Intermed Micro Theory
Intermed Micro Theory ECN 100
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Compensatlng and quvalent Vanatlons Compensating Variation how much money should we give the consumer in order to compensate her for the reduction in her wellbeing due to an increases in the price of a good This amount of money is called the compensating variation in income Note that we may not actually pay the consumer any money we are just trying to nd a monetary measure of a loss in well being Look at Figure 1 Suppose the consumer is initially at point A Then the price of good 1 increases and the BL rotates and becomes blue The consumer is worse off since she moves to a lower IC How much money should we pay her to bring her back to her initial IC In other words how much money should we pay her to shift her BL to the green position This amount of money is called compensating variation Figure l Page 1 of6 Equlvalent Varlatlon This is the reduction in consumer income that results in a reduction in well being that is equaivalent to a loss of well being resulting from a price incresae Look at Figure 2 If the price of good 1 increases the consumer will move from bundle A to B and will become worse off If instead of this price change we had taken away some income from the consumer and as a result the consumer had moved to the same indifference curve that amount of income would have been called the equivaelnt variation Figure 2 Equivalent Variation Page 2 of6 Algebra of Compensating and Equivalent Variations Cobb Douglas Utility Functions Suppose we have UX18X22 M 100 P1 2 P2 4 Therefore Xf 8 X 1002 40 X5 2 X 1004 5 These are point A in Figure 8 Now PlT to 4 If we don t do anything the consumer will move to point B in Figure 8 She will be worse off because she will be on a lower IC How much money should we pay the consumer to compensate her for this price increase Well what is her utility at the initial bundle point A U 40852 2640 If we give her some money to raise her income to M how much will she consume at the new prices X18 gtlt M 4 Xzquot 2 X M 4 We want this new bundle to give the consumer the same amount of utility as the initial bundle So U 8 X M 43982 gtlt M 42 2640 01 M 152640 M 17411 So the compensating variation is M M 17411 100 7411 Now the equivalent variation As a result of the increase in the price of good 1 to 4 the consumers optiomal bundle becomes Xf 8 X1004 20 Xi 2 X1004 5 So her utility becomes Page 3 of6 U 20852 1516 Now what level of income with the old prices would give the same amount of utility Let this level of income be M Then U 8M 23982M 392 1516 M 5743 So the equivalent variation is 100 5743 4257 Note in graphing that in the above cases when P1 changes this change only affects X1 and not X2 See Figure 10 below for the compensating va1taion The case of equivalent variation is similar This is because of the nature of the utility function If we had a different utility function we would get a different result Compensating and Equivalent Variation Quasi Linear Utility Functions A consumer has the following utility function u 2X10395 X2 QuasiLinear Utility Function Assume that initially m 100 p1 2 and p2 10 Then p1 increases to 5 Find the CVandEV l MRS 6X1 ri l 7 7 MRS Slope ofbudget line From this we find 2 2 1 p2 10 X1 i E 25 Plug this in the budget line to get P1 Page 4 of6 X2ampE 2 P2 P1 10 2 5 verify this u 2255 5 15 Now let p1 5 Increase consumer income to m such that with the new price the consumer can reach u 15 2 x 4 5 quot245 y 215 u 10 m39 130 VC 130 7100 30 NowEV 102 x1 t 100 10 xzzi i 10 5 u 245 8 12 How much income should we take away from the consumer so that at the old prices the consumer reaches u 12 u 2255 3 912 10 2 so m 70 and EV 100 70 30 same as CV Page 5 of6 University of CalifomiaDaVis TA Jason Lee ECN lOOSpring 2008 Quarter Email jawleeucdaVisedu Handout 4 I The Budget Constraint We saw with indifference curves that the typical consumer would have higher utility the more goods she had The problem is that there is a constraint on the bundles we can consume and that constraint is our income We start off with two important assumptions 1 We only consume 2 goods in our consumption bundle 2 We spend our entire income no saVings or borrowing is allowed As our result we get a condition as follows PXX PyY M This equation tells us that the total expenditure of good X plus the total expenditure on good Y must equal our income If we solve for Y we can get our general budget line equation Y M iX PY PY Where 7PxPy slope of the budget line Example 1 Suppose Jason has an income of 18 and he buys only two goods Bottles of Jack Daniels J and magazines M Suppose PJ 9bottle while PM 3 In drawing your graphs assume that Jack Daniels is on the x axis and magazines are on the y axis a Find the equation of the budget line b Draw the budget line Important Facts about budget lines 1 If income changes but prices of the goods remain unchanged then the budget line will shift outward if income increases or it will shift inward if income decreases The slope of the budget line will be unchanged 2 If the price of one of the goods changes but income and the price of the other good remain constant then the budget line will pivot 3 If prices and income increase or decrease by the same factor the budget line will not shift Example 2 Suppose that Jason s income increases to 36 Everything else is held constant Draw the budget line Example 3 Return to the situation in Example 1 Now suppose that the price of magazines doubled to 6 but everything else remains constant Draw the budget line Not all budget lines are nice straight lines with a constant slope as the following example illustrates Example 4 Jane consumes two goods Electricity E and Food F PGampE charges 010 per kWh for the first 1000kWh of power each month but only 005kWh for all additional kWh The price of food is 1 a pound Suppose Jane has an income of 400 Graph the budget line In your graph place electricity on the horizontal axis and food on the vertical axis II Consumer Choice A Graphical Representation Now that we have studied indifference curves and budget lines we are ready to determine which consumption bundles consumers will actually choose In order to determine this optimal consumer choice we must determine which bundle maximizes consumer utility subject to a budget constraint In mathematical notation we say that we wish to Max UXY subject to PXXPYY I In other words we want to reach the highest indifference curve possible without exceeding our budget Figure 1 illustrates this point wquot Which bundle will the consumer choose A B or C 9 Bundle C would clearly be preferable since it is on a higher indifference curve The problem is that it is not affordable Bundles A and B are both affordable since they are on the same budget line But Bundle B is on a higher indifference curve than Bundle A Thus B is the preferred bundle One way we can nd the optimal bundle graphically is to apply what the book calls the no overlap rule The no overlap rule states that the area above the indifference curve that goes through the consumer s optimal bundle must not overlap with the area below the budget line Note from our graph that the solution to the consumer s choice problem is found where the budget line is tangent to an indifference curve This has an important implication because at that point both the slope of the indifference curve and the slope of the budget line are equal to each other Implication MRSXY PKPy Most of the time the solution will be an interior solution We ll de ne an interior solution as an optimal bundle that has both goods in it Example 5 Mark is a big guy with an income of 300 and he spends all of it on two items Fritos F and Cheetos C Costco sells a box of Cheetos for 15 a box and sells a box of Fritos for 10 a box His MRSCF 3F2C What is Mark s optimal bundle Sometimes the optimal solution is what we call a corner or boundary solution A corner solution is an optimal bundle in which the consumer does not consume one of the goods Figure 2 illustrates an example Note that in Figure 2 our 53 usual tangency condition no Y I39M NI longer applies To gure out the optimal bundle we should use our nooverlap rule Notice how the area above U1 and U2 overlaps with the area below the budget line Only at U3 do we see no overlap It must be the case that A is the optimal bundle At bundle A the slope ofthe indifference curve is clearly steeper than the slope of the budget line A s The condition now is MRSKy gt PKPy Let s try to understand why in this case a consumer would be better off NOT consuming any of good Y First off recall that the MRSKy tells us how much of good Y a consumer is willing to give up in order to gain an extra unit of good X Suppose we start from bundle A and we tell the consumer than she must give up 1 unit of good X In that case the consumer must be compensated by getting MRS Ky units of good Y But according to P the budget line Y Pg P7X if a consumer gives up 1 unit of good X he can buy only Y Y PxPy units of good Y Since we have shown that MRS Ky gt PxPy the consumer cannot buy the necessary amount of Y to compensate him for losing 1 unit of X He would thus be worse off if he had to sacri ce any amount of good X Thus he will consume only good X and no good Y Many students might have trouble understanding this point Try reading p 136137 in your text a couple of times to see if you understand the very important intuition B Marginal Utility and Consumer Choice Let s introduce the concept of marginal utility Marginal utility of good X MUX is de ned as the extra utility a consumer gets if the consumer consumers one more unit of good X The marginal utility of Y is de ned in a similar manner In notation we have MUx AUAX and MUy AUAY We can easily apply these de nitions to our tangency condition to see how we can use marginal utility to determine consumer choice Recall that the tangency condition is MRSxy PxPy By de nition MRSxy AYAX We can rewrite our marginal utility equations above to get the following AX AUMUx and AY AUMUy Plug these into our MRS formula AU MU jVjRSXy AUy after some rearrang1ng you should get MRSxy MUxMUy M U I Thus our optimal choice equation is now a M U y Py If you are ambitious you can rewrite the above equation to get the following M U I M U y PX Py This equation tells us that at the optimal bundle the marginal utility gained from an extra dollar of expenditure is the same for both goods Example 6 Pat is a very immoral person and is currently dating two people Jamie and Sam Pat s utility function is UJS J S Where J number of romantic dates with Jaime umber of romantic dates with Sam Pat s marginal utility is as follows MUJ S and MUS J Jamie is a high maintenance person and it costs 30 to take Jamie on a date PJ 30 while Sam is a cheap date P5 15 Assume that Pat s income is 300 What is the optimal level of dates Pat will go on with each person 111 O timization under Price and Income Chan es A Chan es in Pr39ces This section looks at what happens to the optimal bundles if price and income were to change Figure 3 illustrates a price change in good X Suppose we start at L1 as our budget line which is associated with a Price of Pm In such a case the optimal bundle will be bundle A Now suppose the price of X increases to sz The budget line will pivot inward and we will be at a new bundle B Notice how at this new price the consumer s optimal bundle contains fewer units of good X than before Suppose instead that the price ofX decreases to PX The budget line will pivot outward and the new bundle will be a C At this bundle the amount of X consumed is greater than in bundle A We can easily plot the points of the price of good X and the corresponding amount of good X consumed in the optimal bundle see the bottom graph You have your regular downward sloping demand curve WM 7x0 MEX If you trace out all the optimal bundles given various price changes of a good you ll end up with a priceconsumption curve The price consumption curve shows the optimal bundles given a price change holding everything else constant If the price consumption curve is positive then the two goods in the bundle are complements The reason why is that if the price of good X decreases the consumer is buying more of good y See Figure 3 for an illustration If on the other hand the price consurnption curve is negative then the two goods in the bundle are substitutes See Figure 516 on page 149 for an example University of CalifomiaDavis TA Jason Lee ECN lOOSpring 2008 Quarter Email jawleeucdavisedu Handout 1 The Preliminaries 0 Office Hours will be Tuesday 1230l30pm Thursday 200300pm and by appointment My office is located in 120 SSH 0 I ll have a class site set up with notes sample problems and other course related materials and links at httpwwweconucdavisedugraduatejawlee Click on the Economics 100 Materials Tab I Bene ts vs Costs In economics there are two ways of determining the optimal level of an activity 1 Your best choice is the one that maximizes your net bene t We de ne Net Bene t Total Bene t 7 Total Cost Thus you will choose a level of activity such that the difference between the total bene t and the total cost is the greatest Oftentimes however economists are not so much interested in total costs and total bene ts but are interested in decisions based on the last unit of activity activity on the margin In our everyday lives we make decisions on the margin Do I study an extra hour for a test Do I work an extra hour of overtime We therefore ask questions about costs and bene ts on the margin 2 Your best choice is where Marginal Bene t Marginal Cost De nition Marginal Bene t of an activity is the extra bene t received due to an extra unit on an activity Mathematical De nition MB ABAX slope of the bene t function De nition Marginal Cost of an activity is the extra cost incurred due to an extra unit of an activity Mathematical De nition MC ACAX slope of the cost function Decision Chart If MB gt MC You should increase the level of activity If MB MC This is the optimal level You should stop the level of activity If MB lt MC You should decrease the level of activity 11 Practice Problems Question 1 The following chart looks at the costs and bene ts of spending time xing your car Hours Total Total Cost Net Marginal Bene t Marginal Cost Worked X Bene t B C Bene t MB AMAB MC ACAX 130 0 0 0 1 6 15 1 50 2 1 15 0 3 80 3 1600 690 4 1975 1080 5 2270 1550 6 2485 2100 a Fill in the Net Bene t Column Using only the net bene t column determine how many hours should you spend xing your car b Calculate the MB and MC Using the information from the MC and MB columns how many hours should you spend xing your car The following question requires some calculus This class will be more quantitative than what you were exposed to in Economics 1A However the mathematics will be at a level similar to what you would have seen in Math 16A You will need to know some basic calculus how to take derivatives and simple algebra If you feel that your math background is inadequate you should talk to the professors or one of the TAs early in the quarter Question 2 The bene t for an C average student from studying for his ECN 100 midterm is expressed in the following bene t function B 420H 7 40H2 where H hours studied His cost function from studying for the midterm is C 100H 120H2 What is the optimal amount of hours this student should study for the midterm Question 3 Marcus has a exible summer job working for Walmart He works every day but is allowed to take a day off anytime he wants However Walmart does not pay its employees when they take the day off His friend Samantha suggests they take off work on Tuesday and go to Six Flags Discovery Kingdom in Vallejo Samantha has coupons so the admission charge to get into Discovery Kingdom is only 35 per person and it will cost them 5 each for parking and they will both spend 20 each on food Marcus loves amusement parks and a day at the park is work 110 to him However Marcus loves working at Walmart so much that he would be willing to pay the Walton family 10 per day to work there If Marcus earns 50 a day at Walmart should he go to Discovery Kingdom University of CalifomiaDavis TA Jason Lee ECN lOOSpring 2008 Quarter Email jawleeucdavisedu Handout 6 1 Costs Much of the material on costs you should have been exposed to in your ECN lA course so we ll brie y highlight the main points De nitions Total Cost Expenditure required to produce a given level of output Fixed Cost Cost that does not depend on the level of output produced Variable Cost Cost that does depend on the level of output produced Total Cost Fixed Cost Variable Cost Average Total Cost Total Cost Quantity Produced Average Fixed Cost Fixed Cost Quantity Produced Average Variable Cost Variable CostQuantity Produced Average Total Cost Average Fixed Cost Average Variable Cost ATC Marginal Cost E The marginal cost 1s the change 1n total cost associated w1th the production of an extra unit of output Practice Problem 1 The following table gives a hypothetical set of cost gures for a rm a Given the information in the table ll in the blanks Output Fixed Variable Casts Cast Fixed Casts C asls C 0st C 0st b Using EXCEL or another graphing program graph the 4 main cost curves MC AFC AVC ATC Once you graph the MC AFC AVC and ATC you should notice the following facts regarding cost curves If you were asked to reproduce the cost curves you would have to make sure the following rules were followed 1 The marginal cost curve intersects the AVC and the ATC at their minimum points Once again this makes sense because as long as MC is below either AVC or ATC those curves are decreasing It is only when the MC is above the AVC or ATC will those curves start increasing When MC AVC 01 MCATC they must be at their lowest point 2 As output increases AFC gets smaller and smaller 3 As output increases the difference between average total cost and average variable cost decreases 4 AVC and ATC are generally UShaped in that they fall initially and then increase AVC falls because of specialization early in the production process However as we increase output it takes more and more labor to get more output which increases costs the law of diminishing returns It is always the case that AVC curve will be below the ATC 5 MC will generally fall in the beginning for a short while and then increase It is generally a Jshaped curve Now let s look at the situation if we allow all the inputs to be variable which is the long run situation In order to understand the longrun average cost curve imagine a single firm deciding on the construction of a single factory In the longrun the firm can choose among various factory sizes each factory size generates its own shortrun average total cost curves What is the optimal size to build That depends on the anticipated output At each given level of output in the long run there is a cost curve that provides the lowest cost The Long run average curve represents the path of points giving the least unit cost of producing any given rate of output szmge c051 22 on 20 cu m on Longrmn Average cm a LOUD 2mm moan some Gummy 01 books salt Der mnnlh Some key de mmhs W ww wuwn mt osts slopes upwardran mcrease m scale and production mereases umt eosts II ocost Lme The isocostlme is very similar to the budgethhe thatwe studmdm Chapter 5 Isoeosthe WLr Where wwage r c K c rental pnee of capital L amount oflabor hhed K amount of capital used ots1 eost In Words The isocostlme eohtams all the mput eombmatzohs thh the same cost C w L Wecanrewritethexsocostlmetothefollowmg Graphically xtlooks exaetly the same as the budgethhe Armed with the isocost line and the isoquant curves we can now nd the level of inputs that will give the rm the leastcost production In the following gure point B is the leastcost production Point C will allow the rm to produce the same level of output but it is on a higher isocost line Point A is on a lower isocost line but it requires the rm to produce less output At the point where the isocost line and the isoquant are tangent that is where the rm can produce output at the lowest cost The goal of the rm is to try to produce a given level of output at the lowest cost L The leastcost point can be found where MRTSLK 3 or MPL 3 or MPL MPK r WK r w k The last condition states that the rm s best choice is where the marginal product from the extra dollar spent on labor is equal to the marginal product from the extra dollar spent on capital Practice Problem 2 Suppose that the production function for making Nintendo Wii s is Q 10L0 2K0 3 Further suppose that w 1500 per week and r 1000 per week Assume that MRTSLK 23KL What is the leastcost input combination for producing 100 Wii s each week What is the total cost 111 Perfect Competmon H vemecquot L 1 Firms are so small and they are facing so much competition that their own behavior cannot determine the mar et price 2 The producm sold are identical 3 It is very easy for new rms to enter the industry 39 L 39 39 r 39 in L L L will have to charge e m l L r J J 39 very uiliei 39 39 39 quot and supply curves not by individual demand and supply Once the market equilibrium has been ompetitive 439 L L L determined At this L c i 4 r ey want quot L 4 Lu rum piouuci is horizontal perfectly elastic Example Susan and ianriei 39 iui 39 39 quot basket Strawberry farming is a perfectly competiti e industry because there are low of strawberry farmers the strawberries are identical and it is relatively easy for someone to grow strawberries If Stan tried to charge 3 a basket for strawberries every would just go to Susan to buy their strawberries Also Stan would a 39 ehe rniild strawberries for ahigher price Thus Stan faces a perfectly elastic demand curve for his strawberries See Figure 1 below Figure 1 T39m N39s antsAria Some Formulas and a Numerical Example 1 The first step is to define Profit Total Revenue 7 Total Cost 2 Average Revenue Total Revenu Quantity However we can L 4 quot 39 39 otal J c cuuc 7 Thus Average Revenue Price under perfect competition 3 Marginal 39 L 39 MR ATRAQ APxAQAQ P Thus Marginal Revenue Price under perfect competition 4 ioget ARmu39 r is r quot MR MC the extra unit produced should equal the cost ofproducing that unit 5 How do we know ifa firm is experiencing profim or not i Use the definition of H TR 7TC ii Divideboth sides ofthe equation by Q E 3 7E 7E P iATC Q Q Q Q Q iii Multiply both sides by Q H QPATC Thus we have the following rule i IfP gt ATC the firm is experienci g profits ii IfP ATC then the firm has 0 profim iii IfP lt ATC then the firm is experiencing losses Let s apply the abovementioned formulas and rules into a numerical example Suppose that a firm sells corn for 1250 a bushel a What is the profitmaximizing output b What is the profit at the output found in point a 0 Draw the MC ATC AVC and MR curves in a graph Since PMRMC when output 6 this is the profit maximizing output The firm should produce at this level At output6 the firm s profit is 2250 25 20 ATC 15 AVC 99 10 MC MR 5 0 0 5 10 Output III Shut Down De01510n As we have seen when P lt ATC the rm experiences losses When the rm has losses it has two choices 1 Continue to produce 2 Stop production by shutting down win a nun Wuulu r simple answer is 39 39 team Au e u 39 quot quotthey output losses The If a rm does not produce any output it will suffer a loss equal to its xed cost This is the maximum loss a rm is willing to accept Ifby producing a rm loses an amount greater than its xed cost the rm will shut down If the total revenue is greater than total variable costs it can use some of the extra money to cover some of the xed costs In this case the rm will continue to produce even though it is suffering losses By the same token the rm will shut down ifthe total revenue is less than its variable cost because its losses will be greater than is xed cosm D I L0 ifTRltx m PXQltx m If the price drops below the AVG the rm will have a smaller loss if it shum down completely and produces no output IV Entry and Exit of Firms in the Long Run 0 Economic Pro t Leads to Entry of New Firms See Figure 2 Mme 391 leth Elm At P the individual rm would be experiencing pro ts since P gt ATC where MR MC Other entrepreneurs will see the rm making pro t and will want to get into the industry This will shift the market supply curve to the right Firms will continue entering until the market reaches a price where there is no pro t where P ATC Economic Losses Leads to Exit ofFirms See Figure 3 Ecn 100 Classnotes RETURNS TO SCALE an SHAPES OF COST FUNCTIONS In these notes following Pindyck and Rubinfeld ch 7 74 on quotEconomies and Diseconomies of Scalequot and 77 on quotCost Functions and the Measurement of Scale Economiesquot we let EC ratio of change in total costs over change in output aTC aTC TC 3Q MC 7 E E ATC Q Q measures the elasticitv of cost with respect to output and we let SCI l Ec be an index of economies to scale We shall see next week that another useful measure of economies of scale is given by the elasticitv of output with respect to the scale of inputs escale ratio of change in output over change in scale of inputs and this turns out to equal ATC MC l EC I Increasing Returns to Scale escale gt 1 SCI gt 0 and Ec lt 1 In this case ATC is decreasing as Q increases MC lt ATC MC quotpullsquot ATC down Rationales Short Run Fixed costs cause ATC to decrease over an initial range of Q Long Run 1 Large Setup Costs quasi xed costs o en called quotIndivisibilities in productionquot These are not xed in the Long Run but they are lumpy a set sum must be invested in order to produce anything at all As output increases this set sum is averaged over more units which decreases ATC Examples building the transcontinental railroad nuclear power plant or the space shuttle Four examples that are especially important now i Research costs for new products RDTE Research Design Testing and Evaluation ii Creation of brandname capital cannot be done in every market like soybeans for example iii Creating and maintaining distribution networks Ben amp Jerry s vs Pillsbury s DoughBoy iv Division of Labor If people can specialize on doing one or a few tasks they can often collectively end up being more productive since they waste less time shifting from task to task Adam Smith39s example was of pin making If one person does all the steps then she wastes time shifting from machine to machine and also wastes machine time as they do not get used intensively or else does everything in less capital intensive ways This is probably best thought of as a special case of quasi xed costs 2 6 Rule Applies where capacity of production depends on the volume of a physical container e g a bottle but the cost of the container depends on the surface area of the container so the capital cost increases less quickly than the volume as the tank gets larger Examples water towers pipelines supertankers airplanes re neries beer production sewage treatment Detailed reasoning Total Fixed Cost of vat constantsurface area of sphere constant 7 d2 7 cl d2 c1 captures the cost of the steelcopperwhatever used to make the vat volume of sphere 167rd3 02613 Thus d Qc213 so Tc c1 Qc2132 c3Q23 Hence ATC TCQ c3Q13 so ATC decreases as Q increases Note that approximately TC Q raised to the power 6 which is where the name of this rule comes from 3 Insurance Principle This means that large rms with eggs in many baskets in many independent investment projects have less risk in their overall return This leads large rms to be able to borrow more cheaply than small rms Whether this leads to any real advantage is debated in the literature of corporate nance A related argument also leads larger rms to have lower average costs of maintaining inventories Safeway has fewer dollars of inventory per dollar of sales than does the local green grocer II Constant Returns to Scale escale Ec 1 and SCI 0 In this case ATC is the same for all Q and so ATC MC Rationale All factors used in production can be duplicated so the recipe for production can be perfectly replicated This means there are no specialized factors of production Examples copy and print shops chocolate chip cookie shops Note that we are discussing long run costs so it is OK to take a lot oftime to replicate a rm III Decreasing Returns to Scale escale lt 1 SCI lt 0 and Ec gt 1 In this case ATC is increasing as Q increases MC gt ATC MC quotpullsquot ATC up Rationales 1 One scarce specialized factor of production which cannot be duplicated Examples Michael Jordan Shaquil O39Neil parts of Kuwait and Saudi Arabia locations in NYC sunny locations in California and Florida for retirement homes 2 Managerial Diseconomies of Scale The complexity of management coordinating supervising and motivating workers increases more than in proportion to the size of output Q increases gt workers increases gt of interactions among workers goes up as workers2 gt larger firms have larger costs of management W This can conceivably set a limit on how large rms become Example dairy farms The increase in management costs as aggregate Q grows will be greater if there are rather different types of output going into quotQquot This is one factor why some large conglomerates have chosen to break up recently since the hoped for quotsynergiesquot gains from combining somewhat distinct businesses did not turn out to big enough to offset the increased costs of management University of CalifomiaDavis TA Jason Lee ECN lOOSpring 2008 Quarter Email jawleeucdavisedu Handout 3 This handout is dedicated to those who couldn t make the makeup lecture on the evening of April 15 The material covered in the lecture corresponds to the Chapter 4 handout slides 821 The handout will try to summarize some of the key points as well as work out some of the examples done in lecture I will refer liberally to the handout slides so you should have them handy I Properties of Indifference Curves The following properties hold true if the moreisbetter principle holds true 1 Indifference Curves Must be ThinRefer to handout slide 8 gure a In that figure we have drawn a thick indifference curve which indicates that any bundles within the shaded area is equally preferred Such as bundle A and bundle B However if we assume the more is better principle holds true such thick indifference curves cannot hold true You have in this case a bundle B in which the consumer has more soup and more bread than bundle A The consumer must prefer bundle B to bundle A Thus A and B cannot be on the same indifference curve 2 Indifference Curves Cannot be Upward Sloping Refer to handout slide 8 figure b We have drawn an upward sloping indifference curve where the consumer prefers bundle C and bundle D However according to the more is better principle bundle D should be preferred since the consumer gets both more bread and more soup Thus indifference curves can t be upward sloping if the moreisbetter principle holds true 3 Consumers prefer bundles 0n indifference curves further away from the origin Refer to handout slide 9 Here we have a series of indifference curves for the same individual based on 5 hypothetical consumption bundles for bread and soup Which bundle would the consumer prefer He would clearly prefer bundle F over any other bundle since he gets more bread and more soup than any other choice If the moreis better principle holds then it must be the case that an indifference curve farther away from the origin represents higher utility happinesssatisfaction 4 Indifference Curves of the same individual cannot cross Refer to handout slide 10 Here we have 2 indifference curves belonging to the same individual Bundle A lies on both indifference curve From the graph we can say that the consumer is indifferent between consuming bundle A and bundle B and likewise the consumer is indifferent between bundle A and bundle C If the transitivity assumption holds we must make the conclusion that the consumer would also be indifferent between bundles B and C However the more is better principle does not allow us to make that conclusion here Bundle C has more soup and more bread than bundle B We must expect that the bundle C would be preferred to bundle B This is why indifference curves of the same individual cannot cross II Indifference Curves quot Good and Bad Up until now we have been assuming that both goods in a bundle are desirable In those cases our moreis better principle would generally hold and we would have our usual convex shaped indifference curves What if one of the goods was a bad In other words what if we had to consume something we normally wouldn t want in our consumption basket such as air pollution loud music at 3am George W Bush presidency etc How would our indifference curve look if we have 1 good good and 1 bad good Figure 1 shows just such a scenario Figure 1 Indifference Curves Representing Good and Bad L Food Garbage Why would the indifference curve take this shape The intuitive explanation is as follows suppose our consumer is given bundle A with so many units of food and garbage in his initial bundle Since we assume that garbage is a bad we assume that a consumer is happier with less garbage In this example as we move from bundle A to bundle B the consumer is unhappy about losing some food which he likes however he is happy that he is also losing some garbage which he hates Overall as we move from A to B the consumer is indifferent between these two bundles The consumer receives higher utility as we move to the northwest where he is consuming more food and less garbage he is happiest We can get around these types of problems by rede ning the bad good Instead of having garbage as the good we can rede ne the good as absence of garbage which is a good good If we do that we get a normal moreis better assumption and our usual convex indifference curves III Marginal Rate of Substitution De nition The marginal rate of substitution MRS is the rate at which the consumer is willing to exchange the good measured along the vertical axis for the good measured along the horizontal axis It is equal to the slope of the tangent line of the indifference curve AY Mathematical Formula MRS XY 7 Note the MRS will be different if you ip the X and the Y variables and you will get an incorrect answer Be sure that the change of the good measured on the horizontal axis is always on the denominator On your homework or on exams we may ask you to nd the MRS for X with Y or alternatively MRS of Y for X It is understood that the good which is preceded by for is the Xaxis good and the other good is the Yaxis good Example Find the MRS for Soup S with Bread B Answer MRSSB amp AS Example Find the MRS of Dogs D for Cats C Answer MRSCD AC Example Refer to Handout Slide 13 What is the MRS for Soup with Bread at Point A Answer We see that the consumer is willing to exchange 1 unit of soup for 2 units of bread at bundle A and move to bundle C MRSSB 2 Handout Slide 14 shows that for very small changes in the xaxis the best estimate for the marginal rate of substitution at a given point is the slope of the tangent line that goes through that point In the example on Slide 14 the slope of the tangent line is 32 as a result the MRS 32 32 One important feature to notice is that the as we move along the indifference curve the marginal rate of substitution changes as well In fact most indifference curves have the property of diminishing marginal rate of substitution When we have diminishing MRS we say that the amount of good Y one must give up to get an extra unit of good X decreases as X becomes more plentiful and Y becomes more scarce Slide 15 illustrates an example of this property To give some intuition consider the following example Suppose you had a shopping cart full of food but you were homeless You would be willing to give up a lot of food in order to get some form of shelter to escape the inclement weather Now suppose you find yourself in a situation where you own a mansion but don t have much food to eat In this case you would be unwilling to give up much food if any at all to add to your shelter by building another room to your mansion This is exactly the situation illustrated in handout slide 15 IV Indifference Curves and Preferences Indifference curves are also useful in that they can illustrate people s preferences for different types of goods Refer to Slide Handout 16 Here we have the indifference maps for two individualsTex Figure a and Mohan Figure b They both are consuming potatoes and rice The only difference is that Tex s indifference curves are atter than Mohan s indifference curves What does this imply Well we know that the MRS is the slope of the tangent line at a given point Suppose we drew a tangent line at Point A for both Tex and Mohan We can easily conclude that the tangent line for Mohan s indifference curve will be steeper than for Tex s indifference curve In other words the MRS at point A is greater for Mohan than for Tex Mohan is willing to sacrifice more potatoes to gain 1 extra unit of rice than Tex In this example we can conclude that Mohan loves rice more than Tex Conversely we can see that Tex is willing to sacrifice more rice to gain an extra unit of potatoes We can therefore conclude that Tex loves potatoes more than Mohan V Special Cases Perfect Substitutes and Perfect Complements A Perfect Substitutes Two goods are perfect substitutes if they are viewed by the consumer as having identical functions A consumer is willing to swap one for another at a fixed rate The example used in the lecture was two types of Advil headache medicine Suppose there are two varieties of the medicine an extrastrength version with 400mg of active ingredients and a regular version of 200mg of active ingredients Clearly taking two of the 200mg pills is equivalent to taking 1 400 mg pill A consumer would be indifferent between a bundle that had 2 200mg pills and 0 400mg pills versus a bundle that had 0 200mg pills and l 400mg pills Both will have an equal effect and the consumer will be equally happy We can continue the thought experiment in a likewise manner for different combinations where the consumer will always substitute at the same 21 ratio Figure 2 shows the indifference curve map What do the indifference curves look like H 20 Dr My I7 M IS M MW 1 900MB Nmammummo mm 7 8 9 w 3 4 5 6 AcleHoomg Do the above indifference curves demonstrate diminishing marginal utility They do not Note that each indifference curve is linear with a constant slope No matter where you are on the indifference curve the slope is the same The consumer will always trade 1 400mg tablet for 2 200mg tablet We can represent this indifferent curve as a utility function UXy 2x y Where X 400mg Advil Tablet and y200mg Advil Tablet Double check to make sure that this utility function gives you the indifference curves you see above B Perfect Complements Two products are perfect complements if they are valuable only when they are used together in xed proportions The obvious example of perfect complement is a right shoe and a le shoe Consider a bundle where a consumer is given 1 right shoe and 1 le shoe What bundles would a consumer nd equally attractive If a consumer has 1 right shoe and 2 left shoes he would be indifferent between that bundle and his original bundle The extra left shoe is worthless by itself Likewise if a consumer has 2 right shoes and 1 left shoes he would be indifferent between that bundle and the original bundle Try drawing the indifference curve for this example See Figure 413 for the answer on how it looks like Refer to Slide Handout 21 In this example Sarah will always use 2 marshmallows m for every cup of hot chocolate This proportion is xed Sarah will be indifferent to extra marshmallows if she doesn t have a proportional increase in hot chocolate and vice versa Figure 3 shows how the indifference map would look like for this example ll
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