First the Basic Background Knowledge especially for SUS students. But going farther:

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1 asic ackground Knowledge: Review of Economics for Economics students. Consumers Economics of the Environment and Natural Resources/ Economics of Sustainability K Foster, CCNY, Sring 0 First the asic ackground Knowledge esecially for SUS students. ut going farther:. More about Consumer ehavior a. Notate demand for a good, x = x (,,m), demand deends on rice of the good, rice of other goods (for now just one other good; it kees down comlexity but is generalizable), and budget m x 0 b. rice rises imly fall in consumtion (excet for Giffen goods) so generally c. can decomose this change in consumtion into income/substitution effects so always x 0 if there is income to comensate for the change. udget a. undles are denoted X, so that a bundle of goods, X=(x,x). Tyically we take the sace of ossible bundles to be the real numbers. We can grahically reresent these bundles as Cartesian coordinates on an x-y lane. Cases with more than goods are analogous, we just move to higher-dimensional Cartesian sace. b. ssume that the budget set is given by x + x m. (So it can be grahed as bordered by a straight line.) x x The intercet of x shows how much x could be urchased if the erson sent all her m money on just x; this is. nalogously the intercet of x is m. c. on the budget line, where x + x = m, the sloe is

2 x (just re-arrange the budget equation to collect the terms involving x x and then solve for x = ) d. note effects of an income shift, m, a arallel shift of line; x x e. effect of rice shifts, i, where the new line is no longer arallel and budget rotates around one intercet (the one whose rice didn't change). In icture below, a fall in the rice of x means that more of x can be bought if the consumer devotes entire budget to it. x x f. and the secial case of an equal roortional change in m and all rices so there is no net change 3. Utility a. From Micro, recall that as long as a consumer's choices satisfy some basic requirements then the choices can be reresented by a utility function b. Define Marginal Utility, MU, MU as MU = ΔU/Δx and MU = ΔU/Δx (For those with U U calculus knowledge, then MU is the derivative of the utility function with resect to the first argument, MU = and MU =. x x If you don't know calculus then ignore this note!)

3 MU c. MRS = MU d. So draw indifference curves x x 4. Choice u x, x a. Given and a budget constraint that x x m max u x, x x b. we want to find the goods, x and x, that solve:, x x x m. MU c. Usually otimal oint is at a tangency, where MU MU MU. or subject to MU MU or x x * x * x d. The last condition is "bang for the buck" n examle from Thomas Friedman's NYTimes column of Set 5, 006: Since the 970 s oil shocks, razil has, with lots of trial and error, made ethanol art of its daily life. It hits you the minute you drive into a gas station in São Paulo, where you need two things: a credit card and a calculator. In rough numbers, sugar ethanol now sells here at a little over $ a gallon and gasoline at a

4 little more than $4 a gallon. ecause sugar ethanol gets only about 70 ercent of the mileage of gasoline, drivers here do the math each day and figure out if ethanol is at least 30 ercent less than the rice of gasoline. If it is, many will fill er u with sugar cane. 5. Effects of Change in Income a. s income, m, rises the consumer could buy more of both; can choose any oint that is between the intersections of the old indifference curve with the new budget (a and b in the figure below): x b c a x or x b c a x or

5 x b c a x Clearly it is ossible that, as income rises, an individual could choose to urchase more of both x and x, or more of x and less of x, or less of x and more of x. It is erfectly reasonable that there are certain goods that eole buy less of, as they get richer. Indeed most marketers of consumer goods offer a whole constellation of brands that aim to hel eole work "u the chain" as they get wealthier. What clothing brands are for oorer (often teen) consumers? For middle-income ones? For the rich? b. We don't want to limit ourselves to only certain sorts of goods so we allow the consumer to have this range of choice. If eole buy more of a articular good as they get richer, economists call this a normal good. Mathematically a normal good, xi, is one where x i x j 0 0 m. n inferior good, xj, is one where m. c. Grahically we can lot the income offer curve or income exansion ath as showing the bundles of x and x that the consumer chooses at each level of income: x income offer curve x It is the locus of tangent oints of the indifference curves and budget line. Of course it is different for every rice combination.

6 d. We also can consider the Engel curve, which is a lot of the quantity of a good, xi, and the level of income, m (with the deendent variable on the horizontal axis, just to confuse you!). m Engel curve x i i. The Engel curve is not always uward-sloing, however. Sloes u only for normal goods. ii. Inferior goods would have an Engel curve sloing downward. (It's also ossible for some goods to be initially normal and then inferior. For instance calorie consumtion: in oor societies the richer eole are fatter because they can afford not to starve; in rich societies the richer eole are more fit because they can afford to exercise and eat better.) 6. Effects of Change in Own Price a. From x x,, m and x x,, m of x i is influenced by changes in its rice, i., we now consider how the otimal choice b. Now if we kee income constant but instead change, we can observe how the consumer's choice of bundles change. Linking these together gives us the rice offer curve: x rice offer curve x

7 c. lot of the rice against the amount of the good chosen (again, utting the deendent on the horizontal axis) gives us the demand curve: i demand curve x i 7. Examles We can get mathematical reresentations for these given functional forms for utility functions. asic common utility functions with easy solutions: Cobb-Douglas, for instance U = sqrt(x,x ); a variant is U = ln(x ) + ln(x ) but you should be able to show that this is a monotonic transformation of the revious case quasi-linear U = sqrt(x ) + x erfect substitutes U = ax + bx erfect comlements U = min(x, x ) Set budget = m, = and = (so is numeraire, without loss of generality) x x Cobb-Douglas case with U xx so MU and MU. x x MU MU MU Solve the bang-for-the-buck condition to find, MU, x x x, x x x x x x x x,, x. Put this into the budget constraint that xx m, x x x x x m x m, x. Then ut this back into the result of the bang-for-the-buck, so m m x. So from this we can sketch the Engel curve and demand curve for good x : the

8 Engel curve is a line from the origin (re-write, since the Engel curve uts m on the vertical axis, as m x ); the demand curve is a rectangular hyerbola (again, re-write with on the left). You should be able to solve the more general case, U x x. x Quasi Linear, U x x so MU, MU. ang-for-buck gives x. Solve for x 4 -- but only if there is enough money to buy this amount! The bang-for-the-buck shows that the consumer first has MU/ > at a low level of x, but then this is satiated and the consumer sends any increase in income on x. The budget constraint gives xx m, x m so x m (as long as this is non-negative). 4 4 I will leave the erfect substitutes and erfect comlements as an exercise, since we don't make much use of those in this course. 8. General Equilibrium: Edgeworth oxes U to now we have been looking at artial equilibrium solutions: taking some factors as exogenous, asking how demand and suly in a articular market would change. ut of course demand and suly in any articular market deend on demand and suly in other markets, as the news regularly reminds us: the roerty rice bubble led to unusually high demand for certain goods; high gasoline rices restrain consumer demand; there are always linkages. We want to build a model that can account for these, that is a general equilibrium (G.E.). Walras was one of the early economists to devote himself to this sort of theorizing, so these models are sometimes referred to as Walrasian general equilibrium. We first look at an economy with eole trading different goods. We assume that they are endowed with a certain amount of each good and then can trade to achieve their otimum level. So let's lay out the notation first. The eole are and ; the goods are and ; so that x is the consumtion, by erson, of good, x is the consumtion of good by erson (the suerscrit does not mean that x is squared). We denote the endowments of each good with. So given rices, and set in the market, each consumer wants to maximize their utility given their budgets; so erson will max U x, x subject to x x, x x and will

9 , x x max U x, x subject to x x, where as long as "more is better" the less-than-or-equal-to signs are equalities. Changing rices will give incentives for the consumers to change their consumtion choices we sometimes exlicitly show that the final consumtion choices, x, x, x, x deend on the rices by writing them as,,,,,,, x x x x. Each consumer might choose to consume either more or less than she is endowed with, so either demanding more than she has or sulying some of what she has. We define the "excess demand" function of each erson, i=,, as: e, x, e, x,. So we could re-write each and i i i i i i erson's maximization roblem as: max, e, e, 0. i, i x x i i i U x x subject to i i How would we define equilibrium here? Certainly we need the condition that the same quantity that is bought is equal to the quantity sulied (sulied by the endowments of goods). These conditions, the global resource constraints, for each good are: x x and x x. One of the key considerations of this model is that consumers don't need to know the global resource constraints, just the rice of a good, when making their choice. In the -erson case the informational savings do not seem that big consumer only needs to know and rather than erson 's choices of goods and, but as the number of consumers increases, the efficiency of the rice signals increases. Imagine if there were just 00 eole trading the goods a consumer in a decentralized market economy needs only to know and rather than the 98 choices of 99 eole of goods! However the details of rice formation are left rather dangling. In the Walrasian case we assume that there is some agent that is the "auctioneer" who calls out rices, gets information from each agent about how much they would demand or suly at those rices, nets u the totals and then changes the rices as necessary. In the real world there is no such auctioneer although maybe someday eay will be! What are the conditions that these rices need to fulfill? We can re-write the global resource constraint conditions, that x x and x x, by instead utting them in terms of net demands, so giving us the aggregate excess demands for each good as z and z, where:,, and z, e, z e i i i i

10 or, droing the extra notation about deendence on rices as well as the summation notation, this is: z e e and z e e. So equilibrium is a set of rices that give both z = 0 and z = 0. However those equality conditions are actually redundant. The maximization constraints for each agent imly Walras' Law, that: z, z, 0 for all, > 0. This "Law" can be verified by substituting in the definitions of z and z as excess demands, e, e, 0. then noting the budget constraint for each consumer, that i i So if we have k goods then the k rices, in order to be equilibrium values, must set (k ) aggregate excess demands to zero (and Walras' Law will take care of the last). The intuition of this result is that, given the way we have secified each consumer's roblem, their choices of consumtion deend entirely on relative rices, not the absolute levels. So the equilibrium values would not be changed if the consumers used different currencies, or if all rices were increased by a factor of 0, or if there were general inflation. In other words, these consumers have no money illusion. Under what circumstances, or what restrictions on the references of the individuals, can we assert that an equilibrium must exist? asically, if the functions z and z are continuous, then this will assure that there is a decentralized cometitive market equilibrium. Continuity can be assured if all consumers have convex references (so that small changes in rices give small changes in their demands) or if there are many consumers (so that, in aggregate, even if a few consumers make large changes in demand in resonse to tiny changes in rice, the total demand does not swing too far). The Edgeworth ox diagrams rovide an elegant visual exlanation for the market rocesses (although it is difficult to diagram more than eole with goods; unlike the functions above where adding more eole or more goods is straightforward). The Edgeworth ox uts a different set of axes starting at the oint,,, so that every single oint in the diagram gives 4 ieces of information: the horizontal axis gives the consumtion of good by erson x (the remainder is consumed by erson so it also shows x since the total consumtion must sum to the total endowment, x x ) and the vertical axis shows consumtions of good by and.

11 + x x x x + So the indifference curves and budget line of consumer would look like this: + + while the indifference curves and budget line of consumer would be as below (turn the icture uside down if it doesn't look familiar):

12 + + So the combination is: + x x x x + where the diagram now shows the initial endowments,,, equilibrium consumtion,,,, and the final x x x x whew! are there enough lines in that icture yet?! The equilibrium market rice is how we get from the initial endowments to final consumtion.