James has a utility function given by , where is the amount of product 1 consumed per period and is the amount of product 2 consumed per period. Derive James’ Marshallian demand functions for and , expressed as functions of the prices of goods 1 and 2 and money income.
Are James’ demand functions homogeneous of degree 0 in prices and money income? Will James always spend all of his income?
a) Derive expressions for James’ price elasticity of demand for good 1, James’ income elasticity of demand for good 1 and James’ cross price elasticity of demand for good 1 with respect to the price of good 2? Calculate the values of these elasticities at and I=$100. Explain the meaning of these elasticity values in words. Show that the sum of these three elasticities must be zero. Comment. Derive James’ indirect utility function and explain its meaning.
b) What is the minimum income necessary for James to achieve 100 utils when and ? Illustrate your answer with a diagram. Derive James’ compensated demand functions for goods 1 and 2. Suppose now that I=$100, .
c) Use your answers to part (a) to find the optimal values of and . Calculate James’ gain in compensating consumer surplus if declines to $.5. Use James’ compensated demand curve for good 1 to illustrate this. Illustrate the “income” and “substitution” effects of this price reduction using James’ indifference curve map and his budget lines.