"My preferences are represented by the utility function u = ln(x 1 ) + ln(x 2 ). One unit of good 1 and one unit of good 2 are perfect substitutes

1. “My preferences are represented by the utility function u = ln(x1) + ln(x2). One unit of good 1 and one unit of good 2 are perfect substitutes for me.”  Is this statement true, or is it false? Explain why.
2. When my income doubles, my demand for water does not change. Draw my income offer curve when water is good 1, and “other goods” is good 2.
3. Consider the utility function u = ln(x1) + x2. Write the Lagrangean function and the first-order conditions (setting its partial derivatives equal to zero). Using these conditions, obtain the expression (formula) for demand for good one and good two.
4. Using the demand functions you found in question 3, draw the Engel curves for the two goods, and draw the demand curves, separately.
5. Draw indifference curves corresponding to perfect complements type of preferences and draw a budget line showing the optimal demands of the consumer. (i) Consider an increase in the price of good 1. Is it possible that the demand for good 1 increases? Why yes or why not? (ii) Consider a fall in income. Is it possible that the demand for good 1 increases? Why yes or why not?
6. Consider the utility function u = x1 + x2. Fix the price of good 2 at p2 = 1. (i) Using indifference curves, identify the optimal choices of the consumer for p1 = 2, p1 =  1 and p1 = ½, and draw the price offer curve (put p1 on the vertical axis). Then, draw the demand curve for good 1. Explain briefly what you do in drawing the price offer curve and the demand curve.
7. Suppose that good 1 is a Giffen good. Using indifference curves and budget lines, obtain two points on the price offer curve for two different levels of p1. Then, represent these points (p1 and x1 combinations) on another figure, for the demand curve of good 1.
8. “When the price of good 2 increases, demand for good 1 increases.” Is this true if my preferences are of the perfect complements type? Illustrate your answer in a figure with x1 and x2 on the two axes, by changing p2 only.
9. For each of the two changes in the price of good 1 below, find the income level mI such that the consumer can just afford buying the quantities she was buying before the price change. In each case, indicate whether in the new situation with income level mI the demand x1* will rise or fall. Illustrate your answer separately in figures with x1 and x2 on the two axes, by drawing budget lines and indifference curves.

 Income is m = 1000 and  p2 = 20 is fixed, and p1 = 10,  x1* = 60, x2* = 20, (i) the new price is p1 = 20;  (ii) the new price is p1 = 5.