions. Since we have assumed U to exist in the first place, we know that A will also exist and that it will be a function of the variables and parameters describing the individual's demand functions. Although i itself is not directly observable, it is, in principle, derivable from information that is known. Further, an infinity of i's could be derived, each one generating a preference function that is a monotonic transformation of any other in the family. The drawback to this procedure is that there exist no general rules for deriving even one of the many integrating factors. As a result, a long and tedious search procedure may have to be endured.

The equivalence of markets and games concerns the relationship between two sorts of structures that appear fundamentally different -- markets and games. Shapley and Shubik (1969) demonstrates that: (1) games derived from markets with concave utility functions generate totally balanced games where the players in the game are the participants in the economy and (2) every totally balanced game generates a market with concave utility functions. A particular form of such a market is one where the commodities are the participants themselves, a labor market for example. But markets are very special structures, more so when it is required that utility functions be concave. Participants may also get utility from belonging to groups, such as marriages, or clubs, or productive coalitions. It may be that participants in an economy even derive utility (or disutility) from engaging in processes that lead to the eventual exchange of commodities. The question is when are such economic structures equivalent to markets with concave utility functions. This paper summarizes research showing that a broad class of large economies generate balanced market games. The economies include, for example, economies with clubs where individuals may have memberships in multiple clubs, with indivisibile commodities, with nonconvexities and with non-monotonicities. The main assumption are: (1) that an option open to any group of players is to break into smaller groups and realize the sum of the worths of these groups, that is, essential superadditivity is satisfied and :(2) relatively small groups of participants can realize almost all gains to coalition formation. The equivalence of games with many players and markets with many participants indicates that relationships obtained for markets with concave utility functions and many participants will also hold for diverse social and economic situations with many players. These relationships include: (a) equivalence of the core and the set of competitive outcomes; (b) the Shapley value is contained in the core or approximate cores; (c) the equal treatment property holds -- that is, both market equilibrium and the core treat similar players similarly. These results can be applied to diverse economic models to obtain the equivalence of cooperative outcomes and competitive, price taking outcomes in economies with many participants and indicate that such results hold in yet more generality.

The functional form of the aggregate import demand equation: A comparison of three European economies

This study empirically estimates some critical parameters of the aggregate import demand function for Bangladesh for Bangladesh using quarterly time series data and by applying the Johansen–Juselius multivariate cointegration technique. Cointegration results indicate that the import demand function is dominated by income and relative prices. The income elasticity is significantly positive and exceeds unity indicating that aggregate imports are to be considered as ‘luxury’ goods. The effect of the relative price variable is significantly negative but its elasticity coefficient is less than unity.

The review of the existing literature on empirical money demand functions presented in Chapter 2 suggests that very little confidence is derived from existing single equation estimates, either with respect to the existence of a stable aggregate demand function for real balances, or with respect to the magnitude of the parameter values of such a function, should it exist. In Chapters 3 and 4 newly developed techniques for multiequation analysis are discussed. The purpose of this and subsequent chapters is to apply these techniques to investigations into the existence of long-run aggregate demand functions for real balances. The analyses involve testing for cointegrating vectors among macroeconomic data series and identification of a long-run demand function for real balances among those cointegrating vectors. The identified long-run money demand functions are examined for stability across various sample periods, particularly the periods of “missing money” and those of velocity trend shifts. The robustness of the long-run money demand function is examined with respect to the dimension of the multivariate process and the number of cointegrating vectors. Finally, various techniques of identifying permanent and transitory “economic shocks” are applied to the vector error correction (VECM) models containing the money demand equation, and the resulting dynamic patterns are analyzed.

Testing exact aggregation in income and household characteristics: the effects of aggregation across goods

This paper deals with the analysis of the relation between aggregate demand for a consumption good and the distribution of income across consumers. We obtain sufficient conditions under which changes in income inequality lead to an increase or decrease in the market demand elasticities. The conditions are satisfied for individual demand functions commonly used in economic models, in particular, for the typical demand functions on luxury goods and necessities.

More on Ausubel's Dynamic Double Auction

Non-standard concave utility functions

MONEY DEMAND FUNCTIONS, as used in macroeconomic theory and in applications to monetary policy, have their theoretical underpinnings in models of portfolio optimization for the individual wealth holder. The use of these functions in an aggregative framework, in the absence of specific premises regarding aggregation, suggests the assumption that all cash holders respond in a uniform way to the same set of variables. Such an assumption is open to question. Separable categories of decision makers can be defined and cash holding behavior may vary across categories. For instance, households, non-financial firms and financial institutions may all determine their demands for cash according to the fundamental principles embodied in the standard portfolio models, but attempts to capture this behavior in money demand functions could properly involve the use of different explanatory variables and structural relations for each sector. Put another way, predictions of the aggregate amount of money demanded based on separate functions for each sector could be more accurate than those based on a single aggregative function. Little attention has been paid to this problem in the extensive literature on the demand for money. The small number of hypotheses that are widely used to generate aggregate money demand functions suggests that it is not difficult to evaluate the empirical importance of the problem. That is, functions considered useful at the aggregate level can be tested at the sectoral level. As well as subjecting these hypotheses to the standard statistical tests on data for a particular sector, their performance can be compared to that of money demand functions that reflect specific sectoral characteristics. The thesis abstracted here reports on such an analysis for the non-financial business sector. Quarterly data on manufacturing corporations in the U.S., from 1947 to 1963, were for the study. A priori suspicions turned out to be justified. Two forms of money demand functions widely accepted at the aggregate level, those based on the permanent income hypothesis and the wealth hypothesis, performed badly on this set of data. The third and simplest of the commonly used hypotheses, that money demand depends on income and the interest rate, fared somewhat better, but not as well as those functions that incorporated additional hypotheses specifically relevant to the firm. The latter hypotheses are based on the argument that the portfolio models normally used to determine optimal cash balances involve an array of assets that is too narrowly defined to represent the firm's behavior. The firm's 'portfolio' involves a complex balance sheet. Thus, the rates of return on assets and liabilities other than the standard marketable ones (e.g., accounts receivable and payable, inventories, bank loans and bonded debt), along with other variables that relate the firm's cash flow to its income flow, are considered in the analysis as well.

Recursive aggregate demand and supply functions are used to simulate the ability of the farm sector to adjust during the 1970's to three policy alternatives. Different output demand elasticities and shifts in the supply and demand for farm output were assumed. Within reasonable bounds, agriculture could remain economically viable during the 1970's under policies diverting about 6 percent of potential output. An average of 6 percent was diverted from the market by Government productiqn centroI, storage, and subsidized exports in 1962-69. Returning to a free market immediately or by 1980 would place severe financial strain on the farm sector.