A graphical introduction to the budget constraint and utility maximization

Transcription

1 EC 35: ntermediate Microeconomics, Lecture 4 Economics 35: ntermediate Microeconomics Notes and Assignment Chater 4: tilit Maimization and Choice This chater discusses how consumers make consumtion decisions given their references and budget constraints. A grahical introduction to the budget constraint and utilit maimization A erson will maimize their utilit subject to their budget constraint. That is, the will do the best that the can given the amount of mone the have to send and the rices that the face. f a erson has income and consumes goods and and the rices of these goods are and, her budget constraint is written as: + That is, the amount that she sends on ( multilied b ) lus the amount that she sends on ( multilied b ) must be less than or equal to her income,. For eamle, if a erson has income of $0, the rice of is $0 er unit and the rice of is $5 er unit, then the budget constraint is written as: $0 + $5 $0 n terms of a grah, the budget constraint looks like:

2 Two grahs. The first shows a general budget line and the second shows the budget line for the situation where income is 0, the rice of is 0 and the rice of is 5. The line reresents the set of bundles that this erson can afford if she sends all of her income on goods and. The sloe of the budget constraint is. Now, given that a erson is constrained to choose a oint on her budget line, she will tr to get onto the highest indifference curve ossible. This will occur at a oint where the budget line is tangent to an indifference curve. n a diagram, this looks like: A grah showing the utilit maimizing oint on a budget line.

3 where (*, *) is the utilit maimizing bundle of goods given the indicated references and budget constraint. Were the erson at some other oint on the budget line, she could make herself better off (that is, she could achieve a higher level of utilit) b choosing a different combination of goods. Now, here s the thing. At the oint (*, *), the sloe of the indifference curve is equal to the sloe of the budget constraint,, or the marginal rate of substitution. A oint that maimizes a erson s utilit must (with a few ecetions) be a oint at which the two sloes are equal. So, at a utilit maimizing oint, it must be true that the sloe of the budget line equals the sloe of the indifference curve, which is also known as the marginal rate of substitution (MRS): d d MRS Eamle: magine that a erson faces rices of $0 and $30 and has the utilit function (,). The sloe of her budget line would be The sloe of her budget line would be M MRS MY giving us 3 3 So, at a utilit maimizing oint, the quantit of that she has will be equal to three times the quantit of that she has. We don t know eactl how much of each she ll have without knowing her income or how much she has to send, but we do know that ossibilities include: three units of and one units of 3

4 si units of and two units of twelve units of and four units of and so on. Some ecetions There are a few ecetions to the otimization rule stated above. That is, there are a few cases in which the utilit maimizing oint is not one where the sloe of the indifference curve is equal to the sloe of the budget line. Ecetion : Corner solutions t ma be that the indifference curve is either alwas steeer or alwas flatter than the budget line. n this case, the utilit maimizing bundle is entirel comosed of one good or the other. The diagram for this looks like: Two grahs showing indifference curves and budget lines in situations where the consumer chooses to consume either onl good or onl good. Ecetion : Perfect substitutes f two goods are erfect substitutes, then the indifference curve is siml a straight line and the analsis is similar to that of Ecetion. The result will either be that all of one good is consumed or that all of the other is consumed. magine two brands of gasoline that ou consider to be identical. f the two brands have different rices, ou would onl consume the cheaer brand, other things being equal. 4

5 The one ecetion is when the sloe of the indifference curve (which, remember, is a straight line) is the same as the sloe of the budget line (also a straight line), in which case the rice ratio is equal to the MRS and an combination of the two goods is utilit maimizing. So if the two brands of gasoline had the same rice, it reall wouldn t matter which ou consumed, or if ou consumed a combination of the two. Ecetion 3: Perfect comlements f two goods are erfect comlements, then the utilit maimizing outcome is to consume them in the aroriate ratio, regardless of their relative rices. So, ou will consume an equal number of left shoes and right shoes and, at an one time anwa, ou will use four times as man tires as ou have automobiles. This diagram looks like: A grah showing a budget line and indifference curves for erfect comlements. The Math Behind tilit Maimization The math behind all this is as follows. The goal is to choose a bundle of goods so as to maimize utilit subject to the budget constraint. This is written as: ma subject to (,,..., ) n 5

6 or... 0 nn This can be rewritten as a Lagrangian (see Chater ) n n ( n ) ( nn L,,..., + λ ) From above, the stuff in arentheses following the λ is equal to zero, so creating the Lagrangian is reall just taking the utilit function and adding zero to it. You might ask, Wh go through all that trouble? Well, it will be worth it. Now, take the artial derivative of L with resect to each of the terms and with resect to λ to get a whole bunch of equations that ou set equal to zero. t s basicall a comlicated maimization roblem: λ 0 λ 0 M λ M λ λn 0 n n... nn λ 0 Mn λn nn All of these equations can then be solved simultaneousl to find the otimal bundle. So what? Well, first consider the relative quantities of an two goods, cleverl named good i and good j. We know that: Mi λi M j λ j So it must also be the case that: Mi M j λi λ j i j or that the ratio of the marginal utilities, the negative of the MRS, is equal to the ratio of the rices, which is the negative of the sloe of the budget line. 6

7 t also turns out that λ M M M... n n that is, λ, is equal to the additional bang for a buck sent on each good, and that this is equal across goods. f this were not true, if one good offered more marginal bang for an additional buck than did some other good, then a consumer could make herself better off b sending less on other goods and more on that good. n an otimal situation, this sort of move is not ossible because she is alread as well off as she can be. n fact, in this sort of roblem, λ has the interretation of being the marginal utilit of income. t is the increase in the level of utilit that would be achieved if income were to increase b one unit. Eamle: magine that the utilit function is (,)5, and 8 and 40.. Set u the Lagrangian. Solve for the otimal bundle 3. Calculate the resulting level of utilit 4. Grah out the relevant curves 5. Calculate the marginal utilit of income at the otimum L (, ) L 5 + λ + λ ( ) ( 40 8) λ 5 λ 0 0 8λ 0 0 From the first two first-order conditions (the first two derivatives) we get: and the budget constraint is: 5 λ 0 8λ

8 f we substitute into the budget constraint we get: 40 () (0) 40 We can confirm that this satisfies the budget constraint: (40) + 8(0) The resulting utilit level is (40,0) 5(40)(0) 80,000 n a icture, this looks like: A grah showing the solution to the receding utilit maimization eamle. Now, the marginal utilit of income, λ, is equal to: λ M λ M

9 n Eamle 4., the tetbook goes through a more general eamle along these lines. This form of utilit function is called a Cobb-Douglas utilit function. The general form is ( ), You should look through this eamle. n articular, ou should go through the calculations that get ou from ( ), and + to the demand functions for and, * * ( + ) ( + ) or, as eressed in the book, when +, * * ndirect tilit Functions So, the underling belief is that eole maimize their utilit given their references and income and the rices the face. Another wa of stating this is that the quantit of each good that a erson consumes is a function of references, income and rices. 9

10 Now, because utilit is a function of quantities consumed, and quantities consumed are functions of references, income and rices, then utilit can be eressed as a function of references, income and rices, assuming that a erson maimized their utilit. This sort of utilit function, where utilit is a function of references, income and rices is called an indirect utilit function. Put somewhat differentl, the usual utilit function is: (,,..., ) n but, forgetting about references for a moment, the otimal quantit of each good consumed can be eressed as a function of rices and income: * (,,...,n,) * (,,...,n,) * n n (,,...,n,) So, maimum utilit can be eressed as * (,,..., ) V(,,...,,) * * n n n terms of the utilit function given above, (,) 5, the demand functions for and are: * * ( + ) 3 ( + ) ( + ) ( + ) 3 So the indirect utilit function is: V (,,) ( * (,,), * (,, )

11 We can confirm that for and 8 and 40 the resulting utilit level is 80,000: * * 3 ( + ) ( + ) V (,,) 80, The Lum Sum Princile OK, ou ve suffered through enough theor with no obvious olic imlications, so here s something that can be alied to the real world. The idea is that if a ta is going to be imosed on a erson, it is better to imose it as a lum sum ta (ou a $X, regardless of our behavior) rather than taing one thing or another. To state this more secificall, the same amount of ta revenue can be raised with less of a decrease in utilit with a lum sum ta than with a ta on one good or another. This statement can be established based on onl the simlest rinciles of consumer references and utilit maimization. Again, t his statement can be established based on onl the simlest rinciles of consumer references and utilit maimization. You don t need to know anthing more. Here s the stor in ictures: A ha consumer is minding her own business, with income level, facing rices and, and achieving utilit level 3 as a result.

12 A grah showing the general utilit maimization solution. Now, for reasons we don t need to go into here (refer to Chater 0), the Government decides that it needs some ta revenue. As such, it must imose a ta. To start with, imagine that the ta good. The choice of which good to ta is comletel arbitrar, but ou might imagine that some sort of ecuse is given for choosing over. As a result, the rice of good rises to + t, or the rice lus the ta. Anhow, with a ta on, the icture changes to: A grah showing the effect on a consumer s budget line and otimal choice when a ta is added to good. can t believe m utting a footnote in lecture notes, but it is worth saing that a ta equal to t won t necessaril raise the rice b t. n general it will raise the rice b less than t, but under some conditions the rice might rise b the full amount of the ta.

13 So, now she s at a lower utilit level and the Government is collecting some amount of ta revenue. What if the government collected these taes through a lum sum ta rather than a ta on. That is, what if the ta revenue staed the same, but the rices of the goods staed the same? The new budget line would have the same sloe as the original budget line, but would ass through the otimal oint that the consumer achieved with the ta on. The budget lines would look like: A grah showing the imact on the consumer s budget line of a ta on and the imact of a lum sum ta. Now, here s the oint. With the lum sum ta instead of the ta on, this consumer can achieve a higher level of utilit without ta revenues changing. n the tetbook, Figure 4.5 shows this new utilit level as. 3

14 A diagram showing that a consumer can achieve greater utilit under a lum sum ta than under a revenue-equivalent ta on good. The imlication is that lum sum taes will be more efficient than will an other sort of ta, including taes on goods, sales, income, roert or labor. That is, the same amount of revenue can be raised with less of a decrease in consumer utilit with a lum sum ta than with an other kind of ta. Now, let s tr this with a articular utilit function and income levels. magine that a consumer s utilit function is (, ), her income is 480 and the rices of and are and. With no ta in lace, the maimization roblem is: The first derivatives are: ( 480 ) L + λ λ 0 λ 0 λ λ λ Taking the ratio of the first to first derivatives gives us: 4

15 λ 0 λ 0 λ λ λ λ λ This is combined with the budget constraint with the result being that 40 and 40 and the utilit level is Now, imagine that a ta of $ is ut on, raising the rice of to. The new result will be 0 and 40 with a resulting utilit level of That ta revenue will be $ er unit of, collected on 0 units of for total ta revenue of $0. Now, imagine that instead of a ta on, there was just a lum sum ta of $0 imosed, bringing this erson s disosable income from $480 down to $360. With the original rices of and, we get a utilit maimizing bundle (ou should calculate this ourself and make sure ou can do it) of 80 and 80 and a utilit level of , instead of the utilit level of 69.7 with the ta on. Now, this can be eressed in terms of the indirect utilit function. That is, when utilit is eressed as a function of income and rices instead of as a function of the quantities of goods consumed, ou can reeat the analsis. For the utilit function given above, (, ), the indirect utilit function can be found b calculating the demand functions for and then inserting these into the utilit function. The demand functions for and, in terms of and and income can be found b maimizing utilit subject to the budget constraint: L + λ The first derivatives are: ( ) 5

16 L L L λ λ λ λ λ The first two of these can be combined to give: and These can be combined with the third equation to give: * * + + And this can be combined with the utilit function itself to give: ( ) 5 0.,, V Now, at long last, we return to this nast issue of lum sum taes. The initial situation was 480,,, for a utilit level of: ( ) ,,480 V When the rice of rose to $, the level of utilit was: ( ) ,, V Finall, when disosable income fell to $360 due to a lum sum ta of $0, the level of utilit was: ( ) ,,360 V 6

17 n a diagram, this all looks like: A grah showing the numerical results of an eamle demonstrating that a consumer can achieve greater utilit under a lum sum ta than under a revenue equivalent ta on good. Eenditure Minimization So, there are two was to look at a consumer s otimal decision, but it s all the same roblem. The first wa is to imagine that the consumer has some budget constraint and tries to maimize her utilit level given that budget constraint. The second wa is to imagine that the consumer will achieve some level of utilit and tries to minimize the cost of doing this. The answer to both of these roblems is a oint where the budget constraint is just tangent to an indifference curve. n terms of a Lagrangian, these two roblems are written as: (,,...,n ) + λ( nn L + ) and L ( ) ( ) + λ (,,..., ) n n n 7

18 The eenditure function is a function giving the minimum eenditure needed to achieve some level of utilit,, given the rices for goods: E(,,, n, ). Now, let s calculate an eenditure function for the utilit function. ) (, The Lagrangian is: ) ( L + λ + Taking first derivatives gives: λ λ λ λ λ 0 L 0 L 0 L Now, taking the ratio of the first two of these gives: Combining these with the other derivative gives: + + * + + * 8

19 These functions are terribl ugl unless, as with Eamle 4.4 in the tetbook, ou assume that, then the become * * Now, the eenditure function will be: (,, ) E (,, ) E (,, ) 0. 5 (,, ) E E * + * + + Proerties of eenditure function. Homogeneit Homogeneit means that if one of the comonents of a function increases b a certain ercentage, the value of the function will increase b that ercentage raised to some ower. For eamle, if f() is homogeneous of degree 6, then the following relationshi will be true: f() 6 f() Now, eenditure functions are homogeneous of degree with resect to changes in rices. This is a fanc wa of saing that if rices rise b 0%, eenditures will rise b 0%, holding utilit constant. n an equation, this is: E(.0,.0. ).0 E(,, ) f rices double, eenditures will double.. Eenditure functions are increasing in rices. f rices go u, eenditures will rise, holding utilit constant. 9

20 3. Eenditure functions are concave in rices. As the rice of one good rises, holding other things constant, eenditures will rise at a slower rate than the rate at which rices rise because, in some sense, eole will substitute toward the other good whose rice hasn t risen. 0

21 Eercises. Do the following for the utilit function (,) A. Solve for the otimal bundle if,, 0. B. Solve for the otimal bundle if,, 40. C. Solve for the otimal bundle if,, 0. D. Solve for the otimal bundle if,, 0. E. Solve for the otimal bundle if,, 40. F. Solve for the indirect utilit function V(,, ). G. Solve for the eenditure function E(,, ).. Do the following for the utilit function (,) A. Solve for the otimal bundle if,, 0. B. Solve for the otimal bundle if,, 40. C. Solve for the otimal bundle if,, 0. D. Solve for the otimal bundle if,, 0. E. Solve for the otimal bundle if,, 40. F. Solve for the indirect utilit function V(,, ). G. Solve for the eenditure function E(,, ). 3. Do the following for the utilit function (,) A. Solve for the otimal bundle if,, 0. B. Solve for the otimal bundle if,, 40. C. Solve for the otimal bundle if,, 0. D. Solve for the otimal bundle if,, 0. E. Solve for the otimal bundle if,, 40. F. Solve for the indirect utilit function V(,, ). G. Solve for the eenditure function E(,, ). 4. Do the following for the utilit function (,) A. Solve for the otimal bundle if,, 0. B. Solve for the otimal bundle if,, 40. C. Solve for the otimal bundle if,, 0. D. Solve for the otimal bundle if,, 0. E. Solve for the otimal bundle if,, 40. F. Solve for the indirect utilit function V(,, ). G. Solve for the eenditure function E(,, ). 5. Do the following for the utilit function (,) A. Solve for the otimal bundle if,, 0. B. Solve for the otimal bundle if,, 40. C. Solve for the otimal bundle if,, 0. D. Solve for the otimal bundle if,, 0. E. Solve for the otimal bundle if,, 40. F. Solve for the indirect utilit function V(,, ). G. Solve for the eenditure function E(,, ).