# This problem works through the Slutsky decomposition for a particular price change and some quasilinear preferences. Suppose a consumer has…

1. This problem works through the Slutsky decomposition for a particular price change andsome quasilinear preferences. Suppose a consumer has preferences represented by the utilityfunction u(:1:1, \$2) = 31n(:1:1) + 232 and income given by m = 60. (a) Initially, suppose that prices are given by p1 = 2 and p2 = 24. Find the optimal bundle.This is point “A.” (b) Suppose that the price of \$1 increases to 4. In the notation we have used, this meansPi = 4. Continue to assume that 102 = 24. Find the optimal bundle at the new prices.This is point “B.” (c) Calculate m’, the income level necessary to compensate the consumer for the increasein prices. It will be exactly enough income to buy the original bundle at the new prices. (d) Find the optimal bundle on the intermediate budget line deﬁned by m’, 103, and p2.This is point “C.” (6) Draw a graph showing all three budget lines (original, intermediate, and new) alongwith the bundles “A,” “B,” and “C.” You do not need to draw the indifference curves. (f) What are the income and substitution effects of the price change on the quantity of 31?(g) What are the income and substitution effects of the price change on the quantity of \$2? 2. A consumer has well—behaved preferences over consumption today, 331, and consumptiontomorrow, \$2. The consumer has an endowment igiven by (mhmg): they receive ml inincome today and m2 in income tomorrow. They can borrow money today at interest rate’r: they must pay back (1 + 7′) times the amount tomorrow. They can also save money todaythat will give them (1 + 5&quot;) times the amount tomorrow. Assume that ‘r &gt; 0. Is the consumer better off with an endowment of (5, 10), an endowment of (3, 12), or can youtell? Justify your answer. (Hint: focus on the budget constraint to answer this question.) 3. Consider a consumer who has well—behaved preferences over leisure (L) and consumption(3). They have nonlabor income m and have 24 hours in the day that must be dividedbetween leisure and working. They are initially paid a wage to for each hour of work. Theprice of x is 1. (a) Suppose they optimally choose to work 8 hours. Draw the consumer’s budget set andan indifference curve showing this situation. (b) Now suppose that they are paid an overtime wage that is higher than to for each hourworked in excess of 8 hours. (The wage is w for the ﬁrst 8 hours of work and greaterthan to for any additional hours.) Draw the new budget set and indicate any impactthis has on their choice of hours worked.

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