Anne likes to have bagels and orange juice for breakfast. A consumption bundle consists of a quan- tity of bagels, x (number of bagels), and a quantity of orange juice, y (liters of orange juice). For all of the following questions you can assume that the consumption set is R2 . This means that bagels and orange juice can be consumed in any non-negative quantity. Anne has a wealth of $10 to spend. At supermarket-ABC the price of bagels is pX = 2$/bagel and the price of orange juice is pY = 4$/liter. At supermarket-XYZ the price of bagels is qX = 4$/bagel and the price of orange juice is qY = 2$/liter. Supermarket-ABC is on the East Side; supermarket-XYZ is on the West Side; Anne lives in the mid- dle. Suppose that Anne can travel for free to either one of the two supermarkets, but she can only go to one of them to do her shopping. Anne knows what the prices of bagels and orange juice are at both supermarkets before she decides where to shop. This means that the budget set is now the set of all consumption bundles that Anne can afford given her wealth, and given the restriction that she can shop at only one of the two supermarkets. Draw Ann’s budget set in an appropriate diagram (this is just repeating the exercise from Problem Set 2). Now suppose that Anne’s preferences over consumption bundles are represented by the utility function: u(x, y) = xay(1−a) , where x is the quantity of bagels, y is the quantity of orange-juice, and 0 < a < 1 is a parameter of her utility function. Suppose that a = 3/5. In which supermarket should Anne do her shopping now? For what value of a is Anne indifferent about which supermarket she shops at?
Anne likes to have bagels and orange juice for breakfast. A consumption bundle consists of a quan- tity of bagels, x (number of bagels), and a quantity of orange juice, y (liters of orange juice). For all of the following questions you can assume that the consumption set is R2 . This means that bagels and orange juice can be consumed in any non-negative quantity. Anne has a wealth of $10 to spend. At supermarket-ABC the price of bagels is pX = 2$/bagel and the price of orange juice is pY = 4$/liter. At supermarket-XYZ the price of bagels is qX = 4$/bagel and the price of orange juice is qY = 2$/liter. Supermarket-ABC is on the East Side; supermarket-XYZ is on the West Side; Anne lives in the mid- dle. Suppose that Anne can travel for free to either one of the two supermarkets, but she can only go to one of them to do her shopping. Anne knows what the prices of bagels and orange juice are at both supermarkets before she decides where to shop. This means that the budget set is now the set of all consumption bundles that Anne can afford given her wealth, and given the restriction that she can shop at only one of the two supermarkets.
- Draw Ann’s budget set in an appropriate diagram (this is just repeating the exercise from Problem Set 2).
- Now suppose that Anne’s preferences over consumption bundles are represented by the utility function:
u(x, y) = xay(1−a) ,
where x is the quantity of bagels, y is the quantity of orange-juice, and 0 < a < 1 is a parameter of her utility function.
- Suppose that a = 3/5. In which supermarket should Anne do her shopping now?
- For what value of a is Anne indifferent about which supermarket she shops at?
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Setting pX = qX and pY = qY, we have:
2x + 4y = 4x + 2y
WHY? This is just the equalization of the two budget lines. What does the prices have to do with it?
2x + 4x = 10
x = 2
It doesn't look correct: 6x = 10 is not x = 2
So the price ratio at which Anne is indifferent is 2 bagels per dollar and 0.5 liters of orange juice per dollar, which corresponds to a slope of -2 in the budget line. To find the corresponding value of a, we need to solve for a in the equation:
-2 = -pX/pY = -2/4 = -1/2 = (1-a)/a
Solving for a, we get:
1/2a = 3/2
a = 2/3
Not clear how the logic and the calculation
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