James has a utility function given by U=X^0.5+Y^0.5 , where X is the amount of product 1 consumed per period and Y is the amount of product 2 consumed per period.
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.Px=Py=1
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