Final answer:
To find Aidan's best bundle, we can use his utility function and the budget constraint equation. Substituting the given values into the equations, we can find the optimal quantities of goods 1 and 2 that maximize his utility. Aidan's best bundle consists of 179 units of good 1 and 1/3 units of good 2. His demand function for good 1 is x = -360 + 3p₂.
Step-by-step explanation:
A utility function represents a person's preferences for different combinations of goods or services. In this case, Aidan's utility function is U = x² + 2y, where x represents the quantity of good 1 and y represents the quantity of good 2.
To find Aidan's best bundle, we need to maximize his utility subject to his budget constraint. The budget constraint is given by p₁x + p₂y = income, where p₁ and p₂ are the prices of goods 1 and 2 respectively, and income is Aidan's income. In this case, we have p₁ = 1, p₂ = 3, and income = 180.
Given these values, we can now solve for the optimal quantities of goods 1 and 2 that maximize Aidan's utility. Substituting the values of p₁, p₂, and income into the budget constraint equation, we get x + 3y = 180. Rearranging this equation, we have x = 180 - 3y.
Now, we can substitute this expression for x into Aidan's utility function and differentiate with respect to y to find the optimal quantity of good 2, y.
Taking the derivative of Aidan's utility function with respect to y, we get dU/dy = 2 - 6y.
Setting this derivative equal to zero and solving for y, we find that y = 1/3.
Substituting this value of y back into the budget constraint equation, we can solve for x. Plugging in y = 1/3, we get x = 180 - 3(1/3) = 180 - 1 = 179.
Therefore, Aidan's best bundle consists of 179 units of good 1 and 1/3 units of good 2.
The demand function represents the relationship between the price of a good and the quantity of that good that a consumer is willing to purchase. In this case, Aidan's demand function for good 1 can be represented as x = f(p₁, p₂, income). Using the budget constraint equation x + 3y = 180, we can solve for x in terms of p₁, p₂, and income.
Substituting x = 180 - 3y and solving for x, we get x = 180 - 3(180 - p₂) = 180 - 540 + 3p₂ = -360 + 3p₂.
Therefore, Aidan's demand function for good 1 is x = -360 + 3p₂.