Consumers live on bread x and cheese y. They face the following pricing schedule

Question

Consumers live on bread x and cheese y. They face the following pricing schedule. If consumption is below A loaves of bread then each loaf costs $1. If consumption is A or more then the price falls to $1 2 (on every loaf and not just those in excess of A). The price of cheese is $1 per unit, regardless of the amount consumed.

1. What is the maximum amount of cheese a consumer with a total budget of M can aord if she buys less than A loaves of bread? And if she buys A or more loaves?

2. Draw the budget constraint for someone with total budget of $10, labelling all rele- vant slopes and intercepts. Is the budget set convex? Justify your answer.

3. Raul has preferences represented by the function uR(x; y) = min[x; y] and a total budget of $10.

(a) How many loaves would he buy if he paid a constant price of 1 for every loaf of bread, regardless of the total amount of break purchased? (Hint: Draw Raul's indierence map. Then argue that he will optimally choose to consume equal quantities of cheese as bread: x = y.)
(b) How many loaves would he buy if he paid a constant price of 12 for every loaf of bread, regardless of the total amount of break purchased? (Hint: follow the same line of argument as in the previous question.)
(c) Hence determine how many loaves Raul will buy subject to the nonlinear constraint described in (2.) if A = 5.

4. Now consider Gabriela with the same budget as Raul, but preferences uG(x; y) =min[2x; y]. How does her demand di er from Raul's when facing the nonlinear constraint described in (2.) if A = 5? Give an intuition for this result.

Explanation / Answer

ANSWER:

The Pricing Schedule.

Indifference curves for Perfect Complements are L-shaped or right angled at the kink. Note that the slope of the indifference curve (MRS) for perfect complements is undefined since substitution is not allowed and two commodities are consumed together. The utility function for two commodities that are perfect complements can be given by: U(X,Y) = min{aX, bY}. Where the optimal ratio Y/X = a/b, which is a constant and is unchanged along the indifference curve

In this case the ratio in which x and y are used is 1:1. For a price of 1 and with a budget of $10, a total of quantity of x and y should be 5 each so that the ratio 1:1 is maintained.

b) For a price of 1/2 and with a budget of $10, a total of quantity of x and y should be 10 each so that the ratio 1:1 is maintained.

c) Given that if consumption is A or above A, price is 1/2. Here A = 5. Hence the consumption is 5 or above 5. So price is 1/2. He can buy a maximum of 20 loaves with an income of $10 but to maintain the combination of x:y = 1:1, the consumption will be 10 units for each good,

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