can you please answer v), vi) and vii)
Suppose that the demand for units of some beverage comes from two types of household. They have the following utility functions representing their preferences of a household over units of the beverage (xt) and expenditure on all other goods (x2),
(Z4(x1,x2) = %i1/2%21/2 and i/B(x1,x2) = 500 ln(Xi) 4- x2
The households have exogenous incomes of IA and IB. The second good is referred to as a composite good and is an amount of money. We assume throughout that p2 = 1.
i) (6 marks) For each household type, derive the households ordinary demand functions, x1(p1,1, Z) and x2(p1# 1, Z). You can assume that both household types have enough income to reach an interior solution.
ii) (2 marks) Suppose there are 10 households of each type. Find the market demand function for the beverage.
iii) (3 marks) If the market supply is provided by 16 identical firms with cost function, C(x) = 4x2. Find the market supply function.
iv) (2 marks) Calculate the market equilibrium price and quantity in terms of the households exogenous incomes.
v) (3 marks) Suppose the exogenous incomes changed so that they became (IA 4- 100) and (IB — 100). Did aggregate income of the households change? Did the market equilibrium price and quantity change? If so, by how much?
vi) (2 marks) If we were to have a market model where the demand side was composed of 20 households of only one household type (A or B) but allowing that households have different parameters in their utility function (exponents or coefficients), ie. they are not identical, which type of household would allow the market demand to be insensitive to changes in the distribution of income (while aggregate income was fixed)? That is, show that the distribution of incomes is irrelevant to the market demand if we choose the correct preference type even when the households are not identical.
vii) (6 marks) Explain this result by illustrating the effect of a change in income for each type of household (as in v)) using an indifference curve analysis for each household.