EC 203 - INTERMEDIATE MICROECONOMICS Bo˘gazi¸ciUniversity Department of Economics Fall 2019 Problem Set 1 - Solutions 1. Draw indifferences curves to represent each of the following type of preferences. (a) Quarters-dollars: A consumer is always happy to change 4 quarters for 1 dollar. Solution: quarters 8 4 dollars 1 2 (b) Bicycle frames-tires: A consumer always wants 2 tires to go with 1 frame. Solution: frames (y) 2y = x 2 2 bikes 1 1 bike tires (x) 2 4 2. Tom always begins his day with a strawberry milkshake. He makes it by mixing milk (x) with five strawberries (y). The secret of a really good milkshake lies in the optimal proportion of milk and fruit: one glass always comes with five strawberries. (a) Plot Tom's representative indifference curves. Depict three indifference curves that pass through the following bundles (5, 1), (10, 10) and (15, 4). What is the MRS at each of these points? Solution: See the graph below. MRS at each of these bundles is zero. (b) What utility function represents these preferences? On the graph from (a), indicate the level of utility corresponding to each indifference curve. Solution: A utility function that represents these preferences would be u(x; y) = minf5x; yg. The utility levels would be u(5; 1) = 1, u(10; 10) = 10, and u(15; 4) = 4 1 (c) Multiply your utility function by ten and add two to it. How did the indifference curves change? How was the level of utility associated with each indifference curve affected? Solution: We have, v(x; y) = 10 minf5x; yg + 2. Since v is a monotonic transformation of u, shape of indifference curve and preference rankings will not be affected. But, the numbers utility levels assigned on each indifference curve will change. We have v(5; 1) = 12, v(10; 10) = 102, and v(15; 4) = 42. strawberries (y) y = 5x (10,10) 10 2 milkshakes u = 10 v = 102 (15,4) 4 4/5 milkshakes u = 4 v = 42 (5,1) 1 1/5 milkshakes u = 1 v = 12 milk (x) 1/5 4/5 2 5 10 15 3. Kate has two favorite kinds of apples: Fuji (x) and Gala (y). Kate loves them both and actually does not distinguish between the two kinds. (a) In a graph, show Kate's indifference that pass through (2, 3) and (3, 3). Solution: gala (y) 6 5 3 2 fuji (x) 2 3 5 6 (b) Suggest two different utility functions that represent Kates preferences. Solution: Here are some examples: u(x; y) = x + y, u(x; y) = (x + y)2, u(x; y) = 5(x + y), u(x; y) = x + y + 1000 (c) Find the marginal rate of substitution. Solution: MRS is negative of the slope of IC which is 1 in this case. 4. Suppose the set of bundles is given by B = fB1;B2;B3;B4;B5g. Also suppose that the preferences of a consumer are given by B1 B2 ∼ B3 and B4 B5. Can you represent these preferences by a utility 2 function? If you can give a utility function that does so. If not, explain why you cannot. Suppose now we also have the information B5 B1: Now, can you represent it with a utility function? If so, provide one. If not explain. Solution: No, because the preferences are not complete as we don't know the preferences between any of B1;B2;B3 and any of B4;B5. Once we have the information B5 B1, now we have complete preferences. These preferences are also transitive as we have the ranking B4 B5 B1 B2 ∼ B3. Since the preferences are both complete and transitive, there exits a utility function that represents these preferences, for instance, u(B4) = 20, u(B5) = 15, u(B1) = 1, u(B2) = 0 and u(B3) = 0. 5. Suppose a consumer's preferences can be represented by the following utility function; u(x; y) = x + y. (a) Show that the utility function v(x; y) = 10x + 10y represent the same preferences with u, that is, v and u are monotonic transformations of each other. Solution: Arbitrarily pick two bundles x = (x1; x2) and y = (y1; y2), and show that both u and v represent the same preference relation. Assume that x y, then u(x1; x2) > u(y1; y2), then x1 + x2 > y1 + y2, then 10x1 + 10x2 > 10y1 + 10y2, that is v(x1; x2) > v(y1; y2). (b) Show that the utility function z(x; y) = x+y2 does not represent the same preferences with u, that is, z is not a monotonic transformation of u. Solution: Consider (1; 1) and (0; 2). According to u, u(1; 1) = 2, u(0; 2) = 2, then (1; 1) ∼ (0; 2). According to z, z(1; 1) = 2, z(0; 2) = 4, then (1; 1) ≺ (0; 2). So they are not the same preferences, hence z does not represent the same preferences as u does. 2 6. Suppose a consumer's preference relation, % is defined over the points in R+ and is given as follows: x % y iff x1 > y1 or [x1 = y1 and x2 ≥ y2]. Suppose this is a strongly monotone preference relation. How do the indifference curves look like? Can you represent this preference relation by a utility function? 2 Solution: This is known as \lexicographic" preferences. It ranks all of the points on R+; but does so in a way that is impossible to represent using any utility function, and also very difficult to \draw" indifference curves for. All points with more of good 1 are preferred to all points with less of good one, so on the graph we have the rule \the further to the right, the better". This alone, if the consumer did not care at all about good 2, would be easy. We could just write u (x1; x2) = x1; and indifference curves would be linear and completely vertical. However, the consumer uses the amount of good 2 as a sort of tiebreaker when two bundles have exactly the same amount of good 1. Therefore, the vertical indifference curves are not indifference curves at all! The higher we move on one, the more preferred the bundle. So 2 every single point on R+ is its own indifference \curve"{the consumer is not actually indifferent between any two points. The order of the indifference \curves" (or, more accurately, points) is starting with the origin, going all the way straight up to infinity(!), then back to zero slightly (how slightly?) to the right and straight up to infinity again, and so on. You can see that it would be impossible to come up with a utility function (a real number for every bundle) that respects this ranking. It can also be proven 2 mathematically that representing this type of preferences as a utility function u : R+ ! R is simply impossible. 3 7. Jonas's utility function is uJ (x; y) = xy. Martha's utility function is uM (x; y) = 1000xy. Noah's utility 1 function is uN (x; y) = −xy. Ulrich's utility function is uU (x; y) = −(xy+1) . Egon's utility function is uE (x; y) = xy − 10; 000. Claudia's utility function is uC (x; y) = x − y. Helge's utility function is uH (x; y) = x (y + 1). Which of these people have the same preferences as Jonas? Solution: We must check which of the utility functions (uM ; uN ; uU ; uE; uC ; uH ) are positive monotonic transformations of uJ : We find that uM (x; y) = 1000xy = 1000uJ (x; y) increases as uJ increases, 1 1 uU (x; y) = = − (xy + 1) −uJ (x; y) − 1 increases as uJ increases, and uE (x; y) = xy − 10; 000 = uJ (x; y) − 10; 000 increases as uJ increases, and so Martha, Ulrich and Egon all have the same preferences as Jonas. On the other hand, uN (x; y) = −xy = −uJ (x; y) is a negative monotonic transformation of uJ ; so Noah has the exact opposite preferences as Jonas, while uC and uH cannot be written as transformations of uJ at all, and so neither Claudia nor Helge share Jonas's preferences. 8. For each of the following utility functions, find the marginal utilities (MUx and MUy) and the marginal rate of substitution between x and y, MRSxy, and explain whether MRSxy is diminishing or not. (a) u(x; y) = 3x + y Solution: MUx = 3, MUy = 1, MRSxy = 3, constant (b) u(x; y) = x1=2y 1 −1=2 1=2 1 y Solution: MUx = 2 x y, MUy = x , MRSxy = 2 x , diminishing (c) u(x; y) = x1=3y2=3 1 −2=3 2=3 1=3 −1=3 1 y Solution: MUx = 3 x y , MUy = 2=3x y , MRSxy = 2 x , diminishing (d) u(x; y) = x2 + 3y − 2 2x Solution: MUx = 2x, MUy = 3, MRSxy = 3 , increasing (e) u(x; y) = x1=2 + y 1 −1=2 1 −1=2 Solution: MUx = 2 x , MUy = 1, MRSxy = 2 x , diminishing (f) u(x; y) = (x + 2y)2 Solution: MUx = 2(x + 2y), MUy = 4(x + 2y), MRSxy = 1=2, constant xy (g) u(x; y) = x+y y2 x2 y2 Solution: MUx = (x+y)2 , MUy = (x+y)2 , MRSxy = x2 , diminishing (h) u(x; y) = px2 + y2 1 2 2 −1=2 1 2 2 −1=2 x Solution: MUx = 2 (x + y ) 2x, MUy = 2 (x + y ) 2y, MRSxy = y , increasing 4 9. Consider the following utility functions: u(x; y) = xy u(x; y) = x2y2 u(x; y) = lnx + lny Show that all three utility functions have the same marginal rate of substitution, MRSxy.