If two people have identical Cobb-Douglas utility functions, then in every Pareto optimal allocation, they must consume all goods in the same proportions as each other.

Question

Quizplus · Accepted Answer

This is a direct application of the First Welfare Theorem, which states that in a competitive equilibrium, the allocation of goods will be Pareto optimal and will also satisfy individual rationality (i.e. each person is at least as well off as they would be in a different allocation). Since the Cobb-Douglas utility function exhibits constant returns to scale and is quasi-concave, it satisfies the conditions for the existence of a competitive equilibrium. In such an equilibrium, the relative prices of goods will be equal to the marginal rates of substitution of each individual. Since the two individuals have identical utility functions, their marginal rates of substitution for goods will be identical as well, meaning that they will consume all goods in the same proportions in every Pareto optimal allocation.

If Two People Have Identical Cobb-Douglas Utility Functions, Then in Every