Answered: marginal utilities

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Existence of representative consumer Suppose households 1 and 2 have one-period utility functions u(c₁) and w(c₂), respectively, where u and w are both increasing, strictly concave, twice-differentiable functions of a scalar consumption rate. Consider the Pareto problem:

Subject to the constraint c¹ + c² = c. Show that the solution of this problem has the form of a concave utility function v_θ(c), which depends on the Pareto weight θ. Show that v_θ(c) = θ_u (c₁) = (1 − θ)w (c₂). The function v_θ(c) is the utility function of the representative consumer. Such a representative consumer always lurks within a complete markets competitive equilibrium even with heterogeneous preferences. At a competitive equilibrium, the marginal utilities of the representative agent and each and every agent are proportional.

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