I am only interested in part c,
1. 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 Jamesâ price elasticity of demand for good 1, Jamesâ income elasticity of demand for good 1 and Jamesâ 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 Jamesâ 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 Jamesâ 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 Jamesâ gain in compensating consumer surplus if declines to $.5. Use Jamesâ compensated demand curve for good 1 to illustrate this. Illustrate the âincomeâ and âsubstitutionâ effects of this price reduction using Jamesâ indifference curve map and his budget lines.
I am only interested in part c,
1. 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 Jamesâ price elasticity of demand for good 1, Jamesâ income elasticity of demand for good 1 and Jamesâ 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 Jamesâ 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 Jamesâ 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 Jamesâ gain in compensating consumer surplus if declines to $.5. Use Jamesâ compensated demand curve for good 1 to illustrate this. Illustrate the âincomeâ and âsubstitutionâ effects of this price reduction using Jamesâ indifference curve map and his budget lines.
Related textbook solutions
Related questions
(a) Explain why the cross elasticity of demand for substitute goods is positive and the cross elasticity of demand for complements is negative.
(b) Explain why the price elasticity of demand changes along a linear demand curve.
(c)
| Income | Px | Py | Quantity of good x | Quantity of good y |
| $30,000 | $6 | $3 | 2 | 20 |
| $50,000 | $6 | $4 | 5 | 10 |
(i) Using the information in the table, calculate the income elasticity of demand for good X and characterize the good. Use the midpoint formula.
(ii) Can you calculate the income elasticity of demand for good Y? If you can, show your calculation and characterize the good. If you cannot, explain why.
