Preferences. Preferences. Budget Constraint and Line 9/10/2014


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1 9/0/04 references Chapter 4 At this point, we know a lot about preferences and their representation with utilit functions references tell us how a consumer ranks a given bundle compared to another bundle, and the utilit function tells us how much utilit a consumer derives from different bundles. references However Different bundles do not alwas cost the same amount, and therefore, all of the available bundles ma not be reasonable given a consumer s budget. Therefore, in order to determine how a consumer chooses amongst different bundles, we need to take into account not onl consumer preferences but also the budget constraint. Budget Constraint and Line The budget constraint is the set of bundles that a consumer can purchase with a limited amount of income. The budget line is the set of bundles that a consumer can purchase when spending all of his or her available income. Eample.
2 9/0/04 Budget Constraint Suppose a consumer purchases onl two tpes of goods, food and clothing. Let = food = clothing = income To construct the budget line, we must mathematicall represent that the total ependiture of and coincides with the consumer s income = Eample X Y Let s assume that p = 0, p = 40, = G is unattainable at the current prices and income p + p = = $,000 > $ 800 Budget Line: p + p = where p total ependiture on food p total ependiture on clothing At F: p + p = illustrates the budget constraint because it states that ou can onl spend as much mone as ou have To find the slope of the budget line, we must solve for (the good in the vertical ais) p + p = p = p Notice that this equation has the linear format of =b+m, and thus, the slope of the budget line is Note that if =0, then the consumer can afford (vertical intercept), while if =0, the consumer can afford (horizontal intercept) The slope of the Budget Line tells us how much of the good on the ais we must give up to bu one more unit of good on the ais, as we move from left to right on the figure. From our eample with food and clothing 0 40 Thus, the slope indicates that our consumer must give up ½ a unit of clothing () to acquire one more unit of food (). ntuitivel this makes sense because food is half as epensive.
3 9/0/04 ncome Change But how does the budget line change when income changes? Given constant prices, the budget line will make a parallel shift outward with a raise in income, or inward with a reduction in income. The budget line remains parallel because the slope, (p /p ), remains unchanged, assuming onl income is changed Slope of BL = arallel Shift in the Budget line Slope of budget line unchanged at rice Changes What about a change in prices? A change in the price of either good changes the slope of (p /p ). For instance, a change in the price of food () from 0 to 5 changes the slope to Thus, the consumer must give up more clothing to obtain more food rice Change 0 5 Steeper budget line BL ivots nward Reduction in the set of affordable bundles
4 9/0/04 What if both income and all prices are doubled? s this good/bad news for the consumer? A proportionate change in income and prices (let s sa b doubling each) causes no change in the budget line Vertical intercept /p = /p Horizontal intercept /p = /p Slope = p /p = p /p The s cancel out and leave B.L. unchanged Hence, the consumer s purchasing power is unaffected. utting references and B.L. Together Given preferences and budget constraints, we can find a consumer s optimal choice, the optimal amount of each good to purchase. His or her optimal choice bundle will be a bundle that fulfills two conditions () Maimize Utilit (preference) () s affordable (lies on the budget line) mmediatel we can see that the optimal choice bundle must be located on the budget line. Wh? Because an bundle inside the B.L. would leave the consumer with etra mone that could be spent to further increase utilit And an bundle outside the B.L. wouldn t be affordable n this continuing eample, A is the optimal bundle. t is the onl affordable bundle that reaches a utilit level of U. 4
5 9/0/04 Wh other baskets are not optimal? D unaffordable C mone unspent could increase our utilit b spending it (generall, it could be future consumption of the good, or savings) B lower.c. than at A Notice that at the optimal bundle A, the budget line and the utilit curve U are tangent. That is, the have the same slope (This is crucial!) From chapter, we know that the slope of an indifference curve is the marginal rate of substitution, MU MU Slope of.c.= Slope of B.L. Therefore, MU MU or MU MU This is called the Tangenc Condition and it reveals nterior Optimums to us, where the optimum bundle is at a tangenc between B.L. and.c. B rearranging the terms we get MU MU (The marginal utilities per dollar of both good and at bundle A are equal, so ou get the same BANG FOR YOUR BUCK for each good) Etra utilit per dollar spent on =etra utilit per dollar spent on f this were not the case, the bundle would not be optimal because ou could take some dollars awa from the good with less marginal utilit per dollar and put it into buing more of the good with a higher marginal utilit Eample U(,) = Hence, MU =, MU = = 800, p = 0, p = 40 Tangenc condition ) p + p = = 800 ) MU 0 MU 40 MU MU p p A 0() + 40 = = 800 = 0 = = 0 = 0, 0,0. 5
6 9/0/04 oint B=(8,6) was not optimal because Thus, at point B MU MU MU MU MU Utilit Maimization roblem MU and (At bundle B, the marginal utilities per dollar show us that spending more mone on good and less on good will increase overall utilit) So, mathematicall we represent the constrained maimization problem as Ma U(,) subject to p + p = n words: Maimize the consumer s utilit level subject to his budget constraint We can hence use the Lagrange Multiplier Method of Constraint Maimization to solve the utilit maimization problem. (,;λ) = U(,) + λ [ p p ] B taking F.O.C.s, we come to the same two conditions as before for a bundle to be optimal (ma U subject to B.L.) Let s tr to show this using Cobb Douglas Utilit Function Eample: Cobb Douglas Utilit function:, MAX U (, ) S. T. XY (,, ) [ p p ] Taking F.O.Cs with respect to u(, ) p 0 p u(,) () And with respect to, ( ) p ( ) p ( ) u(, ) () 6
7 9/0/04 And taking FOCs with respect to 0 () Using the tangenc condition p p and From () and (), u(,) ( ) u(,) ( ) rearranging it, we obtain p =p Which is eactl the tangenc condition = since MU = p p lugging this into (), ields ( ) And MU = lugging into the tangenc condition ields Budget Shares After finding the demand for and, we can find their budget shares (dividing b the income level, ), as follows X Budget share for, Hence, demands are and Budget share for, ( ) Y Budget shares are, hence, constant in income e.g. if α = /, the budget share of good is % while that of good is 66% 7
8 9/0/04 Approach #: Using the Lagrange Method Approach #: Using the tangenc condition Approach #: Using the Lagrange method FOC FOC u (, ) (, ; ) s. t [ ] FOC We now set FOC=FOC FOC FOC Which simplifies to Further rearranging, ields =. 0 0 Use the FOCλ FOC Therefore, And the optimal consumption of good is. 8
9 9/0/04 Approach #: Using the Tangenc Condition Rearranging the tangenc condition ields MU MU p p We can now plug our result from the tangenc condition, =, in the budget line And good is ==. units. Budget Shares Let us evaluate the budget shares in each good Good, Good, p p Recall that these budget shares coincide with the eponents of the Cobb Douglas utilit function. 9
10 9/0/04 After going over this numerical eample of utilit maimization problems (UM), let us now continue with our discussion of UMs: nterior solutions. We have analzed interior solutions (positive amounts of both goods), but Corner solutions. What if the solution happens to be in a corner (impling a zero consumption of one of the two goods?) Corner Solutions What do we do in optimization cases where the B.L. is never tangent to an indifference curve? That is, if Eample MU MU Numerical Eample of Corner solutions U(,) = + 0 MU = + 0 MU = = 0, p =, p = Notice that utilit can be positive where zero units of good are consumed. This is because the.c. crosses the ais. This didn t happen in the Cobb Douglas utilit function where, if =0 or if =0, utilit was zero. Let s approach this like an interior optimum () +=0 (B.L.) X Y () MU 0 0 MU The tangenc approach gives us a negative amount of clothing which shows us that the solution is not an interior optimum. The solution will be a corner point, where =0. (+0) + = 0 4 = 0 =.5 (NO!! cannot be negative, thus we have a corner solution where = 0) B.L. 0 Since 0, then 0 Negative amounts indicate that the solution must be in a corner. 0
11 9/0/04 Then, a consumer wants to increase his consumption of as much as possible. That is, the consumer spends all his income in,, and no income in. 0 0 Bundle R is optimal At R, MU MU Y MU MU FURTHER NCREASE. Corner oint Solution with erfect Substitutes U(,) a b Let s consider the case where a consumer is perfectl willing to substitute one good for another at a constant ratio. For instance, Sara likes both chocolate and vanilla ice cream, and is alwas willing to substitute one scoop of chocolate for two scoops of vanilla. MRS choc.,vanilla = (constant MRS because of erfect substitutes) Hence, c MU MU v c v MU c MU v, MU v v and c v MU c c MU per dollar spent on v MU per dollar spent on c MRS choc.,vanilla = MU c MU v (constant MRS because of erfect substitutes) c v MU c MU v MU c MU v v c Consequentl, since the marginal utilit per dollar of vanilla ice cream is greater than that of chocolate ice cream, Sara should onl consume vanilla ice cream. B [What if price ratio was lower than MRS? Then BL becomes flatter than.c., and the consumer bus onl chocolate] Figure in net slide
12 9/0/04 When MRS,, we have: Application : Coupons vs Cash Subsidies At oint B, MU MU MU MU so the consumer spends all his mone on good alone. Before we start with applications, one note: Composite Goods are goods that represent the composite ependitures on ever other good besides the commodit being considered. That is, we use composite goods when we wish to isolate how a consumer selects one specific good. The price of composite goods is p = so that it measures not onl units of consumed, but also total ependiture on (p =total ependiture on ) The government might consider that the amount of housing individuals select, h A, is insufficient, and seeks to promote a larger amount of housing, h B. t can achieve this goal b providing a subsid S (which shifts individuals budget line to EG), or with a coupon that can onl be used to purchase housing (which ields the new budget line KFG). n the previous case, the consumer was equall happ with either program because both achieve.c. of U. However, in the situation below the consumer is worse off with the coupon, reaching U, than the subsid, where he reaches U4. Cash subsid of S shifts budget line out b S, to B.L. EG Coupon for housing of S units onl increases the budget with respect to housing, so the new budget line with the coupon is KFG B.L. is MN SubsidBundle T, with Utilit U 4 Coupon B.L. is KRG. Bundle R with utilit U Cash: B.L. is EG. Bundle B with Utilit U Coupon: B.L. is KFG. Bundle B with Utilit u
13 9/0/04 Consider instead that the government, still seeking to induce h B units of housing, does not provide a coupon, but rather provides a voucher V, which gives an amount of cash to the individual so that his budget line shifts until the point in which he can afford to bu h B units of housing at point R. That is, budget line EG emerges from the voucher polic. However, will the individual bu bundle R? No! While R and F are both affordable, he reaches a higher utilit level, U, with bundle F than with bundle R (where he onl reaches U). Hence, the voucher polic, despite making the individual happier than the coupon polic (U>U), does not help the government achieve a minimal housing level of h B Application : Joining a Club Man consumers can join clubs to receive member discounts. Yet, the must also pa a membership fee. So is it worth joining? Let s look at an eample Before joining: = 00, p = $0 00 5units Horizontal ntercept of B.L 0 $ After joining: = (fee) = $00, p = $0 p = $ 00 00units Vertical ntercept of B.L 00 0units 00 00units Horizontal intercept of BL Vertical intercept of BL At BL He chooses basket A Consumer chooses bundle A before joining club A similar analsis is applicable to: Cell phone service: ou pa a monthl subscription fee in order to enjo calls at lower prices per minute Gm, golf clubs, etc. After joining, consumer chooses bundle B and is better off than before because B is on.c. U
14 9/0/04 Eample: AT&T plans for its Apple ihone g (Chicago, 009) lan A: $40 fee, 450 minutes each additional minute $.40 lan B: $60 fee, 900 minutes each additional minute $.40 Drawing the B.L. for lan A: ncome: $500 After fee, ncome = $500 40=460 (vertical intercept) Flat for the first 450 minutes, then slope of.40 Drawing B.L. for lan B: After the $60 fee, income is $500 60=440 (vertical intercept) Flat for B.L. for the first 900 minutes, then slope of.40 AT&T hone Service Eample Application : Borrowing and Lending Without borrowing or lending, a consumer consumes toda and tomorrow, where is the consumer s salar toda, and is his salar tomorrow. However, introducing borrowing and lending allows a consumer to put off consuming toda so that toda s income can be lent, so tomorrow he or she can consume + (+r), with (+r) being the interest earned from lending income. e.g. 00, r 0.05 Consume 00 r 00 r.05 toda b borrowing, and not consuming anthing tomorrow. 4
15 9/0/04 How do we find the slope of the BL in this contet? ntuitivel, the price of giving up one consumption unit tomorrow ( ais) in order to gain one more unit of consumption toda ( ais) is measured b the opportunit cost of ever dollar borrowed, (+r). We can state this more formall as We know that the B.L. is = a + m = + (+r) + m a and, in addition, at = 0 we have = + /(+r), then 0 ( r) m r m ( r) ( r) ( r) ( r) ( r) r Here it is better to borrow some mone this ear to be paid back net ear, because the optimum bundle went from being on U before borrowing to U after borrowing. World Without Banks! Borrowing n this eample both the lending interest rate and borrowing interest rate are 0 percent. So far we assumed that the lending and borrowing interest rates coincide, but. What if the lending and borrowing interest rates are different? Lending, r L =.05 Net ear he will have,00 from tomorrow s income =,00 0,000 (+0.05) from savings = 0,500,700 Borrowing, r B =.0 Toda can have : 0,000 from toda s income= 0,000,00 0.0,000,000 5
16 9/0/04 Here the interest rates for borrowing and lending are different. Borrowing ( rl ( rb ) ) Here the.c. of a saver would reach tangenc on segment AE, while the.c. of a borrower would reach tangenc on segment AG Application V: Quantit Discounts Sometimes consumers are offered quantit discounts, so that the price of a good is not constant but changes after some defined quantit (i.e., a discount on electricit after number of units). Notice this will mean the budget line has a kink where the slope changes because of the new price ratio. Let s consider the case of electricit where the prices are as follows p = $ p = $ for the first 9 units (e.g., kw/h) $5.5 for all additional units How to find the vertical and horizontal intercepts in this case? Vertical ntercept = $440 $ 440 Horizontal intercept of BL (no discounts) = Horizontal intercept of BL (with discounts) = Slope is on segment on MA otential Util. gain from the quantit discount if preferences are like these Slope is 5.5 on AR where we subtract the amount of mone alread spent on good before eperiencing the price discount, 9 units $ per unit, to the consumer s initial income At point A, we can use the BL in order to find its height: +(9*)=440, which ields a value of =440 99=4 6
17 9/0/04 BL for Frequent Fler rograms Another Eample: Frequent fler programs, e.g, after 00k miles the price of another ticket decreases. For this eample, we can actuall have man quantit discounts, as the net figure illustrates. Utilit maimization problems and Ependiture minimization problems (Appendi to Chapter 4) Eample : Utilit Maimization roblem (UM) with the Cobb Douglas utilit function. Let us first consider an eample of a utilit maimization problem. Take the Cobb Douglas function epressed b u,, The budget constraint : 00 So we know that: price of good is $, that of good is $, and income is $00. 7
18 9/0/04 Setting up the Lagrangian: L(, ; ) (00 ) We can take first order conditions: L 0 L 0 L 00 0 Taking the ratio of the first two terms shows that: or rearranging, (Note that this eactl coincides with the tangenc condition between the.c. and the budget line, that s wh the tangenc condition and the Lagrangian method ield the same results.) u ndeed, the MU is u while the MU is Hence, the MRS is MRS, Substituting the tangenc condition budget set 00 gives 00 5 units into the Therefore, the utilit level that the consumer reaches is given b plugging 5 and 50 into the utilit function u, And about good, we have 5 50 units This utilit level from selecting utilit maimizing bundles is often referred to as indirect utilit 8
19 9/0/04 Eample : Ependiture minimization problem (EM) with a Cobb Douglas utilit function. Another wa to think about optimum bundles is as Ependiture Minimization roblems: the bundle that will give ou a given level of utilit at the lowest ependiture. Min Ependiture p + p Subject to U(,) = (reaching a given level of utilit) u Eample : Ependiture minimization problem (EM) with a Cobb Douglas utilit function. Consider a consumer with Cobb Douglas utilit function, who tries to reach a utilit level of u=5.5. That is, u (, ) and p, p, u5.5. To find the compensated demands for goods and, let us solve the ependiture minimization problem (EM) min ( ) s.t. 5.5 u Setting up the Lagrangian in this case, ields, ; 5.5 L where μ denotes the Lagrange multiplier of this ependiture minimization problem (we use μ rather than λ, which is the Greek letter we used in the UM). n the case of interior solutions, the above first order conditions become L 0 L 0 L
20 9/0/04 Let s solve the previous sstem of equations, 4 4 From, 4 From, 4 Since 4 From, lugging 4 into Using 4, 5 50 Our old friend!! The same eact tangenc condition we found when finding the optimal consumption bundle using the UM approach. 4 For the Cobb Douglas utilit function u (, ) considered above, we just found that the compensated demands for goods and are ( p, u) 5, ( p, u) 50 c c which coincide with the optimal consumption bundles that we found when solving the UM!! Therefore, the total ependiture required to purchase these compensated demands is e( p, u) p ( p, u) p ( p, u) c c which coincides with the budget constraint of the consumer in the UM (where we started saing the consumer had an income of $00 to spend). 0
21 9/0/04 Revealed references We have now learned how to find the optimal consumption bundle when we know both a consumer s preferences and his or her budget lines. But what if we do not know a consumer s preferences? We can infer her preferences b analzing her actual choices in different situations. () When facing BL, consumer chose A instead of an point inside BL, such as B. Then A B () A and C were equall costl, but he chose A. Hence A C () Since C lies to the northeast of B, then C >B A >B B Transitivit AC B A B Alternative rocedure to check if a consumer s choice maimizes utilit ) At final prices ~ ~ and, he chooses Basket nitial Bundle: ( X, Y ) Final Bundle: ( X, Y ) Since he alread revealed a preference for Basket, Basket must be cheaper than Basket at the new prices. Otherwise, he would have selected Basket ) At initial prices, Basket costs X Y At initial prices, Basket costs X Y Let s suppose that, at initial prices: X Y X Y ~ X ~ Y ~ X f this condition wasn t true, the consumer would be revealing a strong preference for Basket over Basket, contradicting his revealed preference at the initial prices. ~ Y Since at initial prices he chose Basket (despite Basket was affordable), it must be that he prefers Basket to basket
22 9/0/04 Here is an eample where a consumer fails to maimize utilit ncome = $4 ( p, p ) = ($4, $) (, ) = (5, ) A ~ ~, = ($, $) (, ) = (, 6) B Let us see wh he fails to maimize utilit. BL : () when facing BL, the consumer chose A instead of C, in spite of being available (affordable), A C AC BA B () Since C is to the northeast of B, C>B A >B BL : when facing BL, he chose B in spite of D being available, B D Since D is to the northeast of A, D>A BD AB A B >A Contradiction! Bundle A cannot be strongl preferred to B, and B strongl preferred to A in a case where both are affordable. Alternative Approach, using total ependiture Let us tr a few more BL : Consumer prefers basket A. Cost of basket A, $4 5 + $ = $4 Cost of basket B, $4 + $ 6 = $0 X Y X Y BL : Consumer prefers basket B. Thus B >A ~ ~ Cost of basket B, $ + $ 6 = $4 X Y ~ ~ Cost of basket A, $ 5 + $ = $ X Y We need X Y X Y ($4 > $0), and ~ ~ X Y X Y ~ ~ Hence, he is NOT maimizing utilit (but in this case we obtain ($4$). /
23 9/0/04 Case : () C>A since it is to the northeast of A () B C since B was chosen when C was affordable Case : Affordable () BL : From above, we know that B >A since A is inside BL Under BL () BL : Additionall, A B since both were affordable when facing BL Contradiction, not utilit maimizing behavior Case : () BL : the consumer chose A when both A and B were affordable, A B () BL : the consumer chose B when A wasn t affordable Case 4: () BL : chose A but B wasn t affordable () BL : chose B but A wasn t affordable B Transitivit B C A B A B >A Ranking A B is all we can sa We can t infer anthing from these choices because we don t know which bundle the consumer would choose if both were affordable
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