Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deision-makers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some goal a onsumer seeks to maximize some measure of satisfation from his onsumption deisions while a firm seeks to maximize its profits. We first onsider the miroeonomis of onsumer theory and will later turn to a onsideration of firms. The two theoretial tools of onsumer theory are utility funtions and budget onstraints. Out of the interation of a utility funtion and a budget onstraint emerge the hoies that a onsumer makes. Utility Theory A utility funtion desribes the level of satisfation or happiness that a onsumer obtains from onsuming various goods. A utility funtion an have any number of arguments, eah of whih affets the onsumer's overall satisfation level. But it is only when we onsider more than one argument an we onsider the trade-offs that a onsumer faes when making onsumption deisions. The nature of these trade-offs an be illustrated with a utility funtion of two arguments, but is ompletely generalizable to the ase of any arbitrary number of arguments. 2 Figure 2 illustrates in three dimensions the square-root utility funtion u ( 1, 2) 1 2, where 1 and 2 are two different goods. This utility funtion displays diminishing marginal utility in eah of the two goods, whih means that, holding onsumption of one good onstant, inreases in onsumption of the other good inrease total utility at ever-dereasing rates. Graphially, diminishing marginal utility means that the slope of the utility funtion with respet to eah of its arguments in isolation is always dereasing. The notion of diminishing marginal utility seems to desribe onsumers preferenes so well that most eonomi analysis takes it as a fundamental starting point. We will onsider diminishing marginal utility a fundamental building blok of all our subsequent ideas. 2 An advantage of onsidering the ase of just two goods is that we an analyze it graphially beause, reall, graphing a funtion of two arguments requires three dimensions, graphing a funtion of three arguments requires four dimensions, and, in general, graphing a funtion of n arguments requires n+1 dimensions. Obviously, we annot visualize anything more than three dimensions. Spring 2014 Sanjay K. Chugh 19
Figure 2. The utility surfae as a funtion of two goods, 1 and 2. The speifi utility funtion here is the square-root utility funtion, 1 2 1 2 utility axis. u (, ). The three axes are the 1 axis, the 2 axis, and the The first row of Figure 3 displays the same information as in Figure 2 exept as a pair of two-dimensional diagrams. Eah diagram is a rotation of the three-dimensional diagram in Figure 2, whih allows for omplete loss of depth perspetive of either 2 (the upper left panel) or of 1 (the upper right panel). The bottom row of Figure 3 ontains the diminishing marginal utility funtions with respet to 1 ( 2 ), holding onstant 2 ( 1 ). Spring 2014 Sanjay K. Chugh 20
u( 1, 2 ) u( 1, 2 ) Stritly inreasing total utility in eah of the two goods 1 2 u 1 ( 1, 2 ) Keeping 2 fixed, ompute first derivative with respet to 1 u 2 ( 1, 2 ) Keeping 1 fixed, ompute first derivative with respet to 2 Diminishing marginal utility in eah of the two goods Figure 3. Top left panel: total utility as a funtion of 1, holding fixed 2. Top right panel: total utility as a funtion of 2, holding fixed 1. Lower left panel: (diminishing) marginal produt funtion of 1, holding fixed 2. Lower right panel: (diminishing) marginal produt funtion of 2, holding fixed 1. For the utility funtion u ( 1, 2) 1 2, the marginal utility funtions are 1( 1, 2) (1/2) 1/ 1 u (, ) (1/2) 1/ 1 2 (lower right panel). 1 u (lower left panel) and 2 2 2 Spring 2014 Sanjay K. Chugh 21
Indifferene Curves Figure 4 returns to the three-dimensional diagram using the same utility funtion, with a different emphasis. Eah of the solid urves in Figure 4 orresponds to a partiular level of utility. This three-dimensional view shows that a given level of utility orresponds to a given height of the funtion u ( 1, 2) above the 1 2 plane. 3 u (, ), where eah solid urve Figure 4. An indifferene map of the utility funtion 1 2 1 2 represents a given height above the 1-2 plane and hene a partiular level of utility. The three axes are the 1 axis, the 2 axis, and the utility axis. If we were to observe Figure 4 from diretly overhead, so that the utility axis were oming diretly at us out of the 1 2 plane, we would observe Figure 5. Figure 5 displays the ontours of the utility funtion. In general, a ontour is the set of all ombinations of funtion arguments that yield some pre-speified funtion value. In our appliation here to utility theory, eah ontour is the set of all ombinations of the two goods 1 and 2 that deliver a given level of utility. The ontours of a utility funtion are alled indifferene urves, so named beause eah indifferene urve shows all ombinations 3 Be sure you understand this last point very well. Spring 2014 Sanjay K. Chugh 22
(sometimes alled bundles ) of goods between whih a onsumer is indifferent that is, deliver a given amount of satisfation. For example, suppose a onsumer has hosen 4 units of 1 and 9 units of 2. The square-root utility funtion then tells us that his level of utility is u(4,9) 4 9 5 (utils, whih is the fitional measure of utility). There are an infinite number of ombinations of 1 and 2, however, whih deliver this same level of utility. For example, had the onsumer instead been given 9 units of 1 and 4 units of 2, he would have obtained the same level of utility. That is, from the point of view of his overall level of satisfation, the onsumer is indifferent between having 4 units of good 1 in ombination with 9 units of good 2 and having 9 units of good 1 in ombination with 4 units of good 2. Thus these two points in the 1 2 plane lie on the same indifferene urve. Figure 5. The ontours of the utility funtion 1 2 1 2 1 2 u (, ) viewed in the two-dimensional plane. The utility axis is oming perpendiularly out of the page at you. Eah ontour of a utility funtion is alled an indifferene urve. Indifferene urves further to the northeast are assoiated with higher levels of utility. A ruial point to understand in omparing Figure 4 and Figure 5 is that indifferene urves whih lie further to the northeast in the latter orrespond to higher values of the utility funtion in the former. That is, although we annot atually see the height of the utility funtion in Figure 5, by omparing it to Figure 4 we an onlude that indifferene urves whih lie further to the northeast provide higher levels of utility. Intuitively, this means Spring 2014 Sanjay K. Chugh 23
that if a onsumer is given more of both goods (whih is what moving to the northeast in the 1 2 plane means), his satisfation is unambiguously higher. 4 One we understand that Figure 4 and Figure 5 are onveying the same information, it is learly muh easier to use the latter beause drawing (variations of) Figure 4 over and over again would be very time-onsuming! As suh, muh of our study of onsumer analysis will involve indifferene maps suh as that illustrated in Figure 5. Marginal Rate of Substitution Eah indifferene urve in Figure 5 has a negative slope throughout. This aptures the idea that, starting from any onsumption bundle (that is, any point in the 1 2 plane), if a onsumer gives up some of one good, in order to maintain his level of utility he must be given an additional amount of the other good. The ruial idea is that the onsumer is willing to substitute one good for another, even though the two goods are not the same. Some refletion should onvine you that this is a good desription of most people s preferenes. For example, a person who onsumes two pizzas and five sandwihes in a month may be just as well off (in terms of total utility) had he onsumed one pizza and seven sandwihes. 5 The slope of an indifferene urve tells us the maximum number of units of one good the onsumer is willing to substitute to get one unit of the other good. This is an extremely important eonomi way of understanding what an indifferene urve represents. The slope of an indifferene urve varies depending on exatly whih onsumption bundle is under onsideration. For example, onsider the bundle ( 1 3, 2 2 ), whih yields approximately 3.15 utils using the square-root utility funtion above. If the onsumer were asked how many units of 2 he would be willing to give up in order to get one more unit of 1, he would first onsider the utility level (3.15 utils) he urrently enjoys. Any final bundle that left him with less total utility would be rejeted. He would be indifferent between his urrent bundle and a bundle with 4 units of 1 that also gave him 3.15 total utils. Simply solving from the utility funtion, we have that 4 2 3.15, whih yields (approximately) 2 1.32. Thus, from the initial onsumption bundle ( 1 3, 2 2 ), the onsumer is willing to trade at most 0.68 units of 2 to obtain one more unit of 1. 4 You an probably readily think of examples where onsuming more does not always leave a person better off. For example, after onsuming a ertain number of pizza slies and sodas, you will have probably had enough, to the point where onsuming more pizza and soda would derease your total utility (beause it would make you sik, say). While this may be an important feature of preferenes (the tehnial name for this phenomenon is satiation ), for the most part we will be onerned with those regions of the utility funtion where utility is inreasing. A way to justify this view is to suppose that the goods that we speak of are very broad ategories of good, not very narrowly-defined ones suh as pizza or soda. 5 The key phrase here is just as well off. Given our assumption above of inreasing utility, he would prefer to have more pizzas and more sandwihes. Spring 2014 Sanjay K. Chugh 24
What if we repeated this thought experiment starting from the new bundle? That is, with ( 1 4, 2 1.32 ), what if we again asked the onsumer how many units of 2 he would be willing to give up to obtain yet another unit of 1? Proeeding just as above, we learn that he would be willing to give up at most 0.48 units of 2, giving him the bundle 5, 0.84 ), whih yields total utility of 3.15. 6 ( 1 2 The preeding example shows that the more units of 1 the onsumer has, the fewer units of 2 the onsumer is willing to give up to get yet another unit of 1. The eonomi idea here is that onsumers have preferenes for balaned onsumption bundles they do not like extreme bundles that feature very many units of one good and very few of another. Some refletion may also onvine you that this feature of preferenes is a good desription of reality. 7 In more mathematial language, this feature of preferenes leads to indifferene urves that are onvex to the origin. Thus, the slope of the indifferene urve has very important eonomi meaning. It represents the marginal rate of substitution between the two goods the maximum quantity of one good that the onsumer is willing to trade for one more unit of the other. Formally, the marginal rate of substitution at a partiular onsumption bundle is the negative of the slope of the indifferene urve passing through that onsumption bundle. Budget Constraint The ost side of a onsumer s deisions involves the prie(s) he must pay to obtain onsumption. Again maintaining the assumption that there are only two types of onsumption goods, 1 and 2, let P 1 and P 2 denote their pries, respetively, in terms of money. We will assume for simpliity for the moment that eah onsumer spends all of his inome, denoted by Y, (more generally, all of his resoures, whih may also inlude wealth) on purhasing 1 and 2. 8 We further assume (for now) that he has no ontrol over his inome he simply takes it as given. 9 The budget onstraint the onsumer must respet as he makes his hoie about how muh 1 and 2 to purhase is therefore P 1 1 P 2 2 Y. 6 Make sure you understand how we arrived at this. 7 When we later onsider how onsumers make hoies aross time (as opposed to a speifi point in time), we will all this partiular feature of preferenes the onsumption-smoothing motive. 8 Assuming this greatly simplifies the analysis yet does not alter any of the basi lessons to be learned. Indeed, if we allowed the onsumer to save for the future so that he didn't spend of all of his urrent inome on onsumption, the additional hoie introdued (onsumption versus savings) would also be analyzed in using exatly the same proedure. We will turn to suh intertemporal hoie models of onsumer theory shortly. 9 Also very shortly, using the same tools of utility funtions and budget onstraints, we will study how an individual deides what his optimal level of inome is. Spring 2014 Sanjay K. Chugh 25
The term P 1 1 is total expenditure on good 1 and the term P 2 2 is total expenditure on good 2, the sum of whih is equal to (by our assumption above) inome. If we solve this budget onstraint for 2, we get P Y 1 2 1 P2 P2, whih, when plotted in the 1 2 plane, gives the straight line in Figure 6. In this figure, for illustrative purposes, the pries are hosen to both equal one (that is, P 1 P 2 1) so that the slope of the budget line is negative one, and inome is arbitrarily hosen to be Y 5. Obviously, when graphing a budget onstraint, the partiular values of pries and inome will determine its exat loation. Figure 6. The budget onstraint, plotted with 2 as a funtion of 1. For this example, the hosen pries are P 1 = P 2 = 1, and the hosen inome is Y=5. We disussed in our study of utility funtions the idea that we needed three dimensions the 1 dimension, the 2 dimension, and the utility dimension to properly visualize utility. We see here that utility plays no role in the budget onstraint, as it should not beause the budget onstraint only desribes expenditures, not the benefits (i.e., utility) a onsumer obtains from those expenditures. That is, the budget onstraint is a onept ompletely independent of the onept of a utility funtion this is a key point. We ould graph the budget onstraint in the same three-dimensional spae as our utility funtion it simply would be independent of utility. The graph of the budget onstraint (whih we all a budget plane when we onstrut it in three-dimensional spae) in our u spae is shown in Figure 7. 1 2 Spring 2014 Sanjay K. Chugh 26
Figure 7. The budget onstraint drawn in the three-dimensional 1-2 -u spae. The budget onstraint is a plane here beause it is independent of utility. Optimal Choie We are now ready to onsider how onsumers make hoies. The benefits of onsumption are desribed by the utility funtion, and the osts of onsumption are desribed by the budget onstraint. Graphially, the deision the onsumer faes is to hoose that bundle ( 1, 2 ) that yields the highest utility (i.e., lies on the highest indifferene urve) that also satisfies his budget onstraint (i.e., lies in the relevant budget plane). Spring 2014 Sanjay K. Chugh 27
2 slope = -P1 /P 2 5 optimal hoie 5 1 Figure 8. The optimal onsumption hoie features a tangeny between the budget line and an indifferene urve. The optimal hoie must lie on the budget line and attain the highest possible utility for the onsumer. Imagine that both the budget onstraint and the utility funtion were plotted in the three dimensions of Figure 7 and then imagine that we are observing that figure from diretly overhead, so that the utility axis were oming straight out of the 1 2 plane at us, so that we lose perspetive of the utility axis. What we would see are an indifferene map and a budget line. Figure 8 shows that the optimal deision (the one that yields the highest attainable utility) features a tangeny between the budget onstraint and an indifferene urve. Consider what would happen if the optimal hoie did not feature suh a tangeny. In this ase, it must be that the indifferene urve through whih the hosen bundle passes also rosses the budget line at another point. Given that indifferene urves are onvex to the origin, this must mean that there is another onsumption bundle that is both affordable and also yields stritly higher utility, so a rational onsumer would hoose it. 10 At the point of tangeny that desribes the onsumer s optimal hoie, the slope of the budget line must equal the slope of the indifferene urve. The slope of the budget line, as we saw above, is simply the prie ratio P1/ P2. And reall from our disussion of utility funtions that the (negative of the) slope of an indifferene urve is the marginal rate of substitution the maximum amount of one good that the onsumer is willing to give up in order to obtain one more unit of the other good suh that his total utility remains the same. These two points lead us to a very important desription of a 10 The assumption of a "rational" onsumer must also be augmented with other strong assumptions, some of whih are that there is no inome unertainty, pries are fixed and the onsumer has no bargaining power, and there is no unertainty about the quality or nature of the produts. We will disuss some of these strong assumptions later. Spring 2014 Sanjay K. Chugh 28
onsumer's optimal hoie, one that we will refer to as the onsumer's optimality ondition: P 1 MRS P. 2 When markets are funtioning well (and we have yet to disuss what funtioning well means), this optimality ondition is what guides the deisions of onsumers. When markets are not funtioning well, poliy disussions at both the miroeonomi level and the maroeonomi level an use this optimality ondition as a benhmark to strive to ahieve when onsidering intervening in markets. 11 The eonomi logi of the optimality ondition is as follows. Without regard to pries, the MRS desribes the onsumer s internal (i.e., utility-based) willingness to trade one good for another. The prie ratio desribes the market trade-off between the two goods. To understand this last point, suppose that P 1 = $3 per unit of good 1, and P 2 = $2 per unit of good 2. 12 The prie ratio, therefore, keeping expliit trak of units, is P $3/ unit of good 1 3 units of good 2 1 P 2 $2 / unit of good 2 2 units of good 1. Notie the units here it is units of good 2 per unit of good 1, whih is exatly what it must be in our two-dimensional graph with 1 on the horizontal axis and 2 on the vertial axis. This demonstrates that the prie ratio does indeed desribe the market trade-off between the two goods. Suppose the onsumer had hosen a bundle at whih his MRS was higher than the market prie ratio. This means that he is willing to give up more units of 2 for a little more 1 than the market requires him to so he should use the markets to trade some of his 2 for 1 beause he would be made unambiguously better off! Now suppose he has traded himself in this way all the way to the point where his MRS equals the prie ratio. Should he trade yet more units of 2 to obtain a little more 1? The answer is no, beause doing so would now mean having to give up more units of 2 than he willing to for a little more 1. Thus, one he has traded his way to the bundle at whih his MRS equals the prie ratio, he an do no better he has arrived at his optimal onsumption hoie, the one that maximizes u( 1, 2 ) subjet to his budget onstraint. 11 We will have muh more to say later in the ourse about the role of government intervention in markets. 12 It is very easy to lose sight of the fat that pries have units. That is, when a prie tag on a T-shirt says $10, the impliit units attahed to this are 10 per T-shirt beause obviously if you want to buy 2 T- shirts you will have to pay $10x2 = $20. Unit analysis is often helpful in thinking about how eonomi variables relate to eah other. Spring 2014 Sanjay K. Chugh 29
Lagrange Charaterization Let s now study the optimality ondition using our Lagrange tools. To ast the problem we are studying here into the mathematial form we enountered in our general introdution to the Lagrange method: the objetive funtion (i.e., the funtion that the onsumer seeks to maximize) is the utility funtion u ( 1, 2) ; the variables to be hosen are 1 and 2 ; and the maximization of utility is subjet to the budget onstraint P 1 1 P 2 2 Y. To ast the budget onstraint into the form g(.) 0, let s write g (, ) Y P P 0. 13 The Lagrange funtion is thus 1 2 1 1 2 2 L (,, ) u (, ) Y P P, 1 2 1 2 1 1 2 2 where is the Lagrange multiplier. The first-order onditions with respet to 1, 2, and are, respetively, and u P 1 0, 1 u P 2 0, 2 Y P 1 1 P 2 2 0. From the first two of these expressions, we an obtain the optimality ondition we obtained qualitatively/graphially earlier. The first expression immediately above an be u/ 1 solved for the multiplier to give us. Next, insert this in the seond P1 expression immediately above, giving us u P2 ( u/ 1). P 2 1 Rearranging this in one more step gives us P1 u/ 1, P u/ 2 2 whih states that when the onsumer is making the optimal hoie between onsumption of the two types of goods, his ratio of marginal utilities (the right-hand-side of this last 13 Alternatively, we ould equivalently onstrut g (.) as g ( 1, 2) P 1 1 P 2 2 Y 0, and we would obtain exatly the same result we are about to obtain; it would be a good exerise for you to try the subsequent manipulations for yourself using this alternate definition of the funtion g(.).. Spring 2014 Sanjay K. Chugh 30
expression) is equal to the ratio of pries of the two goods (the left-hand-side of this last expression). Compare this expression to the expression earlier that we named the optimality ondition: inspeting the two reveals that they must be the same and furthermore that the MRS between two goods is equal to the ratio of marginal utilities. This latter very important result (that the MRS between any two goods is equal to the ratio of marginal utilities between those two goods) an be derived more rigorously mathematially, but we defer this. Instead, the important idea to understand here is how to apply the Lagrange method to the basi onsumer optimization problem and how it yields the same intuitive result we qualitatively obtained earlier. To link the result here bak to our introdution to the Lagrange method, note that if we ompute the partial derivatives of the onstraint funtion (the budget onstraint in this ase) with respet to 1 and 2, we have g/ 1 P1 and g/ 2 P2, whih means g / 1 P1 that the ratio of partial derivatives of the onstraint funtion is. But this is g / 2 P2 obviously just the left-hand-side of the optimality ondition above. Reall in our introdution to Lagrange theory we noted that a entral result was that at optimal hoies, the ratio of partials of the objetive funtion (here, the utility funtion) would be equal to the ratio of partials of the onstraint funtion (here, the budget onstraint): here is our first speifi instane of this important result. Spring 2014 Sanjay K. Chugh 31
Spring 2014 Sanjay K. Chugh 32