Question

Utility Maximization with Non-Monotone Preferences Suppose there are two goods, coffee (C) and tea (T). The consumption set is R 2 +, so both goods can be consumed in arbitrary non-negative quantities. Abdul owns 2000 grams of coffee but does not own any tea. He has no other wealth. The price of coffee is pC = 2 (in Dhs per gram) and the price of tea is pT > 0 (in Dhs per gram). Abdul can sell coffee to earn income, which he can then use to buy tea or coffee. He can also throw coffee away (at no cost). Abdul’s preferences can be represented by the utility function u(c, t) = −(800 − c) 2 − (800 − t) 2 , where c is the quantity of coffee (in grams) and t is the quantity of tea (in grams). (a) In separate diagrams, illustrate (i) the map of Abdul’s indifference curves, (ii) Abdul’s constraint set. (b) How much coffee and tea does Abdul decide to consume when (i) pT = 2, (ii) pT = 3, and (iii) pT = 4? (c) Illustrate Abdul’s demand for coffee as a function of the price of tea (pT > 0) in an appropriate diagram

Answer 1

Given Abdul owns 2000 gm of coffee where each gm is worth P_c = 2 Dhs.

Price of tea: P_t > 0

Initial endowment income = 2000 * = 4000 Dhs

Abdul's preferences are given by:

$u(c,t) = -(800-c)^2 - (800-t)^2$

Abdul's indifference curves are given by:

N-(800-x)2 _ (800-1)2-2000 V(s02 (s00 »)2000 V (soo ) (s00 )- 40000 2000 พี่ 800 x 1500 (800 x)2 (800 y)2-100 1000 500 500 1000 1500 2000 250% Settings to activ 300 indows by coffee esmos

where (800,800) is the bliss point giving the maximum utility

ii)

Abdul's constraint set is given by:

$2C + P_t t \leq 4000$

b)

i)

$P_t = 2, \\ \text {Budget constraint:} 2C + 2T \leq 4000 \Rightarrow C+ T \leq 2000$

Since (800, 800) gives the maximum utility to Abdul and can also throw coffee away (at no cost),

hence he will consume (800,800) bundle.

ii)

$P_t = 3, \\ \text {Budget constraint:} 2C + 3T \leq 4000$

Since (800, 800) gives the maximum utility to Abdul and it has entire endowment income is used,

hence he will consume (800,800) bundle.

iii)

$P_t = 4, \\ \text {Budget constraint:} 2C + 4T \leq 4000 \Rightarrow C + 2T \leq 2000$

Now Abdul can't consume (800, 800) bundle and have to move from the bliss point.

The next better point will be when C = 800 and T = 600

This gives a utility of -40000.