3. Suppose John’s preference between goods A and B is represented by a utility f

Question

3. Suppose John’s preference between goods A and B is represented by a utility function u(xA, xB) = xAxB, where xA, xB are amounts of A, B consumed.

I just need E, F, G answered please and thank you~!

(a) Suppose John has 40 units of A and 5 units of B. What’s his utility level?

(b) Draw qualitatively the indifference of John that has the same level of utility as 40 units of A and 5 units of B.

(c) Mary offers to give John 15 units of B if he will give her 25 units of A. Would John have a bundle that he likes better than (40, 5) if he makes this trade?

(d) What is the largest number of good A that Mary could demand from John in return for 15 units of B if she expects him to be willing to trade or at least indifferent about trading?

(e) What’s John’s marginal utility of good x and good y respectively when he has consumption bundle (40, 5)?

(f) What’s John’s marginal rate of substitution when he has consumption bundle (40, 5)?

(g) Suppose the market prices for goods A, B are pA = 1, pB = 2. Imagine John can sell the goods he has, namely 40 units of A and 5 units of B, to a trader at the market prices pA, pB and use the money he makes from selling to purchase back any combination of good A and good B affordable at the market prices pA, pB. What would be John’s optimal repurchase choice?

Explanation / Answer

e. From theory we can say that marginal utility of good x is MUA = B and marginal utility of good y is MUB = A

So,

John’s marginal utility of good x and good y are MUA = 5 and MUB = 40.

f. John’s marginal rate of substitution = MUA / MUB = - 5 / 40 = - 1 / 8.

g. When PA = 1 and PB = 2

then John has income of (40 * 1 + 5 * 2) = $50.

Now,

To maximize utility: MUA / MUB = PA / PB.

From this above mentioned function we can say that B / A = 2 / 1

that means, B = 2A

Now we know that ,

A + 2B = 50

5A = 50

A = 10 and B = 2A = 20.

So the John’s optimal repurchase choice is (10 , 20).

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