Final answer:
To determine the quantity of good 1 that a consumer will buy, we use the utility maximization condition that equates the ratio of marginal utilities to the ratio of prices, along with the budget constraint. The consumer will purchase 50 units of good 1 to maximize their utility given the budget of $100 and the prices of the goods.
Step-by-step explanation:
To solve for the amount of good 1 that the consumer will consume, we need to apply the general rule where the ratio of the prices of the two goods should be equal to the ratio of the marginal utilities of those goods. The utility function given is x₁¹⁰x₂⁹ which implies that the marginal utility of good 1 (MU₁) is 10x₁⁹x₂⁹ and the marginal utility of good 2 (MU₂) is 9x₁¹⁰x₂⁸. With the prices given as $1 for good 1 and $2 for good 2, we set up the ratio rac{MU₁}{MU₂} = rac{P₁}{P₂}.
Apart from this, we also have a budget constraint that the consumer spends their entire budget of $100. Representing the quantity of good 1 as q₁ and good 2 as q₂, the constraint is $1q₁ + $2q₂ = $100. To find the optimal consumption, we set up the following equations: 10q₁⁹q₂⁹/9q₁¹⁰q₂⁸ = $1/$2 and $1q₁ + $2q₂ = $100. Solving these two equations simultaneously gives us the optimal quantities of goods 1 and 2 that the consumer will purchase to maximize their utility subject to the budget constraint.
By simplifying the equation, we can find the amount of good 1 that maximizes utility. In this case, when we simplify we find out that the consumer will purchase 50 units of good 1. Consequently, they will spend the remaining $50 on good 2, acquiring 25 units.