1.Write down Carlos’s demand for apples and bananas as a function of his income
ID: 1194123 • Letter: 1
Question
1.Write down Carlos’s demand for apples and bananas as a function of his income (M), price of apples (pa), and price of bananas (pB).
2. Suppose M = 100 and pa = pb = 1. How many apples and bananas would Carlos buy? Denote the quantities by wa and wb.
3. Immediately after Carlos bought wa apples and wb bananas (but before he eats them), the price of bananas increases to pb. Assume Carlos is free to make further trades at the new prices, pa and pb. What is the Carlos demand for apples and bananas?
Hint 1: it should depend on wa, wb, pa, and pb.
Hint 2: your answer to question 6 should be really useful.
Hint 3: What income would allow Carlos to purchase wa apples and wb bananas.
4. Would Carlos be better off or worse off as result of the increase of price of bananas? Show this graphically on an indifference curve diagram.
5. Calculate the change the quantity demanded of bananas due to the substitution effect, income effect and the total effect, when the price of bananas increases from pb=1 to pb=2. (For this question, forget about Carlos engaging in further trades after buying apples and bananas.
Explanation / Answer
1)
The utility function is
U(a, b) = a3b2.
The constrained maximization problem for each of the agents can be written as
maxa,b U(a,b) subject to Paa + Pbb = M
The Lagrangian for this problem is
L(a, b) = u(a, b) – (Paa + Pbb – M)
i.e. L(a, b) = a3b2 – Paa - Pbb + M
The three necessary conditions for the solution are:
(a) L(a, b)/a = 0
(b) L(a, b)/b = 0
(c) Paa + Pbb = M
Solving (a), we have
L(a, b)/a = 0
3a2b2 – Pa + 0 + 0 = 0
3a2b2 – Pa = 0
3a2b2 = Pa
= 3a2b2/Pa …..(A)
Solving (b), we have
L(a, b)/b = 0
2a3b – Pb + 0 + 0 = 0
2a3b – Pb = 0
2a3b = Pb
= 2a3b /Pb …..(B)
From (A) and (B), we have
3a2b2/Pa = 2a3b /Pb
b = 2Paa/3Pb
Inserting the above result in the condition (c) solving for a, we have
Paa + Pb (2Paa/3Pb) = M
Paa + 2Pbb/3 = M
5Paa = 3M
a = 3M/5Pa ....(i)
which is the demand function of apples.
Inserting (i) in the condition (c) and solving for y, we have
Paa + Pbb = M
Pa(3M/5Pa) + Pbb = M
3M/5 + Pbb = M
Pbb = M – 3M/5
Pbb = 2M/5
b = 2M/5Pb ....(ii)
which is the demand function of bananas.
2.
When Pa = 1, Pb = 1, and M = 100, then the equilibrium quantity of apples is
a = 3(100)/5(1) = 60
and the equilibrium quantity of bananas is
b = 2(100)/5(1) = 40
Since the quantities are to be denoted as wa and wb, we have
wa = 60
wb = 40
3.
Carlos has not eaten 60 apples and 40 bananas. The price of bananas has increases to pb (where pb>1). Carlos can trade his uneaten bananas and apples. Therefore, his income is
M = 60Pa + 40pb = 60+ 40pb
Therefore, his new demand for apples is
New wa = 3(60+ 40pb)/5(1) = 36 + 24pb
His new demand for apples is
New wb = 2(60+ 40pb)/5pb = 16 + 24/pb
4.
Old U = (3M/5Pa)3(2M/5Pb)2
New U = (36 + 24pb)3(16 + 24/pb)2
Since pb>1, new U is greater than old U. Therefore, Carlos is better off.
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