CES Utility Function Research Articles

Nonlinear dynamics of a Cournot duopoly game with differentiated products

In this paper we investigate the dynamics of a Cournot duopoly game with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update the quantity produced in each period. As in Ahmed et al. (2015), the demand functions are derived from an underlying CES utility function. The present analysis reveals that a higher degree of product differentiation may destabilize the Nash equilibrium. Through local analysis we provide conditions for the stability of the market equilibrium and through global analysis we investigate some bifurcations which cause qualitative changes in the structure of the attractors and of their basins as some parameters are allowed to vary. Since a higher degree of product differentiation tends to reduce competition and may generate undesirable fluctuations, an implication of our findings is that a stronger competition could help in stabilizing the unique Nash equilibrium.

An optimal portfolio, consumption-leisure and retirement choice problem with CES utility: a dynamic programming approach

In this paper, we study an optimal portfolio, consumption-leisure and retirement choice problem for an infinitely lived economic agent with a CES utility function. Using the dynamic programming method, we obtain the value function and optimal investment, consumption, leisure, and retirement strategies in analytic form. Numerically we observe that the threshold retirement wealth level is an increasing function with respect to the elasticity of substitution.

The Intergenerational Transfer Effects on Sport Activity

The intergenerational transfer effects from parents to their children are strong. If father or mother have been or are active sport exercisers, the probability that children are also active sport exercisers is rather large. We are testing the intergenerational transfer hypothesis using a Finnish data set collected in 2007. The survey allows not only studying intergenerational transfer effect in sport but also in movies at a cinema and classical music concerts participation. Consumers’ preferences are modelled using a CES utility function. If sport (movies, classical music) are normal goods, higher incomes result in higher demand but lost income due to time spent in these leisure activities should decrease the demand. It turns out that lost income (Linder’s disease) is important. Older persons who are already pensioners do not suffer from Linder’s disease are more active sport, movies and classical music consumers than younger (e.g. 55 – 64 years old). In Finland women are more active in any leisure activity.

A non-calculus approach to solving the utility maximization problem using the Cobb-Douglas and CES utility function

This paper proposes a new original non-calculus method to solving the utility maximization problem using the Cobb-Douglas and the CES utility functions, and incorporating the weighted arithmetic-geometric-mean inequality (weighted AM-GM inequality) and Jensen’s inequality. Instead of using calculus, the substitution method or the Lagrange multiplier method, the maximum utility and global maximizer for the case of the Cobb-Douglas and CES utility functions are derived in a direct way. The new method does not require checking first and second order conditions, which the substitution method and the Lagrange multiplier method normally require.

On Bertrand duopoly game with differentiated goods

The paper investigates a dynamic Bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period. The demand functions are derived from an underlying CES utility function. We investigate numerically the dynamical properties of the model. We consider two specific parameterizations for the CES function and study the Nash equilibrium and its local stability in the models. The general finding is that the Nash equilibrium becomes unstable as the speed of adjustment increases. The Nash equilibrium loses stability through a period-doubling bifurcation and the system eventually becomes chaotic either through a series of period-doubling bifurcations or after a Neimark–Sacker bifurcation.

Us and Them: Distributional Preferences in Small and Large Groups

We analyze distributional preferences in games in which a decider chooses the provision of a good that benefits a receiver and creates costs for a group of payers. The average decider takes into account the welfare of all parties and has concerns for efficiency. However, she attaches similar weights to small and large groups so that she neglects large provision costs that are dispersed among many payers. This holds regardless of whether the decider benefits from the provision or not. A CES utility function which rationalizes average behavior implies altruism in bilateral situations and welfare-damaging actions when costs are dispersed.

Coalition-enhancing fiscal policies in an open economy: A CES framework of Gale’s transfer paradox

The motivation of our paper comes from David Gale’s seminal work in 1974. He constructed an example of the “transfer paradox” based on three Leontief functions. The transfer paradox is that when there is a set of agents in the home country and that the home country is trading with other countries, then certain public lump-sum tax transfer plans could make all agents in the home country better off. Our contributions are as follows. First, we show that such an example can be constructed with three smooth CES utility functions. Second, we establish the three crucial conditions for the existence of the transfer paradox: (1) the donor (a taxpayer) has stronger preference for the foreign good than the recipient; (2) the donor is ex-ante wealthier than the recipient; (3) the elasticity of substitution of the foreign country’s preference is strictly less than one.

Heterogeneous Consumers in the Monopolistically Competitive Setting

The paper investigates distributional effects and market structure in a one-sector model of monopolistic competition with heterogeneous consumers. By using the CES utility function depending on consumer's personal income the paper shows how the equilibrium prices, firm size and number of firms depend upon income distribution and intensity of competition. The proposed model extends the traditional Dixit-Stiglitz approach and has a wider range of applications.

Currency intervention and consumer welfare in an open economy

This paper investigates whether China can benefit from a trade surplus in one period, using it to pay off the debt in the next period by manipulating the exchange rates. If the marginal utility of income is nonincreasing in the exchange rate, then the equilibrium exchange rates that yield a trade balance in each period maximize the total utility over two periods, regardless of the interest rate. Numerical examples using the Cobb–Douglas and CES utility functions illustrate the main proposition.

Continuity of a model with a nested CES utility function and Bertrand competition

In recent years models with a nested constant elasticity of substitution utility function and heterogeneous firms involved in some form of competition have become popular in the international trade literature. This paper considers one particular model of this class — with firms competing in prices — and shows continuity of the model as the elasticity of substitution between goods goes to infinity. This result contrasts with the conjecture of prior literature. Continuity of the model ensures consistency of its outcomes when the elasticity of substitution approaches infinity. Therefore, researchers who were reluctant to use this model because of the lack of proof of continuity can now rely on this paper’s result to employ the model in their research.

Getting normalization right: Dealing with ‘dimensional constants’ in macroeconomics

We contribute to a recent literature on the normalization, calibration and estimation of CES production functions. The problem arises because CES ‘share’ parameters are not in fact shares, but depend on underlying dimensions—in other words they are ‘dimensional constants’. It follows that such parameters can neither be calibrated nor be estimated unless the choice of units is made explicit. We use an RBC model to demonstrate two equivalent solutions. The standard one expresses the production function in deviation form about some reference point, usually the steady state of the model. Our alternative, ‘re-parameterization’, expresses dimensional constants in terms of a new dimensionless (share) parameter and all remaining dimensionless ones. We show that our ‘re-parameterization’ method is equivalent and arguably more straightforward than the standard normalization in deviation form. We then examine a similar problem of dimensional constants for CES utility functions in a two-sector model and in a small open economy model; then re-parameterization is the only solution to the problem, showing that our approach is in fact more general.

AGGLOMERATION AND GROWTH WITH ENDOGENOUS EXPENDITURE SHARES*

ABSTRACT We develop a New Economic Geography and Growth model which, by using a CES utility function in the second‐stage optimization problem, allows for expenditure shares in industrial goods to be endogenously determined. The implications of our generalization are quite relevant. In particular, we obtain the following novel results: (1) two additional nonsymmetric interior steady states emerge for some intermediate values of trade costs. These steady‐states are stable if the industrial and the traditional goods are either good or very poor substitutes, while they are unstable for intermediate (yet lower than one) values of the intersectoral elasticity of substitution. In the latter case, the model displays three interior steady states—the symmetric and the core‐periphery allocations—which are stable at the same time; (2) agglomeration processes may always take place, whatever the degree of market integration, provided that the traditional and the industrial goods are sufficiently good substitutes; (3) the regional rate of growth is affected by the interregional allocation of economic activities even in the absence of localized spillovers, so that geography always matters for growth and (4) the regional rate of growth is affected by the degree of market openness: in particular, depending on whether the traditional and the industrial goods are good or poor substitutes, economic integration may be, respectively, growth‐enhancing or growth‐detrimental.

Measuring the gains from trade under monopolistic competition

Abstract Three sources of gains from trade under monopolistic competition are (i) new import varieties available to consumers; (ii) enhanced efficiency as more productive firms begin exporting and less productive firms exit; (iii) reduced markups charged by firms due to import competition. The first source of gains can be measured as new goods in a CES utility function for consumers. We argue that the second source is formally analogous to the producer gain from new goods, with a constant‐elasticity transformation curve for the economy. We suggest that the third source of gain can be measured using a translog expenditure function for consumers, which, in contrast to the CES case, allows for finite reservation prices for new goods and endogenous markups.

Analysis of Labor Tax Progression Under Heterogeneous Domestic Labor Markets and Flexible Outsourcing

What are the impacts of labor tax reform on wage setting and employment to keep the relative tax burden per low-skilled and high-skilled workers constant in the case of heterogeneous domestic labor markets, i.e. imperfect competition in low-skilled labor and perfect competition in high-skilled labor in the presence of outsourcing? A higher degree of tax progression by raising the wage tax and the tax exemption for the low-skilled workers will decrease the wage rate and increase labor demand of low-skilled workers, whereas it will decrease (increase) employment of high-skilled workers in CES utility function when the elasticity of substitution between consumption and leisure is higher (lower) than one. A higher degree of wage tax progression for the high-skilled worker will have no effect on the high-skilled wage in the presence of CES utility function.

CALIBRATION OF CES FUNCTIONS FOR REAL-WORLD MULTISECTORAL MODELING

We show how to calibrate CES production and utility functions when indirect taxation affecting inputs and consumption is present. These calibrated functions can then be used in computable general equilibrium models. Taxation modifies the standard calibration procedures since any taxed good has two associated prices and a choice of reference value units has to be made. We also provide an example of computer code to solve the calibration of CES utilities under two alternate normalizations.

Impacts of Labor Taxation with Perfectly and Imperfectly Competitive Domestic Labor Markets Under Flexible Outsourcing

What are the impacts of labor tax reform on wage setting and employment to keep the relative tax burden per low-skilled and high-skilled workers constant in the case of heterogenous domestic labor markets, i.e. imperfect competition in low-skilled labor and perfect competition in high-skilled labor in the presence of outsourcing? A higher degree of tax progression by raising the wage tax and the tax exemption for the low-skilled workers will decrease the wage rate and increase labour demand of low-skilled workers, whereas it will decrease (increase) employment of high-skilled workers in CES utility function when the elasticity of substitution between consumption and leisure is higher (lower) than one. A higher degree of wage tax progression for the high-skilled worker will have no effect on the high-skilled wage in the presence of CES and C-D utility function so this will have no total employment effects.

Time and Income Poverty: An Interdependent Multidimensional Poverty Approach with German Time Use Diary Data

Income as the traditional one dimensional measure in well-being and poverty analyses is extended in recent studies by a multidimensional poverty concept. Though this is certainly a progress, however, two important aspects are missing: time as an important dimension and the interdependence of the often only separately counted multiple poverty dimensions. Our paper will contribute to both aspects: First, we consider time – and income – both as striking and restricting resources of everyday activities and hence account for time and income as important multiple poverty dimensions. Second, the interdependence of the poverty dimensions will be evaluated by the German population to allow an advanced approach to understand possible substitution effects and the respective trade offs between the dimensions. Referring to the time dimension, we follow Sen's capability approach with its freedom of the living conditions' choice and social exclusion and argue, that restricted time might exclude from social participation. In particular, restricted genuine, personal leisure time (not entire leisure time) in particular is associated with a restricted social participation. The crucial question then is how to measure the substitution between income and such genuine leisure time. In our analysis we consider the country population’s valuation with data from the German Socio-Economic Panel and estimate the substitution by a CES-utility function of general utility/satisfaction. Given this quantification we disentangle time, income and interdependent multidimensional poverty regimes characterising the working poor. In addition, we quantify further socio-economic influences for each interdependent multidimensional poverty regime by a multinomial logit based on time use diary data of the German Time Use Study 2001/02. One striking result for Germany: the substitution between time and income is significant and we find an important fraction of time poor who are unable to substitute their time deficit by income. These poor people are ignored within the poverty and well-being as well as the time crunch and time famine discussion so far.

Pareto-Efficiency and Endogenous Fertility: A Simple Model

Pareto-efficiency of the competitive equilibrium is studied for a simple economy with overlapping generations and endogenous fertility. For CES utility and production functions, the properties of the economy are closely related to the two elasticities of substitution. The competitive equilibrium exists and is unique if the sum of the two elasticities is not less than one. For certain parameters, the equilibrium is both in under-accumulation and inefficient. When the sum of the two elasticities is greater than two, an equilibrium converging in under-accumulation is Pareto-efficient.

Consumer's Surplus Without Computer

Compensated demand functions are useful tools to calculate and manipulate not only equivalent and compensating variations, but also ordinary demand functions. As a bonus I will show that the CES utility function admits welfare analysis without computer aid, contrary to the suggestion of Tohamy and Wilson Mixon (2004).

Giving According to GARP: An Experimental Test of the Consistency of Preferences for Altruism

Subjects in economic laboratory experiments have clearly expressed an interest in behaving unselfishly. They cooperate in prisoners’ dilemma games, they give to public goods, and they leave money on the table when bargaining. While some are tempted to call this behavior irrational, economists should ask if this unselfish and altruistic behavior is indeed self-interested. That is, can subjects’ concerns for altruism or fairness be expressed in the economists’ language of a well-behaved preference ordering? If so, then behavior is consistent and meets our definition of rationality. This paper explores this question by applying the axioms of revealed preference to the altruistic actions of subjects. If subjects adhere to these axioms, such as GARP, then we can infer that a continuous, convex, and monotonic utility function could have generated their choices. This means that an economic model is sufficient to understand the data and that, in fact, altruism is rational. We do this by offering subjects several opportunities to share a surplus with another anonymous subject. However, the costs of sharing and the surplus available vary across decisions. This price and income variation creates budgets for altruistic activity that allow us to test for an underlying preference ordering. We found that subjects exhibit a significant degree of rationally altruistic behavior. Over 98% of our subjects made choices that are consistent with utility maximization. Only a quarter of subjects are selfish money-maximizers, and the rest show varying degrees of altruism. Perhaps most strikingly, almost half of the subjects exhibited behavior that is exactly consistent with one of three standard CES utility functions: perfectly selfish, perfect substitutes, or Leontief. Those with Leontief preferences are always dividing the surplus equally, while those with perfect substitutes preferences give everything away when the price of giving is less than one, but keep everything when the price of giving is greater than one. Using the data on choices, we estimated a population of utility functions and applied these to predict the results of other studies. We found that our results could successfully characterize the outcomes of other studies, indicating still further that altruism can be captured in an economic model.