Question

Questions 1 and 2 refer to a consumer whose equilibrium happens when she maximizes her utility subject to her budget constraint. The consumer loves fast food decides how many pizzas (Product X) and burritos (Product Y) to eat per month based on the following utility function: U=U(x,y)=2×x×y+x^2 where x is the quantity consumed of pizzas per month and y is the quantity consumed of burritos per month. Question 1 Initially, the price per pizza is P_x=$10.00 and the price of one burrito, denoted by P_y, is also $10.00. Furthermore, the monthly consumer’s income, denoted by I, is $100.00. The best affordable bundle is A.(x^*,y^* )=(10,0) B.(x^*,y^* )=(0,10) C.(x^*,y^* )=(10,10) D.(x^*,y^* )=(5,10) E.(x^*,y^* )=(10,5) Question 2 Inflation engenders a new situation where prices and income have doubled, that is, P_x=P_y=$20.00 and I=$200.00 What is the best affordable bundle in this new situation? A.A corner solution B.An interior solution C.(x^*,y^* )=(10,10) D.(x^*,y^* )=(5,10) E.(x^*,y^* )=(10,5)

Answer 1

In Question 1, the consumer's equilibrium occurs when she maximizes her utility while staying within her budget constraint. The utility function provided is U(x, y) = 2xy + x^2, where x represents the quantity of pizzas consumed per month and y represents the quantity of burritos consumed per month.

To determine the best affordable bundle, we need to compare the prices of pizzas and burritos with the consumer's income. In this case, the price per pizza (P_x) is $10.00, the price per burrito (P_y) is also $10.00, and the consumer's monthly income (I) is $100.00.

To find the best affordable bundle, we need to find the combination of pizzas and burritos that maximizes the consumer's utility within her budget constraint.

We can start by setting up the consumer's budget constraint equation:
P_x * x + P_y * y = I
$10.00 * x + $10.00 * y = $100.00

Simplifying the equation:
10x + 10y = 100
Divide both sides by 10:
x + y = 10

Now, we can use the utility function U(x, y) = 2xy + x^2 to find the maximum utility. Since the consumer's utility function is concave (the coefficient of xy is positive), the maximum utility occurs at the point where the marginal utility of x is equal to the marginal utility of y.

To find the marginal utility of x, we differentiate the utility function with respect to x:
dU/dx = 2y + 2x

To find the marginal utility of y, we differentiate the utility function with respect to y:
dU/dy = 2x

Setting the marginal utilities equal to each other:
2y + 2x = 2x
2y = 0
y = 0

Substituting y = 0 into the budget constraint equation:
x + 0 = 10
x = 10

Therefore, the best affordable bundle in this situation is A. (x^*, y^*) = (10, 0).

Moving on to Question 2, the new situation involves doubled prices and income. The price per pizza (P_x) and price per burrito (P_y) are both $20.00, and the consumer's income (I) is $200.00.

We can use a similar approach as in Question 1 to find the best affordable bundle.

Setting up the budget constraint equation:
P_x * x + P_y * y = I
$20.00 * x + $20.00 * y = $200.00

Simplifying the equation:
20x + 20y = 200
Divide both sides by 20:
x + y = 10

Using the utility function U(x, y) = 2xy + x^2, we find the marginal utility of x:
dU/dx = 2y + 2x

And the marginal utility of y:
dU/dy = 2x

Setting the marginal utilities equal to each other:
2y + 2x = 2x
2y = 0
y = 0

Substituting y = 0 into the budget constraint equation:
x + 0 = 10
x = 10

Therefore, the best affordable bundle in this new situation is A. A corner solution, where the consumer only consumes pizzas (x^*, y^*) = (10, 0).