Exercise 1.Income Changes (Cobb-Douglas)

As a consumer you spend part of your income in goods and part in services. Both are essential and take a considerable share of your annual expenditures. Your utility is then u = x_S²x_G, where x_S is the consumption of services and x_G is consumption of goods. Your budget constraint is m = p_Sx_S + p_Gx_G, and you can assume p_S=p_G=1. By normalizing both prices to one, you are implying that inflations would be the same for both goods and services and that m represent real income.

a)Derive the demand curve for both goods and services as a function of income: x_G(m) and x_S(m)

b)Draw the budget constraints and optimal indifference curves for m=100, m'=200 and m''=300. Draw the income offer curve.

c)In two separate graphs illustrate the two Engel curves given the three income levels in part b.

d)Take the derivative with respect to m of both demand curves: x_G(m)/m and x_S(m)/m. Are the derivatives positive or negative? Are x_G and x_S inferior or normal goods?

e)What is the share of expenditure in goods (p_Gx_G/m), and the share of expenditure in services (p_Sx_S/m)? How does it vary with the increase in m? Are these luxury or necessary goods?