Vasco's utility function is U=10 X^2 Z. The price of X is...

Vasco's utility function is $U=10 X^{2} Z$. The price of $X$ is $p_{X}=\$ 1,$ the price of $Z$ is $p_{z}=\$ 20,$ and his income is $Y=\$ 300$. What is his optimal consumption bundle? (Hint: See Appendix 4B.). C

Key Concepts

Utility Function

A utility function represents a consumer's preferences by assigning a numerical value to different bundles of goods, indicating the level of satisfaction or happiness derived from each bundle. It is a key tool in understanding consumer choice and decision-making in economics.

Budget Constraint

The budget constraint describes the combinations of goods a consumer can purchase given their limited income and the prices of goods. It is represented by an equation that sets the total expenditure equal to income, thereby limiting the consumer's choices.

Marginal Rate of Substitution (MRS)

The marginal rate of substitution is the rate at which a consumer is willing to give up one good in exchange for an additional unit of another good while keeping their overall utility constant. In the context of maximizing utility, setting the MRS equal to the ratio of the prices of the goods leads to the optimal consumption decision.

Consumer Optimization

Consumer optimization involves selecting the combination of goods that maximizes utility subject to the budget constraint. This process typically entails equating the marginal rate of substitution to the price ratio and ensures that the consumer is allocating their income in the most efficient manner.

Lagrangian Method

The Lagrangian method is a strategy for solving constrained optimization problems by introducing a Lagrange multiplier, which accounts for the constraint. In consumer optimization problems, it is used to find the optimal consumption bundle by combining the utility function with the budget constraint into a single function and finding where its derivatives equal zero.

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