Question

Suppose your utility function is given by U(x,y)=xy2 . The price of x is Px, the price of y is Px2 , and your income is M=9Px−2Px2.

a) Write out the budget constraint and solve for the MRS.

b) Derive the individual demand for good x. (Hint: you need to use the optimality condition)

c) Is x an ordinary good? Why or why not?

d) Suppose there are 15 consumers in the market for x. They all have individual demand functions equal to the one you derived in part b. The market supply for good x is given by QS=2Px. What is the market equilibrium price and quantity for good x?

Answer 1

Ans.

a) Utility function, U = xy^2

Marginal Utility of x, MU1 = dU/dx = y^2

Marginal Utility of y, MU2 = dU/dy = 2yx

=> Marginal Rate of Substitution, MRS = dy/dx = MU1/MU2 = y/2x

b) At utility maximizing level,

MRS = Px/Py
=> MRS = Px/Px^2 = 1/Px

=> y/2x = 1/Px

=> y = 2x/Px -->Eq1

Substituting Eq1 in budget constraint, x*Px + y*Px^2 = Income = 9Px - 2Px^2, we get,

x*Px + (2x/Px)*Px^2 = 9Px - 2Px^2

=> x = 3 - (2/3)*Px --> Eq2

c) As the demand for x is inversely dependent on the price, so, it is an ordinary good.

d) If there are 15 firms, then the market demand function,

x = 15*(3 - (2/3)*Px) = 45 - 10Px ---> Eq3

Supply function, Qs = 2Px

At equilibrium,

Supply = Demand

=> 2Px = 45 - 10Px

=> Px = $3.75 and thus, x = 7.5 units