. If lim. x 2 x 1. f(x+h) f(x)

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1 Review of Differential Calculus Wen te value of one variable y is uniquely determined by te value of anoter variable x, ten te relationsip between x and y is described by a function f tat assigns a value y = f(x) to eac possible value of x. x is ten called te independent variable, or te argument of te function, and y is called te dependent variable or te value of te function. Te letters x and y are cosen arbitrarily - we could just as well use r, s, or any oter symbols to represent te independent and dependent variables of a function. Similarly, and symbol could be used to represent te function besides f. It s just a convention to use x, y, and f. Te grap of te function f is te set of pairs (x, f(x)) for all x X. X is te domain of te function f, te set of values of x for wic f(x) is defined. Te set {f(x) x X} is called te range of te function. A function f is said to be continuous at x 0 if lim 0 f(x 0 + ) = f(x 0 ). If f is continuous at every point x in its domain, ten f is said to be continuous. We are interested in te effect of a cange in x on a cange in y. Tis effect can be measured by te slope of te grap of te function tat relates y to x if tat slope exists. Calculus is used to define and compute tat slope. Te slope of te line segment joining points (x 1, y 1 ) and (x 2, y 2 ) were x 1 x 2 is given by y 2 y 1 x 2 x 1, tat is te cange in te y coordinate divided by te cange in te x coordinate. Te slope of te line segment joining te points (x, f(x)) and (x +, f(x + )) is ten f(x+) f(x) f(x+) f(x). If lim 0 equals a number, ten tat number is labeled f (x) and called te derivative of f at x. (Sometimes f (x) is also written df(x) or dy ). Te derivative f (x) is te slope of te line dx dx tat is tangent to te grap of f at te point (x, f(x)). If f (x 0 ) > 0 ten a small increase in x at x 0 increases te value of f(x). If f (x 0 ) < 0 ten a small increase in x at x 0 decreases te value of f(x). If f (x) exists for eac x X ten te function f is said to be differentiable. Any differentiable function must be continuous, but a continuous function is not necessarily differentiable. Examples of functions tat are not differentiable: Let g(x) = 5 if x 4 and g(x) = 9 x if x > 4. Ten sow by using te definition of derivative tat g is not differentiable at x = 4. Is g continuous? Examples of functions tat are differentiable: Let f(x) = 3 2x. Ten f(x + ) = 3 2(x + ) and f(x+) f(x) = 2x 2x 2 = 2 = 2. Te slope of tis function f is te same at all values of x. A function f wit a grap tat is a straigt line can be written f(x) = ax + b, were a and b are some constants. Te slope of te segment joining any two points on te line is a, so f (x) = a for all x. As an exercise, sow tat te last sentence is true. Important identities: 1

2 If c is a constant ten dc dx = 0. = nxn 1 were n 0 (n could be a non-integer). Sow as an exercise tat tis is true for n = 2. dx n dx d(ax n +bx m ) dx = anx n 1 + bmx m 1. Note tat n and m need not be integers. Let f and g be differentiable functions at x. Let (x) = f(x) + g(x) and k(x) = f(x)g(x). Ten (x) = f (x) + g (x) and k (x) = f (x)g(x) + f(x)g (x). Example: If r(q) = p(q) Q ten r (Q) = (p (Q) Q) + p(q). Exercise: Sow (using te product rule) tat if g(x) = cf(x) for some number c and for all x, ten g (x) = cf (x). Te cain rule: If k(x) = f[g(x)] ten k (x) = f [g(x)]g (x) Example: k(x) = (2x + 3) 2. Ten k (x) = 2(2x + 3) 2. If (x, f(x)) is te igest point or te lowest point on te grap of a differentiable function f, ten f (x) = 0. If f (x 0 ) = 0, and tere exists ɛ > 0 suc tat f (x) < 0 for x (x 0 ɛ, x 0 ) and suc tat f (x) > 0 for x (x 0, x 0 + ɛ), ten f(x 0 ) < f(x) for all x suc tat x x 0 and x (x 0 ɛ, x 0 + ɛ). x 0 is ten called a local minimum of f. If f (x 0 ) = 0, and tere exists ɛ > 0 suc tat f (x) > 0 for x (x 0 ɛ, x 0 ) and f (x) < 0 for x (x 0, x 0 + ɛ), ten f(x 0 ) > f(x) for all x suc tat x x 0 and x (x 0 ɛ, x 0 + ɛ). x 0 is ten called a local maximum of f. Pindyck and Rubinfeld Capter 3 - Consumer Beavior Te study of consumer beavior is interesting to bot firms, wo use it to determine wat prices to carge, and te government, wo uses it to decide wat policies will elp consumers. One can divide consumer beavior into tree parts. Te first is preferences. Tese determine wat combinations of goods are preferred to oter combinations of goods (witout any constraints). Te second is a budget constraint. Tis says ow muc money a consumer as available to purcase bundles of goods. Te budget constraint determines wic combinations of goods can be bougt by te consumer. Te tird part combines consumer preferences wit te budget constraint to determine wat coice te consumer actually makes. We now formalize te teory of consumer beavior. Several definitions need to be made. A market basket is a list tat specifies quantities of goods. For instance, it could refer to te quantities of food, cloting and ousing tat a consumer buys eac mont. Example of one market basket: 600 units paper, 20 units rice, 4 units pencils, 2 units cars. 2

3 Te word bundle means te same ting as market basket. Here are some assumptions about preferences tat are commonly made: 1. Preferences are complete: Given market baskets A and B, eiter consumer prefers A to B (A B), or prefers B to A (B A), or is indifferent between B and A (B A). If A is at least as good as B we write A B. 2. Preferences are transitive: Given market baskets A, B and C, if A is at least as good as B and B is at least as good as C, ten A is at least as good as C. (If A B and B C, ten A C.) 3. More of any good is preferred to less (nonsatiation). If A contains at least as muc of every good as B and more of one good, ten A B. Tis assumes tat goods are desirable. In tis, we ignore bads - e.g. air pollution. Tese assumptions are believed to old for most people most of te time. Tey impose some rationality on people s preferences. Assume tat a consumer s preferences satisfy completeness and transitivity. Ten from tat consumer s preferences, we can construct indifference sets, wic can be represented in te F-C axis. An indifference set is a set of all te points (F, C) suc tat te consumer is indifferent between (F, C) and (F 0, C 0 ) for some point (F 0, C 0 ). If in addition we assume tat te consumer s preferences are nonsatiated, te indifference sets are indifference curves (if te preferences are not nonsatiated, we could ave an indifference set tat is tick - see drawing). In an indifference curve, to eac F in te indifference curve corresponds exactly one C. Properties of indifference curves 1. Due to our assumptions, an indifference curve must slope downwards. Suppose instead tat an indifference curve passing troug point A = (F 0, C 0 ) sloped upwards. Ten tere must be a point B = (F 1, C 1 ) on te indifference curve suc tat F 1 > F 0 and C 1 > C 0. But ten B as more of bot goods tan A, and te consumer is indifferent between B and A. Tis violates nonsatiation. Similarly, if nonsatiation olds, an indifference curve cannot be orizontal or vertical. 2. Two indifference curves of te same individual cannot intersect. To sow tis, suppose tat two indifference curves of te same individual intersect at point A = (F 0, C 0 ). Ten te individual must be indifferent between any two points on eiter of te two indifference curves. In particular, tere must be a point B = (F 1, C 1 ) on one of te indifference curves and a point D = (F 2, C 2 ) on te oter indifference curve suc tat F 1 > F 2 and C 1 > C 2. Yet te consumer is indifferent between te two points. Tis violates nonsatiation. An indifference map is a set of indifference curves for one individual. Te entire map would sow an infinite number of indifference curves, one passing troug eac point on te F-C axis. 3

4 3. It is likely (toug not necessary) tat an indifference curve becomes less steep as one moves along te F-axis. Tis as a reasonable interpretation. It means tat as te person consumes more food, se is willing to give up less and less cloting for an additional unit of food. Te same is true as we move along te C-axis: As te person consumes more clotes, se is willing to give up less and less food to get an additional unit of clotes. Tus, te more a person consumes of a good, te less valuable or desirable additional units of tat good become in terms of oter goods. Let us formalize tis last point using te concept of marginal rate of substitution. A person s marginal rate of substitution (MRS) of food for cloting is te maximum amount of cloting tat te person is willing to give up in excange for an additional unit of food. If a consumer s MRS is 3, it means te consumer is willing to give up 3 units of cloting in excange for an additional unit of food. Te value of te MRS depends on wic initial point of consumption te consumer is at. If te consumer already as 15 units of food and 1 unit of clotes, te MRS of food for cloting is likely to be different from wen a consumer as 1 unit of food and 15 units of clotes. Te negative of te MRS at a particular point equals te slope of te indifference curve at tat point, if it exists. Most consumers ave a diminising MRS (tis is te same as 3.) Perfect substitutes and perfect complements Consider te following cases of two different consumer s indifference curves over two different goods. 4

5 Apple juice (glasses) Perfect Substitutes Left soes Perfect Complements Orange juice (glasses) Rigt soes In te first case, te consumer (call im Bob) views apple juice and orange juice as perfect substitutes. Tis means tat Bob is always indifferent between one glass of apple juice and one glass of orange juice. Te slope of eac of is indifference curves is 1, because at any amount of orange juice, e is willing to trade 1 glass of apple juice for 1 additional glass of orange juice. Tus te MRS of orange juice for apple juice is always 1. Note tat tese preferences do not satisfy diminising MRS. Goods are called perfect substitutes if te MRS of one for te oter is a constant (it need not be 1). Remember tat we defined two goods to be substitutes (in te eyes of a particular consumer) if wen te price of one increases, te amount demanded of te oter increases. We ave not introduced prices yet, but we can already see tat wit Bob s preferences, if te price of orange juice exceeds te price of apple juice, e will only buy apple juice, and vice versa. Wen te price of orange juice equals te price of apple juice, Bob doesn t care ow muc e buys of eiter. But if te price of apple juice increases from tat level, e will buy only orange juice and no apple juice. So in general is amount demanded of orange juice will increase if te price of apple juice increases, and vice versa. In te second case, te consumer (call er Jane) views left soes and rigt soes as perfect complements. Having an additional rigt soe does not increase er satisfaction unless se as a corresponding left soe. Wen se as one rigt 5

6 soe and at least one left soes, se will not give up any more rigt soes to get anoter left soe. So er MRS of left soes for rigt soes is zero in tis case. Wen se as one left soe and at least one rigt soe, se will not give up any more left soes to get anoter rigt soe, so er MRS of rigt soes for left soes is zero (and er MRS of left soes for rigt soes is infinite). We call two goods perfect complements if te indifference curves for bot are saped as rigt angles. Bads are items suc tat less is preferred to more. Tese include all kinds of pollution, garbage, asbestos in ousing. If an item is a bad in every quantity, ten we can redefine it as te absence of it: clean air, clean streets and lack of asbestos are goods. 6