Question

1. Suppose an agent derives utility out of a consumption good and out of a numeraire good that we call “dollar”. The agent initially owns M = 2 dollars and Q = 16 units of the consumption good. We denote this initial bundle P1 = (2, 16). You offer the agent to pay him 2 dollars for a quantity of the consumption good that you leave up to him. The agent is competitive: he supplies the quantity of consumption good that leaves him indifferent with no trading with you. He gives you 7 units of the consumption good. His new consumption bundle is therefore P2 = (4, 9). You keep buying up the consumption good from him generating the following bundles Pi

"table1"

a. Define an indifference curve and explain why the Pi(i = 1, 2, 3, 4) are on the same indifference curve. Suppose now that the agent initially has 2 dollars and 49 units of the consumption good. Again, you offer money for quantities of the good. You obtain:

"table2"

b. Can we rank the bundles Pi and Qj? Why?

c. Depict all the bundles Pi and Qj on a graph, putting M on the horizontal axis.

You are now being told that the agent has 10 dollars, and that the market price of the consumption good is 1 dollar for 6 units.

d. Write the agent’s budget constraint and depict the set of consumption bundles that he can afford.

e. Find the agent’s optimal consumption choice on the graph.

f. What is the marginal utility of the agent equal to at the optimal consumption choice? We know that the utility function of this agent is represented by the function U(M, O) = M + 2√ O

g. Use this relation to find the optimal quantity Q of the consumption good purchased by the agent.

Answer 1

a) Indifference Curve (IC) - An indifference curve is a graph that shows combination of two goods that give aconsumer equal satisfaction and utility, thereby making the consumer indifferent.

All the combination Pi's lie on the same indifference curve because it is given in the question that the consumer is competitive and he supplies the qty in return of money which leaves him indifferent. Hence, then according to the definition of indifference curve, all those combinations must lie on the same IC.

b) Yes, we can obviously rank bundles pi and qi since quantity of atleast one commodity(i.e money in this case) is constant while the qty of the other good varies. For instance at any point, for the same amount of money the bundle qi consists of more of the consumption good and hence both can be ranked.

Moreover, since the consumer always gets a higher amount of the numeraire good, he will always rank the bundle qi higher than pi.

c) Picture of IC curve has been attached for this part.

d) Income = 10, Price of good (per unit) = 1/6

Budget Constraint :- (1/6)O + m <= 10,

where O is the commodity, m represents the amt of money of the numeraire good he holds and 10 is constraint income. LHS represents what he can purchase with his income, while the RHS shows his total income. Expression <= represents the fact that either his exp on both the goods should be less than or equal to income but certainly cannot be greater than income.

Budget constraint is depicted in the figure attached. In the horizontal axis, 10 is the amount of numeraire good that he can achieve the most whereas, in the vertical axis, 60 is the maximum amount highest he can achieve. Joining this two points we get the budget line with slope 1/6 (slope of budget line is Pm/Po). This line segment shows the combinations that are in the purachsing capacity of our consumer.

e) The consumer achieves his equilibrium at the point of tangency between his IC curve and the budget line. At this point slope of IC is equal to the slope of the budget line. Optimal choice is denoted by the bundle  (4,36).

Answer of part (f) and (g) are attached.

[Just to note MUo can be found by differentiating the utility function with respect to o and similarly for good m. Slope of IC is given by MRS which is equal to MUo/MUm]