Question

Consumer Theory. A consumer has preferences over goods x and y that can be represented by the utility function ?(?,?) = ?+??(?) where ln is the (natural) log function. The consumer has income I (all to be spent on x and y) and the price of x and y are px and py respectively. (You may assume the “at least as good as x” set B(x) is a convex set, so the solution to the consumer’s problem will be a maximum).

a. Solve the consumer’s problem (maximize utility subject to the budget constraint) to derive this consumer’s demand function for x: ?=?(?,??,??).

b. Find the consumer’s income elasticity of demand for good x (as a function of income and prices). What kind of good is x for this consumer?

c. Suppose I = 20 and py = 1. Originally, ???=1 but the price of x then increases to the new price ???=2. Find the substitution and income effect.

d. Calculate the change in Consumer Surplus from the price change.

Answer 1

PArt (A)

substituting in the budget constraint

Part (B)

good x is a normal good since income elasticity of demand is positive

Part (C)

Now price of good y remains the same but price of good x increases from 1 to 2

relative price of good x increases which induces consumer to consume more good y and less of good x while maintaining the same utility level

substitution effect measures the change in the consumption of good that occurs when the consumer moves along the same indifference curve due to a rise in the market price of good.

we have to find the bundle which satisfies new utility maximizing condition and gives the same utility as previous bundle

substitution effect changes consumer bundle from (19,1) to (18.306,2)

increase in price of good x reduces real income of consumer which induces consumer to consume less of both goods

the income effect measures the change in the consumption of good that occurs when the consumer moves to a higher/ lower indifference curve (representing the change in real income).

substituting in the budget constraint

income effect changes consumer bundle from (18.306,2) to (9,2)

Part (D)

Part (4)

price increase reduces consumer surplus from area(AED) to area(ABC)

change in consumer surplus is area(BCDE)