Problems on Intermediate Microeconomic Theory - Review Sheet | ECO 202, Exams of Microeconomics

Download Problems on Intermediate Microeconomic Theory - Review Sheet | ECO 202 and more Exams Microeconomics in PDF only on Docsity! Name Davidson College Mark C. Foley Department of Economics Aug - Dec 2004 Intermediate Microeconomic Theory Review #1 Suggested Solutions Directions: This review is a closed-book, closed-notes exam to be taken in one sitting, no time limit. You may use a calculator. There are 150 points on the exam. Each true-false question is worth 3 points. The problems in Section II are worth 30, 25, 20, and 20 points, respectively. The problems in Sections III are worth 20 points each. You must show all your work to receive full credit. Any assumptions you make and intermediate steps should be clearly indicated. Do not simply write down a final answer to the problems without an explanation. Read the questions carefully, answering what is asked. Think clearly and work efficiently. Carpe diem. Formulas: 0,0,0 21222112211  fffandff 02 21222112 2 211  fffffff Section I: True, False, or Uncertain and WHY. Circle the choice and in the space provided explain why. Unjustified answers earn zero points. 1. True, False, or Uncertain If the prices of food and shelter are $1 per pound and $2 per square yard, respectively, and the corresponding marginal utilities are 6 and 4 at the current consumption bundle, then the consumer is maximizing utility. False. If the person bought 1 square yard (per week) less shelter, he would save $2 (per week) and would lose 4 utils. But this would enable her to guy 2 pounds (per week) more food, which would add 12 utils, for a net gain of 8 utils. Thus, the person could increase his total utility by making this exchange, and cannot be maximizing utility. Another way to do this is to check whether [MUfood / MUshelter] = [Pfood/Pshelter] Plugging in, we see that [6/4 ] is not equal to [1/2] so the optimizing condition,which comes from the first two FOCs, is not met. 2. True, False, or Uncertain If a consumer has homothetic preferences, then their Engel curves are linear. True. If preferences are homothetic, then when income is scaled up or down by “t” percent, the optimal bundles change by the same “t” percent. Intuitively, if the optimal indifference curve is tangent to the budget line at (x*,y*), then the indifference curve through (tx*, ty*) will be tangent to the budget line that has “t” times as much income as originally (and the same prices). That’s what homtheticity means, the slope of indifference curves along any ray from the origin are identical. 3. True, False, or Uncertain Assume the price of apples is $3, the other good is the composite good, and apples are on the x-axis. An ad valorem tax on apples of 5 percent will yield a flatter budget line than a quantity tax of 5 cents. False. The price under the ad valorem tax would be $3(1.05) = $3.15, while the price under the quantity tax would be $3+.05 = $3.05, yielding the flatter budget line. 2 A CCG 3/3 3 50 3/(3.15) 3/(3.05)    XXYYYXX Y PPP I PPP I P P X                   3 1 3 2 3 1 3 2 3 1 * 5 (c) If the consumer’s I = $40, XP = $8, and YP = $1, what is the utility- maximizing consumption bundle and maximum utility? Graph the budget line, optimal consumption bundle, and at least three (3) points on the optimal indifference curve.   4 )88*1( 40 3 1 3 1 3 2 *      XXY PPP I X   8 )11*8( 40 3 2 3 1 3 2 *      YYX PPP I Y Two other points on U* indifference curve: U X Y -5/128 4 8 the optimal point -5/128 5 5.1214 I chose 5, and solved for 5.1214 -5/128 5.1214 5 and the opposite must be a point as well given the symmetric nature of the utility function (d) What is the value of  at the optimum? State its units. What is the economic interpretation of  ? Show that this holds (approximately) by increasing income by $1 and calculating the new optimal level of utility. dollarperutils PY Y 512 1 1*8 11 33  which is how much extra utility another dollar spent on good X (or good Y) will yield. Thus  can be thought of as the marginal utility of an additional dollar of consumption, i.e., the marginal utility of “income”. If I = $41, then X* = 41/10 = 4.1 and Y* = 41/5 = 8.2, thus U*new= - 1/ (2*4.1*4.1) – 1/(2*8.2*8.2) = -.03718 6 X Y 5 40 8 4 U* = -5/128=-.03906 The increase in utility is therefore -.03718 – (-.03906) = .00188 which is approximately equal to 1/512 = .00195. (e) Are the second order conditions satisfied? Justify mathematically and explain intuitively. 02 22  xyyyxxyyxx fffffff ? 4343 3,,0,3,   YfYffXfXf yyyxyxxx 0 48 3 84 3 )(3)0(2)(32 6464 2343323422   XYYXYXfffffff xyyyxxyyxx yes, since this number is negative the optimal values of X and Y MAXIMIZE utility and the critical point defined by the FOC is not a minimum or saddle point. Intuitively, the second total differential needs to be negative so that the rate of increase of utility is decreasing at the optimum. That is, “stepping off the mountain top” in any direction will cause utility level to decline, hence the critical point must be a maximum. 7 (b) Ignoring any rounding differences (i.e., thinking about what this type of utility function means), how much of the total change in demand for Y is due to the income effect? Explain why. None of it since this is quasi-linear utility function. Mathematically, B and C are both at Y = 2.5. 10 4. (a) Derive the expenditure function for }3,2min{ YXU  . The expenditure function takes the form YPXPE YX  First, let’s plot the indifference curves. Say for U = 6 utils. If Y = 2, then X = 3 or 4 or 5 or anything above 3, yields U = 6. If X = 3, then Y = 2 or 3 or 4 or anything above 2 yields U = 6. The expenditure function is found by minimizing E subject to UYXU ),( . So what’s the cheapest way to achieve 6 utils? That is, what are X* and Y*? We know X* =3, so if U = 6, then 2 * U X  would give 3. Notice that the prices do not appear. It doesn’t matter what the prices of X and Y are, the optimal X* is still 3, and optimal Y* still 2. In other words the slope of the budget line can be anything, and the optimal (X*,Y*) still goes to (3,2). Similarly, 3 * U Y  . Thus, 32 * * U P U PYPXPE YXYX  . 11 3 Y U=6 X 2 Minimize in this direction Section III: Problems Do 2 of the following problems (Choice ). If you start more than one, be sure to indicate (by circling the question #) which you want graded. 5. (a) Graph the income-expansion path (income offer curve) for 32YXU  . Assume 21  YX PandP . Use incomes of $10, $20, and $40 to get 3 points on it. Label all axes. The points are at (X,Y) = (4,3) and (8,6) and (16,12). Since this is a Cobb-Douglas utility function, the exponents tell us the percent of income the consumer will spend on that good. This person will maximize utility by spending 2/5 of income on X, and 3/5 on Y. (b) If Y is plotted on the y-axis, what is the slope of the Engel curve for Y ? 3/10 12 10 Y 20 X 5 Income expansion path 40 10 20    10 Y 20 Income 3 Engel curve 40 6 12   