Intermediate Microeconomic Theory
Intermediate Microeconomic Theory ECON 3070
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This 17 page Class Notes was uploaded by Bridie Batz on Friday October 30, 2015. The Class Notes belongs to ECON 3070 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 46 views. For similar materials see /class/232142/econ-3070-university-of-colorado-at-boulder in Economcs at University of Colorado at Boulder.
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Date Created: 10/30/15
Economics 3070 Answers to Practice Problems A Mush q Mobarak Make sure to label axis and indifference curves 1 Fill in the table below Ult172 MU1 MU2 2061 3062 2 3 7 41 62 4 6 7 ml bacg b 7 212 062 00142 1 7 12 ln 1 2 92 11 1 7 061062 062 061 00 fo 106117le bx f cg l 7 1221 21 12 iii E 61 1062 b 62 b 61 a 7121 nail x f x2 ax f l bxg l 7 14 2 2 Charlie s utility function is UxA7 x3 xAxB a U40 5 200 b xAxB 200 So the indifference curve through 407 5 has the equation x3 i jf xB 1040 1020 2010 401 U xA C 31 Yes 03 He Recall Shirley Sixpack and Lorraine Quiche from HWl e 10 Shirley thinks a l6 ounce can of beers is just as good as two 8 ounce cans Lorraine only drinks 8 ounces at a time and hates stale beer so she thinks a l6 ounce can is not better or worse than an 8 ounce can a U x1 2 U 1 2 0 Yes Yes No Reconsider Charlie again Recall that his utility function is U x A7B Suppose that pA 1 p3 2 and m 40 7 AB a 06A 2063 40 b xA7xBMaac acAacB st 06A 2063 40 c xBMaac 40 7 20050003 72B 407 2B 0 4x3 40 x 10 xA4072xB407210 20 cc 20 d W242 200 XB 2010 f MRSWZJE 7 slope 27 5 Calculate f y for the following cases a 3167 065 b y4 0616 6 Consider the utility function U IIllIl417 2 a b ac and ac 0061 2 and ac 8 4m 1714172 7 Consider the utility function UxAxB xAxB Where pA 2113 l7 and m 40 The price of xA suddenly falls to 1 xA are apples and x3 are bananas m PBQCB PB xB PA PA 1393 SE 15 IE 20 b 10 20 14 8 bundle that Charlie would Choose at letter B c 5 more 30 30 m 30 m this income and the new prices With the 15 d e f 003 AB 400 400 40 i m 7 20 21 20 20 400 m m m m m2 T 40 increase 5 more 5 more 5 more 8 Maude spends all her income on delphiniurns and hollyhocks She thinks that del phiniurns and ho yhocks are perfect substitutes one delphiniurn is just as good as one hollyhock Delphiniurns cost 4 a unit and hollyhocks cost 5 a unit Hint sketch graphs a U 13 H MRS 71 406 50 m m 7 506 063 T So the slope of the budget constraint is 7 5 gt1 4 So the budget constraint is steeper than the indifference curve If the price of delphiniurns decreases to 3 a unit budget constraint becomes even steeper 1n the graph the indifference curves are black and the budget constaint is red With the price decrease for delphiniurns the budget constraint swings out to the right Maude buy more of delphiniurns All of the change in consumption is due to the substitution effect 40 1 cu If the prices of delphiniums and hollyhocks are respectively Pd 4 and 1 5 and if Maude has 120 to spend draw her budget line in blue ink Put hollyhocks on the X axis Draw the highest indifference curve that she can attain in red ink and label the point that she chooses as A c see above graph d From graph we can see that new budget constraint goes through both A amp B so her original bundle is still affordable e All substitution effect 32 9 All due to income effect 10 Douglas Corn eld s demand function for good x is xpxpy m 2 7quot His income is 5 1000 pm 5123 20 a 2000 ac 80 55 2000 00 100 5 4 If the price of x falls to 4 then his demand for x will change from 80 to 100 b 580 20y 1000 y 30 So Douglas was spending 30 on good y m 4802030 920 2 920 7 54 I C The substitution effect in this case is 12 and the income effect is 8 11 12 For the case of an inferior but non Giffen good draw a graph that illustrates the SE 1E and TE for an increase in price Use 3 separate colors SEE TEXTBOOK 1 For the case of a Giffen good draw a graph that illustrates the SE 1E and TE for an increase in price Use 3 separate colors SEE TEXTBOOK A Mush q Mobarak Econ 3070003 Chapter Outlines Disclaimer Studying this outline does not ensure that you will do well in the exam In fact just memorizing this outline will not help you solve the types of problems that you will have to solve in the exam It will help you review the material in an organized way Do not view this as a substitute to gaining a deep understanding of all lecture notes Chapter 1 The Market A Example of an economic model 7 the market for apartments 1 models are simpli cations of reality 2 for example assume all apartments are identical 3 some are close to the university others are far away 4 price of outerring apartments is exogenous 7 determined outside the model 5 price of innerring apartments is endogenous 7 determined within the model B Two principles of economics 1 optimization principle7 people choose actions that are in their interest 2 equilibrium principle C Constructing the demand curve 1 line up the people by willingnesstopay See Figure 11 2 for large numbers of people this is essentially a smooth curve as in Figure 12 D Supply curve 1 depends on time frame 2 but we ll look at the short run 7 when supply of apartments is fixed E Equilibrium 1 when demand equals supply 2 price that clears the market F Comparative statics 1 how does equilibrium adjust when economic conditions change 2 comparative 7 compare two equilibria 3 statics 7 only look at equilibria not at adjustment G Other ways to allocate apartments 1 discriminating monopolist 2 ordinary monopolist 3 rent control H Comparing different institutions 1 need a criterion to compare how ecient these dierent allocation methods are 2 an allocation is Pareto effcient if there is no way to make some group of people better off without making someone else worse off 3 if something is not Pareto efficient then there is some way to make some people better off without making someone else worse off A Mushfiq Mobarak Econ 3070003 Chapter Outlines 4 if something is not Pareto efficient then there is some kind of waste in the system 1 Checking efficiency of different methods 1 free market 7 efficient 2 discriminating monopolist 7 efficient 3 ordinary monopolist 7 not efficient 4 rent control 7 not efficient Chapter 2 Budget Constraint A Consumer theory consumers choose the best bundles of goods they can afford 1 this is virtually the entire theory in a nutshell 2 but this theory has many surprising consequences B Two parts to theory 1 can afford 7 budget constraint 2 best 7 according to consumers preferences C Consumption bundle 1 x1 x2 7 how much of each good is consumed 2 p1 pz 7 prices ofthe two goods 3 m money the consumer has to spend 4 budget constraint plxl pzxz 5 m 5 all x1 x2 that satisfy this constraint make up the budget set of the consumer See Figure 21 E Two goods 1 theory works with more than two goods but can t draw pictures 2 often think of good 2 say as a composite good representing money to spend on other goods 3 budget constraint becomes plxl x2 5 m 4 money spent on good 1 plxl plus the money spent on good 2 x has to be less than or equal to the amount available m F Budget line lp1x1pzxz m 2 also written as x mpz plpzx1 3 budget line has slope of 7plpz and vertical intercept of mpz 4 set x1 0 to and vertical intercept mpz set x 0 to and horizontal intercept mpl 5 slope of budget line measures opportunity cost of good 1 7 how much of good 2 you must give up in order to consume more of good 1 G Changes in budget line 1 increasing m makes parallel shift out See Figure 22 A Mush q Mobarak Econ 3070003 Chapter Outlines 2 increasing p1 makes budget line steeper See Figure 23 3 increasing p2 makes budget line atter 4 multiplying all prices by t is just like dividing income by t 5 multiplying all prices and income by t doesn t change budget line 6 e g a perfectly balanced in ation doesn t change consumption possibilities 1 Taxes subsidies and rationing 1 quantity taX 7 taX levied on units bought pl t 2 value taX 7 taX levied on dollars spent pl rpl Also known as ad valorem taX 3 subsidies 7 opposite of a taX apl s b l 0pl 4 lump sum taX or subsidy 7 amount of taX or subsidy is independent of the consumer s choices Also called a head tax or a poll taX 5 rationing 7 can t consume more than a certain amount of some good Chapter 3 Preferences A Preferences are relationships between bundles 1 if a consumer would choose bundle x1 x2 when 011 yz is available then it is natural to say that bundle x1 x2 is preferred to 01 yz by this consumer 2 preferences have to do with the entire bundle of goods not with individual goods B Notation 1 x1 x2 gt 011 yz means the Xbundle is strictly preferred to the ybundle 2 x1 x N 01 yz means that the Xbundle is regarded as indifferent to the ybundle 3 x1 x2 yl yz means the Xbundle is at least as good as preferred to or indifferent to the ybundle C Assumptions about preferences 1 complete 7 any two bundles can be compared 2 re exive 7 any bundle is at least as good as itself 3 transitive 7 ifX Y and YZZ thenX Z a transitivity necessary for theory of optimal choice D Indifference curves 1 graph the set of bundles that are indifferent to some bundle See Figure 31 2 indifference curves are like contour lines on a map 3 note that indifference curves describing two distinct levels of preference cannot cross See Figure 32 a proof 7 use transitivity A Mush q Mobarak Econ 3070003 Chapter Outlines E Examples of preferences 1 perfect substitutes Figure 33 a red pencils and blue pencils pints and quarts b constant rate of tradeoff between the two goods 2 perfect complements Figure 34 a always consumed together b right shoes and left shoes coffee and cream 3 bads Figure 35 4 neutrals Figure 36 5 satiation or bliss point Figure 37 F Wellbehaved preferences 1 monotonicity 7 more of either good is better a implies indifference curves have negative slope Figure 39 2 convexity 7 averages are preferred to extremes Figure 310 a slope gets atter as you move further to right b examples of nonconvex preferences G Marginal rate of substitution 1 slope of the indifference curve 2 MRS Ax2Axl along an indifference curve Figure 311 3 natural sign is negative since indifference curves will generally have negative slope 4 measures how the consumer is willing to trade off consumption of good 1 for consumption of good 2 Figure 312 5 measures marginal willingness to pay give up a not the same as how much you have to pay b but how much you would be willing to pay Chapter 4 Utility A What is a utility function a summarizes preferences b a utility function assigns a number to each bundle of goods so that more preferred bundles get higher numbers c that is ux1 x2 gt uy1 yz if and only if x1 x2 gt 011 yz d only the ordering of bundles counts so this is a theory of ordinal utility B Utility functions are not unique 1 if ux1 x2 is a utility function that represents some preferences and f is any increasing function then ux1 xz represents the same preferences 2 why Because ux1 x2 gt uOl yz only iffux1 xz gtfuy1 yz 3 so if uxl x2 is a utility function then any positive monotonic transformation of it is also a utility function that represents the same preferences A Mush q Mobarak Econ 3070003 Chapter Outlines C Constructing a utility function 1 can do it mechanically using the indifference curves Figure 42 2 can do it using the meaning of the preferences D Examples 1 utility to indifference curves a easy 7 just plot all points where the utility is constant 2 indifference curves to utility 3 examples a perfect substitutes 7 all that matters is total number of pencils so ux1 x2 x1 x2 does the trick 1 can use any monotonic transformation of this as well such as logx1 x2 b perfect complements 7 what matters is the minimum of the left and right shoes you have so ux1 x2 minx1 x2 works c quasilinear preferences indifference curves are vertically parallel Figure 44 1 utility function has form ux1 x2 vx1 x d CobbDouglas preferences Figure 45 1 utility has form ux1 x2 xlcxzd 2 convenient to take monotonic transformation vx1x2 lnux1x2 clnx1 allnx2 E Marginal utility 1 extra utility from some extra consumption of one of the goods holding the other good fixed 2 this is a derivative but a special kind of derivative 7 a partial derivative 3 this just means that you look at the derivative of ux1 x2 keeping x2 fixed 7 treating it like a constant 4 examples a if ux1 x2 x1 x2 then MUl awaxl l b if ux1 x2 xlaxZH then MUl auax1 axlailxzm 5 MU is closely related to MRS which is an operational concept a ux1 x2 k where k is a constant describes an indifference curve b we want to measure slope of indifference curve the MRS c so consider a change dxl dxz that keeps utility constant Then MUlabc1 MU2abc2 0 ora uabc1 6 uabc2 0 6x1 6x Hence amp MUI abc1 M U 2 The slope of the IC MRS Ratio of Marginal utilities for good 1 and good 2 What is the intuition behind this result d so we can compute MRS from knowing the utility function A Mush q Mobarak Econ 3070003 Chapter Outlines Chapter 5 Choice A Optimal choice 1 move along the budget line until preferred set doesn t cross the budget set Figure 5 l 2 note that tangency occurs at optimal point 7 necessary condition for optimum In symbols MRS price ratio plpz a exception 7 boundary optimum Figure 53 3 tangency is not sufficient Figure 54 unless indifference curves are convex and optimum is interior Condition 2 given in class 4 optimal choice is demanded bundle a as we vary prices and income we get demand functions b want to study how optimal choice 7 the demanded bundle 7 changes as price and income change B Examples 1 perfect substitutes x1 mp1 ifpl lt pz 0 otherwise Figure 55 2 perfect complements x1 mp1 pz Figure 56 3 neutrals and bads x1 mpl 4 concave preferences similar to perfect substitutes Note that tangency doesn t work Figure 58 5 CobbDouglas preferences 7 learn how to solve for optimal choice using calculus D Calculus Solutions in the Appendix 1 Using the MRS condition 0 At point of optimal choice budget line is tangent to indifference curve This implies slope of IC slope of budget line MRS p1pz is one equation The budget constraint plxl pzxz m is the other equation We have two equations and two unknowns xbxz Can solve for x1 and x2 2 Constrained Maximization 0 max ux1 x2 stp1x1 p2xz m o This is a two variable maximization problem 0 Can convert this to a one variable maximization problem by substituting from the constraint into the utility function 0 After substitution we are left with maximizing u with respect to x1 d 0 We find the solution by setting 61 0 WHY x1 A Mush q Mobarak Econ 3070003 Chapter Outlines Chapter 6 Demand A Demand functions 7 relate prices and income to choices B How do choices change as economic environment changes 1 Changes in income a this is a parallel shift out of the budget line b increase in income increases demand 7 normal good Figure 61 c increase in income decreases demand 7 inferior good eg Ramen noodles d as income changes the optimal choice moves along the income expansion path income offer curve e the relationship between the optimal choice and income with prices fixed is called the Engel curve Figure 63 2 Changes in price a this is a rotation or pivot of the budget line b decrease in price increases demand 7 ordinary good Figure 69 c decrease in price decreases demand 7 Giffen good Figure 610 d as price changes the optimal choice moves along the price offer curve e the relationship between the optimal choice and a price with income and the other price fixed is called the demand curve C Examples 1 perfect substitutes Figure 612 2 perfect complements Figure 613 D Substitutes and complements abc1 1 increase in p2 increases demand for x1 7 substitutes d gt 0 Z 2 increase in p2 decreases demand for x1 7 complements i lt 0 P2 E Inverse demand curve 1 We usually think of demand curve as measuring quantity as a function of price 7 but can also think of price as a function of quantity 2 this is the inverse demand curve 3 same relationship just represented differently A Mushfiq Mobarak Econ 3070003 Chapter Outlines Chapter 8 Substitution Effects and Income Effects A We want a way to decompose the effect of a price change into simpler pieces 1 Break up into simple pieces to determine behavior of whole 2 When price decreases two things happen a relative prices change one good becomes more attractive to buy relative to the other good b Purchasing power of your money real value of income increases B Break up price change budget line rotation into a pivot and a shift 7 see Figure 82 1 these are hypothetical changes we draw an imaginary line 2 we can examine each change in isolation and look at sum of two changes C Change in demand due to pivot is the substitution effect 1 this measures how demand changes when we change prices keeping purchasing power fixed 2 By drawing the imaginary line the question we are asking how much would a person demand if he had just enough money to consume the original bundle 3 this isolates the pure effect of changing the relative prices 4 substitution effect must be negative if price of a good increases it becomes relatively less attractive to buy so negative means quantity moves opposite the direction of price D Change in demand due to shift is the income effect 1 increase income keep prices fixed 2 income effect can increase or decrease demand depending on whether we have a normal or inferior good E Total change in demand is substitution effect plus the income effect 1 if good is normal good the substitution effect and the income effect reinforce each other 2 if good is inferior good total effect is ambiguous F Speci c examples 1 perfect complements 7 Figure 84 2 perfect substitutes 7 Figure 85 3 quasilinear 7 Figure 86 G Slutsky Equation
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