Abstract
Quasi-linear utility functions are widely used in economics and game theory as convenient tools. Quasi-linearity ignores income effects on individual evaluations of alternative choices; indeed, it is captured by a condition of no-income effects on such evaluations. However, income effects are non-negligible relative to agents’ economic activities in many social/economic problems such as in housing markets. In this chapter, we consider how quasi-linearity holds for large incomes for agents. We employ an axiomatic approach to this problem and study its implications and applications to some economic and game theoretic problems.
The author thanks two referees for many helpful comments. He is supported by Grant-in-Aids for Scientific Research No. 26245026, and No.17H02258, Ministry of Education, Science and Culture.
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Notes
- 1.
He used the term “asymptotic quasi-linearity.” We use “approximate quasi-linearity” to emphasize approximation of a utility function including income effects by a quasi-linear utility function.
- 2.
It is indirectly related but relevant to mention Vives [22]; he showed that in economies that possibly have many commodities, the income effects on demand of each commodity become negligible relative to the number of commodities as the number increases to infinity.
- 3.
When X is an infinite set with some topology, under C0, a sufficient condition for C3 is: for any \(y\in Y,\) \(\{(x,0)\in X\times R:(y,0)\succsim (x,0)\}\) is a compact set in \(X\times R.\) This is proved by using the finite intersection property.
- 4.
This is pointed out by a referee. Let a utility function representing \( \succsim \) be given as \(u(x)+c\) over \(R_{+}^{2}.\) When u(x) is a strictly concave function, utility maximization gives a choice of x, independent of an income if it is large. However, if \(u(x)=x^{2},\) then utility maximization gives a corner solution; a change in income affects this corner solution.
- 5.
This proof is given by a referee and is clearer than the original proof by the author.
- 6.
Miyake [13] provided a result (Theorem 2 in p.561) related to this approach. He studied the behavior of “willingness-to-pay” and willingness-to-accept,” and he provided many results on the behavior of these concepts.
- 7.
Kaneko-Ito [10] conducted an equilibrium-econometric analysis to study how utility functions have “significant income effects,” adopting utility functions of the form \( U(x,c)=u(x)+c^{\alpha }\) \((0<\alpha <1).\) It was shown that this \(\alpha \) is bounded away from 1 using rental housing market data in Tokyo. Since incomes of households are distributed over some interval, we do not need the above modification of a utility function.
- 8.
When the set of commodity bundles is infinite, we need some modifications.
- 9.
The set \(I_{S}(N,v)\) is nonempty under some additional condition (e.g., \( v(S)\ge \sum \nolimits _{i\in S}v(\{i\})\)).
- 10.
This strict version is used in Kaneko [8].
- 11.
This term “normality” is motivated by the following observation. Suppose that \(\succsim \) is weakly increasing with respect to \(x\in X=R_{+}.\) Then, the demand function, assumed to exist here, for the commodity in \(X=R_{+}\) is weakly monotonic with an income. Indeed, let \(p>0.\) Let \((x,I-px)\succsim \) \((x^{\prime },I-px^{\prime })\) and \( x>x^{\prime }.\) By C1 and C2, \((x,I-px)\sim (x^{\prime },I-px^{\prime }+\alpha )\) for some \(\alpha \ge 0.\) Let \(I^{\prime }>I.\) Then, since \( I-px<I-px^{\prime }+\alpha ,\) we have \((x,I^{\prime }-px)\succsim \) \( (x^{\prime },I^{\prime }-px^{\prime }+\alpha )\succsim \) \((x^{\prime },I^{\prime }-px^{\prime })\) by C4\(^{NM}\) and C1. This means that the quantity demanded weakly increases when an income increases.
- 12.
The finiteness of \(Z_{o}\) is assumed to have the uniform convergence result in Theorem 9.4.2. Otherwise, we could take the set of all nonnegative integers \(Z_{+}.\)
- 13.
This is a variant of the method of obtaining the existence of a competitive equilibrium from the maximization of the total social surplus, which was first given by Negishi [16].
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Kaneko, M. (2018). Approximate Quasi-linearity for Large Incomes. In: Neogy, S., Bapat, R., Dubey, D. (eds) Mathematical Programming and Game Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-13-3059-9_9
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