Question

Suppose that a consumer has utility given by U(a,b) = ab + 10b, her income is n, the price of a is Pa, and the price of b is Pb.

a. Write the demand functions for both goods.

In parts b - d you assume interior solution:

b. Is each good normal or inferior? Explain.

c. Is each good a normal good or a giffen good?

d. Are the two goods (gross) complements or substitutes?

e. Prove that these preferences are not homothetic by either using the definition of homotheticity and the utility function to find a counterexample, or by Engel curves.

Answer 1

1)a)MRS=MUa/MUb=b/a+10

at optimality MRS=Pa/Pb=Qb/Qa+10

QbPb=(Qa+10)Pa

and budget QaPa+QbPb=n

2QaPa+10Pa=n

Qa=(n-10Pa)/2Pa

put in Qb={(n-10Pa)/2Pa + 10} Pa/Pb =(n+10Pa)Pa/Pb

b)both goods are normal becauSe when income increases demand of both good increases

c)Both goods are normal because when price of a good increases, demand decreases

d)since dQb/dPa=+ve whcih means an increase in price of A leads to increase in quantity of B. Thus A and B are substitute goods