Economics

For questions 3) - 9), show at least some of the steps of your calculation, not only the end result!

 

1) For a linear preference function u (x, y) = x + 2y, calculate the utility maximizing consumption bundle, for income m = 90, if,

a) px = 4 and py = 2

b) px = 3 and py = 6

c) px = 4 and py = 9

 

2) When an agent maximizes utility given a certain budget, how can we solve the problem graphically? Show a general case for what Banerjee calls ‘Cobb-Douglas preferences’. (In other words, what is the condition that has to be met between budget line and indifference curve in order to maximize an individual’s utility?)

3) For a demand function u (x, y) = xy, show the demand functions for good x and good y. (Remember that MRS = (du/dx) / (du/dy) = px / py in the point of interest, the tangency point of budget line and indifference curve. The budget condition is given by pxx + pyy = m).

 

4) Calculate the own-price elasticities as well as the income elasticities of demand for goods x and y based on your results in 3).

 

5) Follow the same steps as in 3) for u (x, y) = x1/3 y2/3.

 

6) Show the individual’s demand for good x and good y that follows from their utility function u (x, y) = x1/2 y1/2.

 

7) For the values in 5), assume income m = 30, px = 2, py = 1. How many units of each good does the individual consume? Calculate the same using the function from 6).

8) Show the individual’s demand for good x and good y that follows from their utility function

u (x, y) = xay1-a  , with 0 < a < 1. 

 

9) Use your result from 8) to calculate the own-price elasticity of demand for both goods.

 

10) Briefly explain why we assume that it makes sense that demand functions are homogeneous of degree zero in income and prices.