( )! ±" and g( x)! ±" ], or ( )! 0 ] as x! c, x! c, x! c, or x! ±". If f!(x) g!(x) "!,

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1 IV. MORE CALCULUS There are some miscellaneous calculus topics to cover today. Though limits have come up a couple of times, I assumed prior knowledge, or at least that the idea makes sense. Limits are pretty straightforward. If f is a function defined on a neighborhood of a point x, then the limit of the function as it approaches x is!! lim x" x f (x if for all! > 0 there is some! > 0 such that f ( x!! < " whenever x! x but these points are less than distance! away from each other. In other words, some point in the domain, not much different from x, will give you a value which is as close to! as you like. A function f is continuous at a point x if and only if f ( x = lim f x x! x (. This makes finding limits very easy where the function is continuous. The tougher cases are when we have zeros in the denominator, or when taking a limit approaching infinity. These are the conventions: lim x = " x!" 1 lim x!0 x = "# lim x = "# x!"# lim 1 x!0 x = " 1 lim x!" x = 0 These are all fine if you have a problem with only one thing going to a point at which it is not defined; what about two? This is my favorite example: x lim x!" x =? lim x x!0 x =? Judging by the rules above, as the function approaches either of these points, it would see to go off to zero or to infinity. The function in question is, of course, essentially the constant function f ( x = 1. Writing it this way means that it is not defined at these two points, but everywhere else it would equal one so these limits are equal to one. Cases like 0 0, 0! ", and!! are called indeterminate forms. When evaluating limits that take one of these forms, we use L Hôpital s rule: (! ±" and g( x! ±" ], or (! 0 ] as x! c, x! c, x! c, or x! ±". If f!(x g!(x "!, Let f,g : X! R, X! R. Suppose that [ f x that [ f ( x! 0 and g x then f (x g(x!!. Fall 2007 math class notes, page 27

2 This can be used to verify that the limit of f ( x = x x is indeed one at the points in question. Keep in mind that L Hôpital s rule works only for these indeterminate forms. However, it always works in these cases, provided the derivative exists. If the derivative is still an indeterminate form, you can apply the rule again. When you have interesting functions limits may be more complicated. Here are some to keep in mind, as n = 1,2,3, goes to positive infinity: If x > 0, then x 1 n! 1 If x < 1, then x n! 0 For all x, x n n!! 0 If! > 0, then n!" # 0 n If! < 0,! i "! n $ If! < 0, i=1 $! i " 1 i=0 1#! 1#! (lnn n! 0 n 1 n! 1 While on the topic of interesting functions, some polynomials have tricky derivatives. One way to cope with some of these is to use the differential approach mentioned previous. However, consider this wacky function: y = 2 x What is dy dx? You can t use the power rule here that works only when you have a constant (not an independent or dependent variable as the exponent. When you have something to the power of x, try this trick: ln y = x ln2 Then take the differential: dy y = dx ln2 Substituting back in the original y = 2 x and rearranging, dy dx = 2x ln2 Logarithms, exponentials, and other transformations are useful tricks when trying to differentiate difficult functions. Multiple integrals show up a few times. Let s say you have some function z = f ( x, y, and you want to find the volume of the area underneath some area! in the x-y plane. This is written as: V = ""! f ( x, ydx dy Fall 2007 math class notes, page 28

3 First, you need to identify the values that one of the variables runs over, and then express the other variable s values in these terms. For instance, suppose! is the unit circle. Then x runs from!1 to +1. Given a particular x, y takes values between! 1! x 2 and 1! x 2. Let s let f x, y ( = x 2 y 2. This problem is then rewritten: x=1 $ y=# 1# x 2 ' V = "" f ( x, ydx dy = x 2 y 2 dy! " " dx % & ( x=#1 y=# 1# x 2 We first integrate with respect to the innermost term, and evaluate at the limits. x=1 " 1 V = 3 x2 y 3 y= 1! x 2 % 2 ( # $ y=! 1! x 2 & ' dx = ( 3 x2 1! x 2 x=!1 x=1 x=!1 ( 3 2 ( dx At this point, we are simply integrating a function of one variable, which we ve reviewed. This iterative procedure works for when for triple, etc., integrals. The number of years it takes students to complete a PhD is distributed: f ( t = 0.187e!1.87t. The starting salaries of new PhDs in economics is distributed: g( y = ( 2 1! 2" exp $ & # y # µ % 2! 2 ' ( As before, µ = 56,412 and! = 8,273. Expected utility is discounted in the following manner: U ( y = (!y " #y 2 + $ (% " t What is the expected utility of future consumption for a person starting a PhD program? Oliver and I go out rent a movie, not knowing how much we will enjoy it. My valuation of the movie ( v m and my friend s valuation ( v o are jointly distributed uniformly between $0 and $1; that is: ( = 1 f v m,v f What is the expected value of our joint benefit, E[v m + v o ]? Earlier I used the fact that the first derivative can be used to approximate the change in a function:!y " f #( x!x Fall 2007 math class notes, page 29

4 This is sometimes called a first-order approximation. It is particularly good when the function is approximately linear and when!x is fairly small. A better approximation would use the second derivative to compensate for this error:!y " f #( x!x f ## x ( (!x 2 This has the property that it corrects for very nonlinear functions, and corrects more the farther you move from x. This is often called a second-order approximation of the function f. Because it s a quadratic (the derivatives evaluated at x are constants as far as we re concerned, it might be a lot easier to work with than the function f itself. These approximations may come up frequently in macro. They are derived from a Taylor series or a Maclaurin series. This principle is stated in many ways. Let f ( k denote the k-th derivative of f. In the univariate case,! Let f : X! R be C X (infinitely differentiable on X! R. f (x +!x " f (x = # $ k =1 ((!x k % f ( k (x k! Then Let f : X! R be C X m on X! R. Then there exists some point ˆx in the interval from x to (x +!x such that f (x +!x " f (x =! k =1 ((#x k $ f k (x k! +(!x m " f m ( ˆx m! Let f : X! R be C m X on X! R. Then there is some remainder function such that f (x +!x " f (x =! m k =1 (("x k # f ( k (x k!+ R( x,"x,m, with R(x,!x,m (!x m " 0 as!x " 0 or m! ". All of these can be generalized to the multivariate case, for vectors x and!x, by replacing the derivatives with the appropriate matrix of derivatives, and replacing the!x k with a suitable arrangement of multiplications of the vector!x. For instance, the second order Taylor approximation looks like: f (x +!x " f (x! #f #x!x + 1! x $ 2 #2 f #x# x $!x Consult Simon and Blume (Chapter 30 for more on this. Derivatives are also used to determine concavity and convexity of functions. A function f : X! R, where X! Rk, is concave if for any! "[0,1], f (!x + ( 1"! x# $! f ( x + ( 1"! f ( x# for any x and x! in X. This relationship for concave functions is sometimes called Jensen s inequality. Alternatively, the function f : X! R is convex if f is concave (that is, the inequality above is reversed. When Jensen s inequality holds strictly for all x! x " and! "(0,1, the function is strictly concave. My utility function is strictly concave in consumption. I have an initial wealth of $1,089. My friend Oliver offers me a fair deal this time: heads, I get $212; tails, I lose $212. What does accepting this bet do to my expected utility? Fall 2007 math class notes, page 30

5 Oliver and I both have the same strictly concave utility function. However, he has managed to amass much more wealth than I have. His wealth is $12,749, and mine is $1,089. Society s aggregate welfare is the sum of our utilities. Would it be a social improvement to redistribute some money to me? What is the optimal amount of redistribution? An interpretation of Jensen s inequality is that the expected output of a concave function is less than the value of the expected input. This appears often when dealing with expected utility. Here are two properties that are equivalent to the definition of a concave function. concave if and only if: The continuously differentiable function f : X! R, defined on X! R k, is f ( x +!x " f ( x + f # ( x!x (for k = 1 f ( x +!x " f ( x + #f!x (for k > 1 #x provided x and x +!x are both in X. The inequality holds strictly for all!x " 0 if and only if the function is strictly concave. The twice continuously differentiable function f : X! R, defined on X! R k, is concave if and only if: f!! ( x " 0 (for k = 1!x " D 2 f ( x!x # 0 (for k > 1 for all!x in R k. The inequality holds strictly for all!x " 0 if and only if the function is strictly concave. The second theorem is very useful. Often this is called the second derivative test for concavity. In each of these theorems, reversing the sign of the inequality gives the condition equivalent to convexity. Is the utility function U ( c = ln( c concave? Verify this all three ways. Is the utility function U ( c, h = Ac a h 1!" concave in each of its variables? (That is, use the k = 1 conditions. ( " 0 Can a function be both concave and convex? Yes, but this requires that f!! x and f!! ( x " 0, which means pretty clearly that f!! ( x = 0. An affine function on a set X is a function that is both concave and convex on X. From the definition of concavity and convexity, we see that affine functions have the property that ( ( # =! f ( x + ( 1"! f ( x# f!x + 1"! x Fall 2007 math class notes, page 31

6 for! between 0 and 1, inclusive. It follows that a function f : X! R, X! R k, is affine if and only if it can be expressed in the form: f ( x = a! x + b where a is a k-dimensional vector, and b one-dimensional. [You ve doubtlessly heard these called linear functions before, and you ll probably hear them carelessly called linear functions again, but there s a difference.] A linear function is a one that for any real! and!, f (!x + " x# =! f ( x + " f ( x# By setting! = "# and x = x!, you can see that a proper linear function must run through the origin. This means that linear functions take the form: f ( x = a! x It s a picky difference, but one that you should at least be aware of. When people say a linear transformation of x, they mean that x has been multiplied by something. When you are told to take an affine transformation of x, you multiply it by something and then add on something else. A tax is called proportional if the average tax rate (total tax over total income is the same at all levels of income. It is called progressive if this is increasing; regressive if it is decreasing. A linear tax schedule (that is, total tax is a linear function of total income is necessarily proportional. An affine tax schedule need not be. Moving on from technical nitpicking, a property of functions that shows up frequently is homogeneity. A function f : X! R, X! R k, is homogeneous of degree r if for any! > 0 : f (!x =! r f ( x In other words, a function is homogenous of degree one if doubling all the inputs means all outputs are doubled; of degree two if doubling inputs means outputs are quadrupled; of degree three if doubling inputs leads to an eight-fold increase in output; and so forth. The Cobb-Douglas production function, F( K, L = AK! L 1"!, is homogeneous of degree one in capital and labor. A linear function is homogeneous of degree one. An affine function need not be. (Think back to taxes! The utility function U ( c =! lnc isn t homogeneous of degree anything. Fall 2007 math class notes, page 32

7 The demand functions: x 1 w, p 1, p 2 are homogeneous of degree zero in w, p 1, p 2 in( p 1, p 2. ( =!w p 1, x 2 ( w, p 1, p 2 = (1! "w p 2 (. They are homogeneous of degree minus one Homogeneity of degree r is abbreviated hd(r. A very, very important property of homogeneous functions is Euler s theorem: only if: for all x in X. The continuously differentiable function f : X! R, X! R k, is hd(r if and ( k! f x " x i=1 i = r f x!x i ( ( = AK! L 1"!, is hd(1. Profits The Cobb-Douglas production function, F K, L are given by! ( K, L = pf( K, L " wl " rk, where p is the price of output, w the wage rate, and r the rental rate on capital. Given that the equilibrium factor prices are marginal revenue products, what are pure profits? This example illustrates one reason why Euler s theorem is important. It should also indicate that chances are pretty good you ll be working with lots of hd(1 functions in economics. A final note about homogenous functions is this theorem.!i = 1,2,,k,! f x If the continuously differentiable function f : X! R, X! R k, is hd(r, then (!x i is hd(r-1 in X. The Cobb-Douglas production function, F( K, L = AK! L 1"!, is hd(1. What happens to the marginal product of labor when you increase all inputs n-fold? References: Limits: Salas and Hille (Chapter 2; Sydsæter et al. (Chapter 3 Taylor series: Simon and Blume (Chapter 30; Mas-Collel et al. (Math Appendix Convexity and concavity: Simon and Blume (Chapter 21; Mas-Collel et al. (Math. Appendix Linear functions, affine functions, and cousins: Rockafellar. Homogeneity: Simon and Blume (Chapter 20 Fall 2007 math class notes, page 33