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Economics 1011A Section Notes Economics 1011a Section Notes Kevin He January 1, 2015 This file collects together all the section notes I wrote as a Teaching Fellow for an undergraduate micro-economic theory course (Economics 1011a) in Fall 2014. These notes are not self-contained, as they are meant to accompany weekly lectures and the textbook. For instance, the notes provide very few numerical examples, which are already in ample supply in these other sources. Instead, I try to emphasize the “theory” in “intermediate microeconomic theory”. Some of the material in these notes borrow from previous Teaching Fellows and many images are taken from the Internet. Specific sources and acknowledgements appear on the first page of each week’s notes. I am very grateful for the work of these individuals. Contents Page 0. Math Review......................................................................................................................................2 1. Modeling and “Verbal Problems”....................................................................................................8 2. Firm Optimization............................................................................................................................ 12 3. Preference and Utility Representation............................................................................................ 19 4. Duality and Its Implications............................................................................................................ 26 5. Time and Preferences....................................................................................................................... 32 6. Risk and Preferences......................................................................................................................... 41 7. Consumer Welfare, Markets, and General Equilibrium................................................................ 50 8. General Equilibrium, Externalities, and Public Goods................................................................ 58 9. Normal-Form Games......................................................................................................................... 68 10. Extensive-form Games and Signaling Games.............................................................................. 79 11. Thanksgiving – No Section............................................................................................................ 87 12. Summary of Definitions and Results............................................................................................ 89 1 Economics 1011a . Section 0 : Math Review1 9/4/2014 (1) Partial differentiation; (2) Total differentiation; (3) Implicit functions; (4) Optimization TF: Kevin He ([email protected]) 1 Partial differentiation 1.1 What is a partial derivative? Let’s think back to single-variable calculus for a moment. For a function g(t), the derivative g0(t) describes the rate at which the output changes when the input changes. In multivariate calculus, when a function f(x, y) depends on several inputs, we are often interested in the rate at which each of these inputs affects output. Partial differentiation allows us to disentangle the effects of these different inputs. Formally, starting at some point (a, b), define the partial derivative of f with respect to x at (a, b) to be: ∂f f(a + ∆x, b) − f(a, b) (a, b) := lim (1) ∂x ∆x→0 ∆x ∂f In other words, although f actually depends on the pair of inputs (x, y), in computing ∂x we treat ∂f y as a fixed constant. This way, we can interpret ∂x as the marginal contribution of just the first input to the output. Indeed, for small enough ∆x, we have the approximation: ∂f f(a + ∆x, b) ≈ f(a, b) + (a, b) · ∆x ∂x Of course, we could have done the above exercise for y holding x fixed, with analogous interpretations. 1.2 The geometric interpretation of partial derivative. Back in single-variable calculus, we could 0 read off the value of g (t0) as the slope of the tangent line to g(t) at t0. A similar procedure can be performed for partial differentiation. Given function f(x, y) and a point (a, b), consider the curve that f would trace out in R3 if its second argument were fixed at b while its first argument varies (curve C1 in Figure1). Construct the tangent line to this curve at point a (line T1 in Figure1). The ∂f slope of this tangent line is ∂x (a, b). Figure 1: The geometric interpretation of partial derivatives. The curve C1 traces out f with y fixed at b, while curve C2 traces out f with x fixed at a. The slopes of the tangent lines to these two curves ∂f ∂f at (a, b), T1 and T2, equal to ∂x (a, b) and ∂y (a, b), respectively. 1This week’s section material borrows from the work of previous TFs, in particular the notes of Zhenyu Lai. Image credits: Calculus: Early Transcendentals, Wikimedia Commons. 2 1.3 Partial differentiation as a functional operator. It is important to remember that the partial ∂f derivative ∂x (x, y) is a function defined on the same domain as f(x, y). It is not a number, unless evaluated at some specific point (a, b) ∈ R2. Let me make this point clearer by viewing the act of partial differentiation as a functional operator. Most familiar functions take numbers as inputs and return a number as output. But a function could also take a function as input, returning another function as output. Such functions are called functional operators, in that they “operate” on the input function and give back a transformed function2. ∂ ∂ Partial differentiation is a functional operator. Let’s write ∂x and ∂y for partial differentiation against 3 2 ∂ first and second arguments. Take f(x, y) = x y . Then ∂x (f) is another function, satisfying ∂ ! (f) (x, y) = 3x2y2 ∂x ∂ ∂ ∂ ∂ Further, since ∂x (f) is just another function, we can operate on it with ∂x again, obtaining ∂x ∂x (f) , which is yet another function – namely, ∂ ∂ !! (f) (x, y) = 6xy2 ∂x ∂x ∂ ∂ It can get quite tiresome to write expressions like ∂x ∂x (f) , so people have developed some ∂f ∂ shorthands. For example, ∂x is just the shorthand for ∂x (f). Table1 shows some other common shorthands for partial derivatives. No matter how they are written though, always remember that partial derivatives are functions. Equivalent notations ∂ ∂f ∂x (f) ∂x fx f1 ∂ ∂ ∂2f ∂x ∂x (f) ∂x2 fxx f11 ∂ ∂ ∂2f ∂y ∂x (f) ∂y∂x fxy f12 Table 1: All notations in each row are equivalent. One last thing. Using the specific example f(x, y) = x3y2, you may have noticed that ∂ ∂ ! ∂ ∂ ! (f) = (f) ∂x ∂y ∂y ∂x Turns out this is a general result. Young’s theorem states that, provided f has continuous second derivatives, fxy = fyx (2) For the purposes of this class, assume Young’s theorem always holds. 2Also called “higher order functions” by computer scientists. 3 2 Total differentiation 2.1 What is total differentiation? Partial differentiation is useful for isolating the effect of one input variable on the output when the other input variables are held fixed. Sometimes, however, there is another variable that determines the values of the input variables. For instance, suppose in the function f(x, y), x and y are actually functions of s. In that case, the output is really determined as f(x(s), y(s)). When all inputs to a multivariate function are themselves functions of a common variable, we may df ask how this common variable affects the final output. That is, we may want to compute ds . This calculation is called total differentiation, since we are computing the total dependency of f on s without assuming that any of the input variables is held fixed. (Usually, the only time it makes sense to talk about the “total derivative” of a multivariate function is when all of its arguments depend on a single common variable.) Chain rule allows us to write a total derivative in terms of partial derivatives. Suppose f : Rn → R while each of the arguments to f, namely x1, x2, ..., xn, is a function of s. Then chain rule implies: df ∂f dx ∂f dx ∂f dx = · 1 + · 2 + ... + · n (3) ds ∂x1 ds ∂x2 ds ∂xn ds 2.2 An example of total differentiation. Suppose utility is a function of income (Y ) and leisure (L). Further, each of Y and L is a function of hours worked, H. Specifically, Y (H) = wH (for some positive number w representing hourly wage) while L(H) = 24−H. In that case, the utility function, U(Y, L), is totally dependent on H and might be re-written as U(Y (H),L(H)). By above formula, its total derivative with respect to H is: dU dY dL = U · + U · = wU − U dH Y dH L dH Y L What is the sign of the total derivative? Ha, trick question! Remember that the partial derivatives dU UY and UL are functions (of Y, L), so dH is also a function. Its value, hence its sign, depends on values of Y and L, which in turn depend on H. In economic terms, whether working one more hour improves your total utility depends on how many hours you are already working. 3 Implicit functions 3.1 From relation to function, an example. So far, we have dealt with multivariable functions, which are mappings from Rn to R. We next turn to multivariable relations, which are equations of the form R(x1, ..., xn) = 0. Rough speaking, we would like to find some function h(x2, ..., xn) so that R(h(x2, ..., xn), x2, ..., xn) = 0 holds for a “rich” set of (x2, ..., xn). That is, h is a function implicit within the relation R(x1, ..., xn) = 0, which says what the value of x1 has to be for every n − 1 tuple (x2, ..., xn) in order to solve the equation. Said another way, h defines x1 as a function of the other variables, under the constraint that the relation R(x1, ..., xn) = 0 must be maintained. 1 1 An example might clarify the desired construction. Suppose n = 2 and R(x, y) = x 2 y 2 − 6. To 1 1 add economic motivation, view V (x, y) = x 2 y 2 as the utility function of some consumer, so that 4 the (x, y) pairs satisfying the relation R(x, y) = 0 are the set of consumption pairs carrying a utility 1 1 level of exactly 6. We want to find a function h(y) so that h(y) 2 y 2 − 6 = 0 for “many” values of 36 y. Simple enough: we rearrange to get h(y) = y . This is the function “hidden inside” the relation dh R(x, y) = 0, in the sense that R(h(y), y) = 0, at least for y > 0.
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