Introductory to Microeconomic Theory [08/29/12] Karen Tsai What is microeconomics? Study of: Choice behavior of individual agents Key assumption: agents have well-defined objectives and limited resources -> must solve constrained optimization problems Outcomes of interaction among optimizing agents Key assumption: markets reach equilibrium prices (or something else) adjust so that all individual choices are mutually consistent - equilibrium usually described by a system of equations What s the use of formal/mathematical economic model? Allows one to focus clearly on the few main factors affecting some outcome Can be solved to find choices/market outcomes as a function of fundamental environmental factors -> can predict how outcomes will change if environment changes Models, at best predict direction in which outcomes change when environment changes Models don t give predictions about the size of effects -> Need data and empirical analysis Constrained optimization example: Firm sells product in cylindrical canisters and it wants to minimize the surface area of the cylinder (minimize the material cost) subject to the cylinder holding a particular volume. Surface area of cylinder = 2πr^2 + 2πrh Volume of cylinder = πr^2h How to solve? 1. Impose h = V / (πr^2) 2. Lagrangian method: transform constrained problem to an unconstrained problem a. Rewrite constraint: V πr^2h = 0 b. Define a new function (Lagrangian function): L(r, h, λ) = 2πr^2 + 2πrh + λ(v πr^2h) 1
c. Turns out that minimizing this newly defined function is equivalent to solving the constrained problem How do we find (r, h, λ) that minimize L(r, h, λ)? Take derivative and set it to 0 a. Derivative of L with respect to r = 4πr + 2πh - 2πrhλ = 0 b. Derivative of L with respect to h = 2πr πr^2*λ = 0 c. Derivative of L with respect to λ = V πr^2*h = 0 => solve for r, h, λ that jointly satisfy these equations λ = 2 / r h = 2r r = (V / 2π)^(1/3) 2
Chapter 2 Consumer Theory [09/05/12] Karen Tsai Consumers maximize utility (satisfaction) subject to constraints imposed by prices and income (resources) How to model utility? Ch. 3,4 next class How to model constraints? Ch. 2 Consumption bundle: a listing of quantities consumed of all goods In this class, consumption bundles consist of 2 goods o X1 = quantity of good 1 o X2 = quantity of good 2 o Can represent any consumption bundle in the x-y plane Assume that each good has a constant per unit price to consumers o P1 = per unit price of x1 o P2 = per unit price of x2 Assume that consumer has income = M Consumer Budget Constraint: p1*x1 + p2*x2 <= M consumer s expenditure = cost of bundle <= available income affordable bundles cost less than available income the budget line is the set of exactly affordable bundles o all (x1, x2) such that p1*x1 + p2*x2 = M! x2 = M / p2 (p1/p2) * x1! (-p1/p2) is the opportunity cost of good 1 in term of good 2 (i.e. how many units of good 2 must be given up to buy 1 more unit of good 1)! Ex. M = 40; p1 = 2, p2 = 5 M/p2 slope = -> budget line: 4*x1 + 5*x2 = 40 -p1/p2 -> must give up 4/5 of a unit of x2 to buy 1 more unit of x1 M/p1 Factors that affect budget constraints Change in income, M -> parallel shift of budge line o M increase -> outward o M decrease -> inward 1
Change in p1 -> horizontal intercept & slope change -> rotation of budge line with vertical intercept fixed o P1 increase -> inward rotation o P1 decrease -> outward rotation Change in p2 -> rotation of budget line with horizontal intercept fixed o P2 increase -> inward o P2 decrease -> outward Change in p1 & p2 in same direction and by same proportion o Parallel shift of budget line like change in income o Prices increases is like income decrease Change in p1 & p2 and M in same direction and by same proportion o No change o Pure inflation Numeraire Goods Budge line: p1x1 + p2x2 = M Can equivalently express as (p1/p2) * x1 + x2 = M/p2 Written this way, we say that x2 is the numeraire good: prices and income are now expressed in units of good 2 rather than $ units Nonlinear budget constraints So far, we ve considered simple linear budget constraints In real world, sometimes budget constraints are nonlinear Ex. o Rationing (e.g., x2 <= x2 ; x2 is the max quantity of x2 allowed to be purchased) o Quantity discounts (e.g., p1 = p1 if x1 < x1 & p1 if x1 > x1 where p1 < p1 ) o In-kind transfers (e.g., food stamps, housing allowance, Medicaid) 2
Chapter 3 Preferences [09/10/12] Karen Tsai Consumer theory 2 basic ingredients: o what s affordable (budget constraint) o what s preferred (preferences and utility) Reminder: A consumption bundle is just a listing of quantities of each good With 2 goods, a consumption bundle is a point in the non-negative x-y plane Basic assumptions about preferences (defined over consumption bundles) Preferences are complete: consumer can rank any 2 bundles o If A is strictly preferred to B! Notation: A B o If A is weakly preferred to B! Notation: A B o If A is indifferent preferred to B! Notation: A ~ B Preferences are transitive: if A B & B C, then A C Preferences are monotonic: More is better than less Under these basic assumption plus technical assumption continuous preferences, then it turns out that preferences can be represented by a utility function A utility function is a function that assign a score to each consumption bundle such that higher scores <-> better bundles Ex. U(x1, x2) = x1^2 * x2 x1 x2 -> U 1 1 -> 1 2 1 -> 4 1 2 -> 2 Note: the utility function representing any set of preferences is NOT unique o Ex. U (x1,x2) = 2 * x1^2 * x2 ranks all bundles in the same way as U(x1,x2) = x1^2 * x2 o Any transformation that won t change the ranks is monotonic transformation 1
Given a utility function, it is straight forward to create a graphical representation of those preferences, i.e., the consumer s indifference curves Indifference curve (IC): complete set of (x1,x2) bundles that are equally preferred, i.e., give equal utility Mathematically, this is set of (x1,x2) combinations that satisfy U(x1,x2) = k, where k is a constant Ex. U(x1,x2) = x1^2 * x2. Then IC for utility level 100 is the set of (x1,x2) bundles satisfying x1^2 * x2 = 100 <-> x2 = 100/x1^2 x1 x2 1 100 2 25 5 4 10 1 What can we say about the shape of a consumer s indifference curves? Negatively sloped because above is better IC s further northeast give higher utility IC s don t cross Given IC s, we can ask what the slope represents. The slope of an IC represents the marginal rate of substitution (MRS), i.e., the rate at which the consumer is willing to trade x2 for x1, at the margin, given the initial bundle at the margin for a small trade In most cases, we would expect a consumer s marginal willingness to trade good 2 for good 1 to diminish as the amount of good 2 decreases and amount of good 1 increases => usual shape = diminishing MRS = convex preferences <-> IC s are bowed inward towards origin Marginal Utility: the change in utility resulting from a small increase in consumption of one of the goods, all else equal 2