Burt’s utility function is U(x1, x2) = (x1 +2)(x2 +6), where x1 is the number of cookies and x2 is the number of glasses of milk that he consumes.
(a) What is the slope of Burt’s indifference curve at the point where he is consuming the bundle (4, 6)?_____. Use pencil or black ink to draw a line with this slope through the point (4, 6). (Try to make this graph fairly neat and precise, since details will matter.) The line you just drew is the tangent line to the consumer’s indifference curve at the point (4, 6).
(b) The indifference curve through the point (4, 6) passes through the points (10 ,0), (7, _____ ), and (2, _____ ). Use blue ink to sketch in this indifference curve. Incidentally, the equation for Burt’s indifference curve through the point (4, 6) is x2 = ___________
(c) Burt currently has the bundle (4, 6). Ernie offers to give Burt 9 glasses of milk if Burt will give Ernie 3 cookies. If Burt makes this trade, he would have the bundle (1, 15). Burt refuses to trade. Was this a wise decision? _____U(1,15) = 63 < U(4,6) = 72. Mark the bundle (1, 15) on your graph.
(d) Ernie says to Burt, “Burt, your marginal rate of substitution is −2. That means that an extra cookie is worth only twice as much to you as an extra glass of milk. I offered to give you 3 glasses of milk for every cookie you give me. If I offer to give you more than your marginal rate of substitution, then you should want to trade with me.” Burt replies,
“Ernie, you are right that my marginal rate of substitution is −2. That means that I am willing to make small trades where I get more than 2 glasses of milk for every cookie I give you, but 9 glasses of milk for 3 cookies is too big a trade. My indifference curves are not straight lines, you see.” Would Burt be willing to give up 1 cookie for 3 glasses of milk?
_____, U(3,9) = 75 > U(4,6) = 72. Would Burt object to giving up 2 cookies for 6 glasses of milk? _____,
(e) On your graph, use red ink to draw a line with slope −3 through the point (4, 6). This line shows all of the bundles that Burt can achieve by trading cookies for milk (or milk for cookies) at the rate of 1 cookie for every 3 glasses of milk. Only a segment of this line represents trades that make Burt better off than he was without trade. Label this line segment on your graph AB.