Economics Terms Exercise - Utility Maximization

Consider an individual who is deciding to spend $40 on either coffee or snacks. A cup of coffee (x₁) costs $2, while a bag of snacks (x₂) costs $0.50. This person’s utility function can be represented by the Cobb-Douglas utility function U(X) = x₁^1/2 * x₂^1/2. How many cups of coffee and how many bags of snacks maximize this person’s utility?

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  To solve, first we must find the marginal utility of coffee and of a bag of snacks. Taking the first derivative of U(X) with respect to x₁ yields (½)x₁^(-1/2) * x₂^1/2. Doing the same for x₂ yields (½)x₂^(-1/2) * x₁^1/2.

  Then, the ratio of marginal utilities becomes [(½)x₁^(-1/2) * x₂^1/2] / [(½)x₂^(-1/2) * x₁^1/2]. The ½ in the numerator and denominator cancel, and the resulting expression can be simplified to yield: x₂ / x₁.

  Next, we must set the price ratio equal to the ratio of marginal utilities, which gives us:

  2 / 0.5 = x₂ / x₁.

  Simplifying this equation yields 4x₁ = x₂. At this point, we have the optimal product mix, so now we need to figure out how much the consumer can afford. We know that the budget constraint is $40. To solve, we insert the optimal product mix into the budget constraint.

  The budget constraint is $40 = $2x₁ + $0.50x₂. Inserting the fact that at the optimal point, x₂ = 4x₁, we can put the budget constraint into terms of only x₁ and solve:

  40 = 2x₁ + 0.5(4x₁) = 4x₁. Then, x₁ = 10, and so x₂ = 40.

  The consumer chooses to buy 10 cups of coffee and 40 bags of snacks.

   

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