申論題

1. The majority of the study of economics in general consists of solving constrained op- timization problems. In microeconomics, usually a consumer is assumed to be with a set of preferences which can be represented by a utility function. This utility function "measures" how happy she is at any level of consumption of the goods that provide her with satisfaction.

(a) Crystal lives only with two goods, coffee (x₁) and chocolate (x₂). Suppose that Crystal has the following quasi-linear utility function:
u (x₁,x₂) = α₁lh x₁ + α₂x₂.
Each day, Crystal spends w dollars on coffee and chocolate. Suppose that the prices of coffee and chocolate are respectively pi and pz. 'That is, the budget constraint of Crystal can be written as:
p₁x₁+p₂x₂ = w.
Suppose that there is an inner solution. Derive x₁ (p₁, p₂, w) and x₂ (p₁, p₂, w), the demand functions of coffee and chocolate of Crystal. That is, solve the following problem:

and write x₁ and x₂ as functions of p₁, p₂, and I.