Question

Consider an agent with perfectly substitutable utility over R +n :U(x 1 ,…,x n )=x 1+⋯+x n​

. The agent has total wealth w>0. 4. The agent faces the price schedule P(x)=∑ i=1nxi
​. Describe the set of optimal consumption bundles. Show that the KKT conditions are satisfied for each optimal bundle.

Answer 1

The utility-maximizing choice on a consumption budget constraint occurs at the point where the budget constraint touches an indifference curve. To find the optimal consumption bundle, we compare the ratio of the marginal utility to the price of good 1 with the marginal utility to the price of good 2 and ensure that the two ratios are equal. The KKT conditions, including non-negativity, optimality, non-satiety, and budget constraint, are satisfied for each optimal bundle.

Step-by-step explanation:

The utility-maximizing choice on a consumption budget constraint occurs at the point where the budget constraint touches an indifference curve. In this case, the agent with perfectly substitutable utility over R +n seeks to maximize their utility function U(x 1,…,x n )=x 1+⋯+x n, subject to the price schedule P(x)=∑ i=1nxi. To find the optimal consumption bundle, we compare the ratio of the marginal utility to the price of good 1 with the marginal utility to the price of good 2 and ensure that the two ratios are equal.

The KKT conditions are satisfied for each optimal bundle. These conditions include:

1. Non-negativity condition: The goods consumed must be non-negative, x i ≥ 0.
2. Optimality condition: The ratio of the marginal utility to price for each good consumed must be equal, ∂U/∂x i = λP(x), where λ is the Lagrange multiplier.
3. Non-satiety condition: The consumer cannot be satiated, meaning that the marginal utility of each good consumed cannot be equal to zero.
4. Budget constraint condition: The total expenditure on goods must not exceed the agent's total wealth, ∑ i=1npi xi = w, where pi is the price of good i and w is the agent's wealth.