1 Budget constraints, Exams of French

Download 1 Budget constraints and more Exams French in PDF only on Docsity! Practice Problems Fall 2015 Econ 263 1 Budget constraints 1. Each month, Andy gets the following (exogenous) endowments: CA = 5, TA = 3. Andy can exchange these endowments for currency at market prices, which are exogenous to him, in a central marketplace. (a) If the price of coffee is 7 units of currency per unit of coffee, and the price of tea is 5 units of currency per unit of tea, show why Andy’s income/month measured in currency is $50.00 . A: CA︷︸︸︷ 5 × PC︷︸︸︷ 7 + TA︷︸︸︷ 3 × PT︷︸︸︷ 5 = 50. (b) What is his income/month measured in units of coffee? In units of tea? A: 50 7 = 49 1 7 = 7. 142 9; (C/month) 50 5 = 10. (T/month) (c) What is the relative price of coffee? What are the units of this price? A: PC PT = 7 5 = 1.4 The units are units of tea/unit of coffee. (d) Write his budget constraint in standard slope-intercept form with consumption of tea/month on the left-hand-side of the equality sign. A: TA = 10− 1.4CA. (e) With tea on the vertical axis and coffee on the horizontal, draw a schematic diagram of his budget constraint, making sure you identify all relevant features, i.e., slope, intercepts, and endowment point. A:y = 10 − 1.4x. Coffee-intercept is 7. 142 9., endowment point is (5, 3) 1 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 C(A) T(A) (f) What would happen to this schematic diagram if both PC and PT were to double? Triple? Be cut in half? A: nothing. Both slope and intercept remain unchanged. 2. Now consider another scenario. Andy grows coffee for a living, and takes his harvest to market once a year. There, he can sell as much of his crop as he wants at a market price of a certain amount of dollars per pound of coffee. While at the market, Andy can use the money he gets from selling his coffee to purchase the only other good he likes to consume, tea, at a market price of a certain amount of dollars per pound of tea. (a) Suppose Andy grows 80 pounds of coffee per year, and coffee ex- changes in the market place for $2.00/kilo. Tea exchanges in the market place for $4.00/kilo. Let CA symbolize the variable that measures the amount of coffee Andy consumes per year, and TA sym- bolize the variable that measures the amount of tea Andy consumes per year. Describe in an equation with only TA on the left-hand-side of the equality sign all those pairs of kilos of coffee/yr. and kilos of tea/year that Andy could consume at these prices, assuming he spent all of his income. Answer: Let me put the budget constraint in parametric form. First I express in an equation the equality of income and expenditure: Expenditure︷ ︸︸ ︷ PCCA + PTTA = Income︷ ︸︸ ︷ PTTA + PCCA. 2 Answer: First, the tangency conditions for (1) and (2): Note they are identical. dUA = (− CATA(2) (22(CA + TA)2 + CA(2)(CA + TA) (2(CA + TA))2 )dTA +(− 2CATA (2(CA + TA))2 + 2TA(CA + TA) (2(CA + TA))2 )dCA; dUA = 0 : dTA dCA = −−2CATA + 2TA(CA + TA)−2CATA + 2CA(CA + TA) = −−2CATA + 2CATA + 2(TA) 2 −2CATA + 2CATA + 2(CA)2 = − (TA) 2 (CA)2 = −p. Now for (3): dUA = (− CATA(2) (22(CA + TA)2 + CA(2)(CA + TA) (2(CA + TA))2 )dTA +(− 2CATA (2(CA + TA))2 + 2TA(CA + TA) (2(CA + TA))2 )dCA; dUA = 0 : dTA dCA = −−2CATA + 2TA(CA + TA)−2CATA + 2CA(CA + TA) = −−2CATA + 2CATA + 2(TA) 2 −2CATA + 2CATA + 2(CA)2 = − (TA) 2 (CA)2 = −p. Or to put it in a form useful for graphing: (TA) 2 = p (CA) 2 ; TA = √ pCA. Next, use the tangency conditions in the budget constraints. The budget constraint is TA = TA + pCA − pCA So, for both (1) and (2), CA︷︸︸︷ pCA = TA︷︸︸︷ 1 2 + CA︷︸︸︷ 3 2 p− pCA; 2pCA = 1 + 3p 2 ; CdA = 1 + 3p 4p ; For (3) (p) 1 2 (CA) = 1 + 3p 2 − pCA; CA(p+ p 1 2 ) = 1 + 3p 2 CdA = 1 + 3p p+ p 1 2 5 3. 10 points. Now assume Antoines’s endowment is (3, 25), and his prefer- ences are represented by UA = TA + 10CA − (CA)2 for CA ≤ 5; UA = TA + 25 for CA ≥ 5. What is his demand curve for coffee? For tea? A: For coffee: TA = UA − 10CA + (CA)2, CA < 5; dTA dCA = −10 + 2CA = −p; p = 10− 2CdA; CdA = −1 2 p+ 5 ; For tea, we need the budget constraint: TA = TA︷︸︸︷ 25 + CA︷︸︸︷ 3 p− p ( −1 2 p+ 5 ) = 25 + (3− 5) p+ 1 2 p2 = 25− 2p+ 1 2 p2. 3 Importance of substitutability (a diffi cult ques- tion not likely to be on an exam); but still interesting and fun. Consider the following model of a two-person two-good endowment economy based on Radford’s account of a British POW camp . Each individual has preferences over goods x and y that can be represented by the following utility functions: U1 = min{x13 4 , y1 1 4 }; U2 = min{x23 11 , y2 8 11 } In words, for any given permissible value of x1, y1, the utility level for individual one (1) associated with that point is the minimum value of the two numbers {x13 4 , y11 4 }. For example, if individual one (1) were consuming {x1 = 1, y1 = 1}, then this person’s associated level of utility would be given by the minimum of { 13 4 , 11 4 } which is 43 . The graphs of the families of indifference curves associated 6 with such preferences have L-shaped individual indifference curves (with respect to the origin of the (x, y) plane) which have the right-angle apex of each curve lying along rays through the origin with equations for these rays given by: y1 = 1 4 3 4 x1 = 1 3 x1; y2 = 8 11 3 11 x2 = 8 3 x2. Each POW in this camp receives an endowment from the camp commandant of (xi, yi), i = 1, 2. 1. (a) Suppose each of the above two individuals consumes his endowment. True (T) or false (F): i. _____We can unequivocally conclude that individual one (1) is better off than individual two (2). ii. _____We can unequivocally conclude that individual two (2) is better off than individual one (1). iii. _____We can unequivocally conclude that both individuals are equally well off. iv. _____There is no way to tell which individual is better or worse off. Answers: FFFT. No way to make interpersonal comparisons of utility. (b) Denote the relative price of x by p, i.e., p ≡ Px Py . Write down the competitive budget constraint for individual i. Answer: p(xi − xi) = yi − yi; (any variant will do, even in nominal terms). (c) With the above preferences and endowments, the demand curves for each of the above individuals is given by: x1 = px1 a1 + p + y1 a1 + p , a1 = 1 3 ; (xd1) x2 = px2 a2 + p + y2 a2 + p , a2 = 8 3 . (xd2) We get these by noting from a diagram that any most-preferred pair lies on the straight-line ray through the origin with slope, as given above. Subbing this "tangency condition" into the budget constraint then yields the above demand curves. Assume each prisoner receives endowments of one unit of x and one unit of y. Verify, i.e., show that the equilibrium conditions for this model are satisfied, that an equilibrium price for this competitive autarkic economy is given by: p̂ = 11 9 7 where individual 1 has indifference curves with right-angles and a "tangency condition," i.e., ray through the origin that passes through the right angles of the indifference curves, that are be- low those of individual 2. In autarky, each person’s budget con- straint goes through the endwoment point (1, 1) and has slope − 119 . Now imagine rotating the budget constraint through the endowment point (1, 1) so that it now has slope −1. This is depicted in the above figure as the flatter budget line. Clearly, individual one is better off and individual two is worse off. So, TFTF. (f) Now imagine that the camp commandant intercepts the red cross packages and reallocates the two goods so that the endowments for the two POW’s are given by: x1 = 10 7 ; y1 = 10 21 ; (1) x2 = 4 7 ; y2 = 32 21 (2) The two British POW’s can now trade in the larger camp at the exogenous price π̂FT = 1. True (T) or false (F): i. _____Individual one (1) is now better off than in autarky. ii. _____Individual two (2) is now better off than in autarky. iii. _____Neither individual is now worse off than in autarky. iv. _____Both individuals are better off than in autarky. Answer: In the above diagram, imagine rotating each individ- ual’s budget constraint through the autarkic consumption points. clearly, their most-preferred pairs don’t change. Hence, FFTF. 4 Preferences For each question, be prepared to explain the steps used in how you got your answer, and be prepared to illustrate these steps with appropriate schematic diagrams. 1. Consider three individuals, Alex, Bobby, and Don, who have preferences represented by the following utility functions: UA = CATA (Alex’s) UB = CBTB + (CB) 2(TB) 2. (Bobby’s) UD = (CD) 1 2 (Td) 1 2 . (Don’s) 5 points each. True or false with short explanation: 10 (a) _____If both Alex and Bobby consume two (2) units of coffee and two (2) units of tea per unit of time, then we can say that Bobby has a higher level of satisfaction than does Alex, and Alex a higher level than Don. Answer: False. First, though, consider what happens if we simply substitute the values of coffee and tea in each utility function: UA = CATA = CA︷︸︸︷ 2 × TA︷︸︸︷ 2 = 4; UB = 2× 2 + 22 × 22 = 20; UD = √ 2 · 2 = 2. One might be tempted to say this means Bob is better off than Andy, and Andy is better off than Don, when they each consume two (2) units of coffee and two(2) units of tea. But utility numbers are ordinal rankings, and we can’t compare across individuals. (b) _____If Alex’s, Bobby’s, and Don’s consumptions of coffee and tea per unit of time increase from two (2) units of coffee and two (2) units of tea to three (3) units of coffee and three (3) units of tea, then we can say that Bobby’s increase in well-being is greater than Alex’s increase in well-being and Alex’s is greatere than Don’s. Answer: false. Again, first consider a naive approach of considering what happens to changes in utility levels: UA = CA︷︸︸︷ 3 × TA︷︸︸︷ 3 = 9; 9− 5 = 5 utils; UB = 3× 3 + 32 × 32 = 90; 90− 20 = 70 utils; UD = √ 3 · 3 = 3; 3− 2 = 1 util. But these are not useful for making comparisons: utility numbers provide an ordinal ranking. (c) _____Bobby would be classified as a tea-lover relative to Alex. Answer: false, they have the same mrs function, i.e., same slopes of indifference curves at every point in the coffee-tea plane. Calcu- late the mrs′s: (for Bob, you need to use the "take total derivative of U and set equal to zero" approach): once without using total 11 derivatives, once with) TA = UA CA ; dTA dCA = − UA (CA) 2 = − UA︷ ︸︸ ︷ TACA (CA) 2 = − TA CA ; UB = CBTB + (CB) 2(TB) 2 = TBCB(1 + TBCB); dUB = ( TB + 2(CB)(TB) 2 ) dCB + ( CB + 2(CB) 2(TB) ) dTB = 0; dTB dCB = − ( TB + 2(CB)(TB) 2 ) (CB + 2(CB)2(TB)) = − TB (1 + 2(CB)(TB)) CB (1 + 2(CB)(TB)) = − TB CB ; (TD) 1 2 = (UD) 1 2 (CD) 1 2 ; TD = (UD) CD ; dTD dCD = − (UD) (CD) 2 = − UD︷ ︸︸ ︷ TDCD (CD) 2 = − TD CD . 2. In New Jersey, exits on the New Jersey Turnpike start at "1" at the first one north of Delaware, and continue "2," then "3," and so forth, no matter the miles between the exits, as one heads north. In Tennessee, exits on I65 are designated by how many miles they are from the border with Kentucky. Which scheme is analogous to how members of an indifference curve are identified? A: The NJT. It is an ordinal ranking, whereas the TN one is cardinal. 3. Andy has an indifference curve described as TA = 1 CA Bob has an indifference curve described as TB = 1 (CB) 3 . Who is the "coffee-lover" between these two, and why? A: Consider the slopes of indifference curves evaluated at the point at which they intersect, (1, 1): dTA dCA = − 1 (CA) 2 = −1; dTB dCB = −3 1 (CB) 4 = −3. This says that, at teh point (1, 1), Andy is willing to substitute "as small as possible" a unit of tea for a unit of coffee. That is, if offered one infinitesimal unit of coffee in exchange for one infinitesimal unit of tea, he would be willing to make the exchange, because it would leave him on gthe same indifference curve. Bob, though, if offered one unit infinitesimal 12 where RA and RB are Andy and Bob’s real incomes measured in units of tea. (For future reference, note that combining these demand curves with each individual’s respective tangency condition yields the most-preferred choices of tea as functions of p and Ri) : T dA = 3 4 RA; T dB = 1 2 RB . Back to the problem at hand. For the specified endowments, their GE demand functions are CdA = 1 4p CdB = 1 2 Adding up: Cd ≡ CdA + CdB = 1 2 + 1 4p Miller time! (b) 10 points. For this two-person economy, calculate the equilibrium relative price of coffee and the equilibrium quantities of coffee and tea consumed by each individual. Answer: Equilibium condition: CdA + C d B = CA+CB︷︸︸︷ 1 Substituting for Cdi , i = A,B, into the equilibrium condition and solving for p: pa = 1 2 . The subscript a is a mnemonic for "autarky." Subbing this back into the demand equations gives ĈA = 1 2 ; ĈB = 1 2 ; Using the tangency conditions, T̂A = 3 4 ; T̂B = 1 4 . (c) 10 points. In the French POW camp, Alphonse has identical prefer- ences to Andy, and Baptiste has identical preferences to Bob. What is different are the endowments each receives: Alphonse receives an 15 endowment of zero (0) units of tea, and one (1) unit of coffee, while Baptiste receives an endowment of one (1) unit of tea and zero (0) units of coffee: TA = 0; CA = 1; TB = 1; CB = 0. That is, the endowments have been switched relative to Andy and Bob. Construct the equilibrium demand curves for coffee for Alphonse and Baptiste and construct the market equilibrium demand curve. Answer:the tangency conditions remain the same: TA = 3pCA; TB = pCB . The budget constraints are now: TA + pCA = p; TB + pCB = 1. Upon substitution of the tangency conditions into the budget con- straints, we have CdA = 1 4 ; CdB = 1 2p . Aggregate or market demand: Cd = 1 4 + 1 2p The two aggregate demand curves (one for the English and one for the French) differ, even though aggregate endowments are the same and preferences are the same in both scenarios. We depict the two demand curves below, where the thick line represents Andy and Bob’s market demand curve: y = .25 + .5x−1 16 0 1 2 3 4 5 0 1 2 3 4 5 C p The point of this exercise: aggregate (market) demand functions de- pend on income distribution. This is a caveat for most of the models we use, in which we focus on model specifications that have demand functions invariant to income distribution. We focus on these models that don’t have "income effects" because: (1) observation suggests it is not such a strong effect that it overturns qualitative predictions of our models; (2) we want to focus on other features without muddying the water with this complication, i.e., it has a pedagogical purpose. (d) 5 points. Calculate the autarkic equilibrium relative price of coffee in the French camp. Answer: Equating demand to supply: 1 4 + 1 2p∗ = 1 This implies p∗a = 2 3 . Now, substituting this value of p back into the individual demand functions yields Ĉ∗A = 1 4 ; Ĉ∗B = 3 4 T̂ ∗A = 1 2 ; T̂ ∗B = 1 2 . (e) 5 points. Which of these individuals would be described as relatively coffee-loving? 17 coffee-tea pair? By feasible we mean that he could afford it. Explain your reasoning. Answer: IA800. The most-preferred pair should be at that point on the BC at which the slope of an indifference curve is just equal to the slope of the BC, which is .5. Thus, IA800 : dTA dCA = − 800 (CA) 2 = − 1 2 ; (CA) 2 = 1600; CA = 40 If this is true, then the associated value of TA on the I-curve is TA = 800 40 = 20. This pair (40, 20) is feasible (check that it satisfies the BC). For the other I-curve, IA450 : dTA dCA = − 450 (CA) 2 = − 1 2 ; (CA) 2 = 900; CA = 30. Hence, the associated value of TA on this I-curve is TA = 450 30 = 15. The pair (30, 15) is less-preferred than (40, 20) because at that point there is less of both goods compared to the other feasible point. 4. Suppose someone told you that two member’s of Andy’s family of indif- ference curves were described by the following two equations: IA800 : TA = 800 CA ; IA20 : TA = 20 + ln ( CA 40 ) . If Andy’s preferences satisfy the properties economist’s assume in their model of the consumer, why would this person be wrong? Hint: look at the pair (40, 20). Answer: The point (40, 20) lies on both I-curves. I-curves cannot cross, i.e., share a point. 20 5. Suppose Andy’s preferences are such that the preceding two indifference curves IA800 and IA450 are in fact members of his indifference-curve fam- ily. This means that his preferences can be represented by the utility function: UA = CATA where UA can take any value represented by the set of positive real num- bers. (a) Re-write this equation with only TA on the left-hand-side of the equal- ity sign. A: TA = UA CA . (b) Show that Andy’s marginal rate of substitution function, i.e., the negative of the slope of an indifference curve, can be expressed as: mrs = TA CA . Hint: find dTA dCA as a function of UA and CA and then substitute in for UA. A: − dTA dCA = UA (CA) 2 ; − dTA dCA = TACA (CA) 2 = TA CA . You can also take the total derivative approach. (c) An individual general equilibrium demand curve answers the ques- tion: for any given set of prices, what is the quantity/unit of time chosen by an individual? Suppose that Andy produces CA amount of coffee per year (and nothing else), derive Andy’s demand curve for tea, and then coffee. Answer: mrs=slope of BC: TA CA = PC PT ; CA = PT PC × TA; TA = CA × PC PT . 21 Sub this back into BC: TA = CA × PC PT − PC PT ( PT PC × TA ) ; 2TA = CA PT PC ; T dA = CA 2PTPC ; CA × PC PT = CA × PC PT − CA × PC PT ; CdA = CA × PC PT 2CA × PC PT = 1 2 . 7 Trade and arbitrage 1. Consider the following inverse excess supply and inverse excess demand functions for coffee: p∗ = −2ED∗ + 12; p = 2ES + 6. Arbitrageurs have the following cost function: C(A) = c 2 A2; c > 0 with associated marginal cost function MC(A) = cA. (a) What are the autarkic prices? Answer: Set ED, ES to zero: p∗a = 12, pa = 6 (b) Assume c = 2. What are the equilibrium prices at home and abroad, and how much coffee is exported? A: In equilibrium, ED∗ = A; ES = A; p∗ − p = cA(= 2A). 22