Intermediate microeconomics Lecture 1: Introduction and Consumer Theory Varian, chapters 1-5
Who am I? Adam Jacobsson Director of studies undergraduate and masters Research interests Applied game theory Environmental inspections and enforcement Media economics Teaching Microeconomics & game theory 2017-01-18 Adam Jacobsson, Department of Economics 1
Agenda Introduction and overview Budget constraint Preferences Utility Choice 2017-01-18 Adam Jacobsson, Department of Economics 2
Why study intermediate microeconomics? Microeconomics helps us understand human economic behaviour Microfoundations important in modern economics! Mostly same same but different than NEKI Main difference: Now more math but also a deeper understanding of the theory. Why do we need more math? After all we re studying humans, not machines Math helps us to discipline our thinking, formulate testable hypotheses, construct models for analysis etc this will make it easier to apply the theory on real world problems. Ability to read (scientific) articles. 2017-01-18 Adam Jacobsson, Department of Economics 3
Aim of the course The aim of this course is to show how price theory and game theory can be used to explain how different markets work... Using the tools of analysis from price and game theory, this course explains how different economic structures arise and work, and how they can be shaped in a socially and economically efficient fashion... 2017-01-18 Adam Jacobsson, Department of Economics 4
Learning outcomes Upon completion of this course, the student is expected to be able to: Give an account of the central components of modern price and game theory, and describe how these methods of analysis can be used in order to explain consumption and production decisions. Perform practical calculations of problems that consumers and producers may encounter, and intuitively explain how and why the selected calculation method has been used and how this may explain the results. 2017-01-18 Adam Jacobsson, Department of Economics 5
Overview of the course Topic 1 Price theory and general equilibrium Consumer theory Production theory Competitive markets and general equilibrium Idealistic analysis Price P* Supply Demand Q* Quantity 2017-01-18 Adam Jacobsson, Department of Economics 6
Topic 2 - Uncertainty, Insurance, Investment and Asset Pricing Now we introduce some complications to the analysis: Intertemporal choice when should we consume? Uncertainty sometimes bad things happen How should we deal with this? Insurance a way of dealing with uncertainty. Investment and asset pricing another way of dealing with uncertainty the tradeoff between returns and risk... 2017-01-18 Adam Jacobsson, Department of Economics 7
Topic 3 - Welfare Theory Do we care about the distribution of welfare and what about market failures? Social welfare How should we split the pie? Externalities E.g. environmental consequences of economic decisions Public goods E.g. military defence, public broadcasting 2017-01-18 Adam Jacobsson, Department of Economics 8
Topic 4 - Strategic Interaction and Models of Competition Are people and organizations passive drones, just milling about? Game theory: a theory of strategic interaction Imperfectly competitive markets (& more market failures ) 2017-01-18 Adam Jacobsson, Department of Economics 9
Topic 5 - Asymmetric information, Moral Hazard, Adverse Selection and Contracts Oops, people are not only strategic, but also heterogenous and everybody does not know everything Asymmetric information Moral hazard Adverse selection 2017-01-18 Adam Jacobsson, Department of Economics 10
Examination Written exam at the end of the course. Before each exercise, students can hand in solutions to one assignment question (denoted by *), which will be awarded points by the teachers. If all assignments receive a pass grade, you will receive a credit for the exam and the associated re-take exam for this semester only. Note that you cannot save your course credit for later exams. The credit allows you to skip one specified question on the exam (corresponding to 10% of total exam points). If you do not have the credit, you then need to solve all questions on the exam for a full score. Solutions are to be handed in by groups of two or no more than three persons. 2017-01-18 Adam Jacobsson, Department of Economics 11
Three course components Lectures Introduces theory the big picture Group seminars Apply your theory, practice, practice, practice Mathematics lectures Revise your math skills! Math workshops And do not forget to read the book! 2017-01-18 Adam Jacobsson, Department of Economics 12
1. Budget constraint The optimization principle: An individual chooses the best bundle (combination) of goods that he/she can afford. 1. Best: according to his/her preferences 2. Can afford: the budget constraint! Simplifying assumption: The bundle consists of two goods (good 1, good 2). Consumption bundle X = (x 1, x 2 ), where x 1 and x 2 are the quantities of goods 1 and 2 that the individual chooses to consume Example: a bundle X could be composed of 5 apples, x 1 =5, and 10 oranges, x 2 =10, or X=(5, 10). Another bundle, Y, could contain 7 apples and 8 oranges, or; Y=(7, 8). X is the set of all consumption bundles: X X Prices P = (p 1, p 2 ) 2017-01-18 Adam Jacobsson, Department of Economics 13
Individual income: m The budget constraint: p F x 1 + p 2 x 2 m (1) Budget set: set of all X that satisfy the budget constraint Budget line: set of all X, for which or p F x 1 + p 2 x 2 = m (2) x 2 = K L M L F L M x 1 (3) Solve for x 2! Divide equ. (2) lhs & rhs by p 2 L F x L 1 + x 2 = K M L M Subtract L F x L 1 from lhs & rhs M We now have equation (3) 2017-01-18 Adam Jacobsson, Department of Economics 14
Budget set and budget line Good 2 (x 2 ) m p 2 Budget line Slope x 2 = K L M L F L M x 1 Budget set 2017-01-18 Adam Jacobsson, Department of Economics m Good 1 (x 1 ) p 1 15
The slope of the budget line: The opportunity cost of consuming one more unit of good 1 in terms of good 2. Consider the consumption bundle (x 1, x 2 ) on the budget line. Increasing consumption of x 1 by dx 1 means we have to decrease consumption of x 2 by dx 2. Mathematically: p 1 x 1 + dx 1 + p 2 x 2 + dx 2 =m (4) p 1 x 1 + p 1 dx 1 + p 2 x 2 + p 2 dx 2 =m p 1 dx 1 + p 2 dx 2 = m p 2 x 2 p 1 x 1 QR ST UV WXYZ[\ ]SV[ Y^2 = L 1 (5) Y^F L M 2017-01-18 Adam Jacobsson, Department of Economics 16
Preferences Preferences: ordering of consumption bundles. An individual s preferences over X is a ranking such that: X Y indicates that X is preferred to Y (strict preference), X~Y indicates that X and Y are equivalent (indifference), X Y indicates that X is preferred toy or is equivalent to Y (weak preference). 2017-01-18 Adam Jacobsson, Department of Economics 17
Assumptions about rational preferences Completeness. All bundles can be compared, that is, for all X, Y in X it is true that either X Y, X~Y or X Y. Reflexivity. Each bundle is at least as good as itself: X X. Transitivity. If X Y and Y Z then X Z. Indifference curve: Bundles that the individual is indifferent between, that is, for all bundles X, Y along an indifference curve it is true that X~Y. 2017-01-18 Adam Jacobsson, Department of Economics 18
The indifference curve and the weakly preferred set Good 2 (x 2 ) x 2 X Weakly preferred set of the bundle X Indifference curve x 1 Good 1 (x 1 ) 2017-01-18 Adam Jacobsson, Department of Economics 19
Theorem 1. The indifference curves of an individual with transitive preferences cannot intersect. Proof. Consider two indifference curves that intersect. Let bundle X lie on one indifference curve and bundle Y on the other and let bundle Z lie at the intersection of the curves. In this case: X~Z and Y~Z. Transitivity then implies X~Y. However, the two indifference curves represent different levels of utility. Thus it must be that X Y or X Y which contradicts X~Y. 2017-01-18 Adam Jacobsson, Department of Economics 20
Proof theorem 1. Good 2 Y~Z and X~Z By transitivity: Y~X Y Z X But indifference curves represent different levels of utility where it must be either X Y or X Y. This contradicts Y~X! QED Good 1 2017-01-18 Adam Jacobsson, Department of Economics 21
Well-behaved preferences ( simplifies the life of an economist ) The assumption that preferences are well-behaved (=well-behaved indifference curves) is common, although other assumptions can be made. 1. Monotonicity Let X=(x 1, x 2 ) and Y=(y 1, y 2 ). If x 1 > y 1 and x 2 > y 2 (or if x 1 = y 1 and x 2 > y 2 ), monotonicity implies that X Y. That is, more is better (no satiation)! Indifference curves then have negative slopes. 2017-01-18 Adam Jacobsson, Department of Economics 22
2. Convexity Let Z = λx + (1 λ)y, where λ 0, 1. If X Y, then Z X. For λ 0, 1, if Z X then we have strict convexity. That is, averages are preferred to extremes. Weakly preferred sets are convex (or strictly convex). 2017-01-18 Adam Jacobsson, Department of Economics 23
Convexity Good 2(x 2 ) X Z Z = λx + (1 λ)y λ=1 λ=0 λ=0,5 Convex? Monotone? Y Good 1(x 1 ) 2017-01-18 Adam Jacobsson, Department of Economics 24
3. Utility Utility is a way to describe an individual s preferences. Definition 1. A function is a utility function if it assigns a real number (utility) to every possible consumption bundle (i.e. u: X R) such that u(x) > u(y) X Y The symbol means if and only if (iff) 2017-01-18 Adam Jacobsson, Department of Economics 25
Utility X: a set of consump tion bundles X Y u(y) u(x)>u(y) u(x) 2017-01-18 Adam Jacobsson, Department of Economics 26
Note: According to our assumptions individual preferences imply an ordering of consumption bundles. Under these assumptions it is therefore only the rank assigned by different utilities that is important (ordinality). Differences in utility levels (cardinality) do not matter under our assumptions. Comparability of utility levels and differences in utility between individuals do not matter either. 2017-01-18 Adam Jacobsson, Department of Economics 27
Theorem 2. If u is a utility function, then every positive monotonic transformation f of u is a utility function that represents the same preferences. Proof. If u is a utility function that represents the individual s preferences, then u(x ) > u(y) X Y (6) If f is a positive monotonic transformation of u (f(u) is always increasing in u), then f u X > f u Y u X > u(y ) (7) Combining (6) and (7), we obtain f (u(x )) > f(u(y)) X Y (8) Hence, f is also a utility function according to Definition 1. 2017-01-18 Adam Jacobsson, Department of Economics 28
Marginal rate of substitution Good 2 (x 2 ) The slope = Y^M Y^F Indifference curve dx 2 dx 1 Good 1 (x 1 ) 2017-01-18 Adam Jacobsson, Department of Economics 29
Marginal rate of substitution (MRS) The utility function u(x 1, x 2 ) is differentiated in the point (x 1, x 2 ) and the result is set to 0 (why?): Rearrange: du = u(x 1, x 2 ) x 1 dx 1 + u(x 1, x 2 ) x 2 dx 2 = 0 MRS:= Y^M y = Y^F YXQR The MRS measures how much consumption of good 2 a consumer is prepared to give up for one more unit of good 1 at a specific consumption bundle. (MRS is the slope of the indifference curve at a specific point) Note that the MRS is not affected by a positive monotonic transformation! z{ z } z{ z ~ 2017-01-18 Adam Jacobsson, Department of Economics 30
4. Choice The consumer chooses the most preferred bundle from his/her budget set (optimization). Intuition: move along the budget line until the weakly preferred set does not overlap the budget set. Or, find the indifference curve with the highest utility that just touches the budget line With some exceptions, the indifference curve is tangent to the budget line at the optimal point: MRS = - L F L M Exceptions: Corner solutions, kinked indifference curves etc 2017-01-18 Adam Jacobsson, Department of Economics 31
The utility maximization problem Good 2 (x 2 ) m p 2 Note! This is an example of well behaved preferences and an interior optimum! u=3 u=2 Good 1 (x 1 ) p 1 2017-01-18 Adam Jacobsson, Department of Economics 32 m u=4 u=5
Examples where the indifference curve is not tangent to the budget constraint at the optimum Good 2 (x 2 ) m p 2 Perfect substitutes Good 2 (x 2 ) m p 2 Perfect complements u=5 u=3 u=4 u=2 u=3 u=4 u=5 m p 1 Good 1 (x 1 ) u=2 m p 1 Good 1 (x 1 ) 2017-01-18 Adam Jacobsson, Department of Economics 33
Utility maximization putting the pieces together (assume well-behaved prefs. and interior solutions) Maximize utility given the budget constraint. max ^F,^M Set up the Lagrangian: u x 1, x 2 s. t. p 1x 1 + p 2 x 2 = m (9) L x 1, x 2, λ = u x 1, x 2 λ p 1 x 1 + p 2 x 2 m Let the optimal solution be denoted by x F, x M, λ = (x F P, m, x M P, m, λ P, m ) (note that P=(p 1, p 2 )) where (x F P, m, x M P, m ) are the Marshallian demand functions. 2017-01-18 Adam Jacobsson, Department of Economics 34
The first-order conditions (FOC) are given by ˆ ^} ˆ ^~ ˆ = X ^},^~ ^} = X ^},^~ ^~ λ p 1 = 0 λ p 2 = 0 (i) (ii) = p 1x F + p 2 x M m = 0 (iii) Solve (i) and (ii) for λ : u x F, x M λ x = F = p 1 u x F, x M x M p 2 2017-01-18 Adam Jacobsson, Department of Economics 35
At the optimal solution the following is true for the MRS: MRS = u x F, x M x F u x F, x M x M = p 1 p 2 2017-01-18 Adam Jacobsson, Department of Economics 36
The cost minimization problem Good 2 Among the bundles on uš, choose the cheapest. m = 5 m = 7 uš m = 10 Good 1 2017-01-18 Adam Jacobsson, Department of Economics 37
Cost minimization Minimize costs to attain a given utility level uš. min ^F,^M p 1 x 1 + p 2 x 2 s. t. u x 1, x 2 = u (10) Set up the Lagrangian: M x 1, x 2, μ = p 1 x 1 + p 2 x 2 μ u x 1, x 2 Let the optimal solution be denoted by uš x F, x M, μ = (x F P, uš, x M P, uš, μ P, uš ) where (x F P, uš, x M P, uš ) are the Hicksian demand functions. 2017-01-18 Adam Jacobsson, Department of Economics 38
The first-order conditions (FOC) are given by ^} ^~ = p 1 μ X ^},^~ ^} = p 2 μ X ^},^~ ^~ = 0 (i) = 0 (ii) = u x 1, x 2 uš = 0 (iii) Solve (i) and (ii) for μ : μ = u x F, x M x F p 1 = u x F, x M x M p 2 2017-01-18 Adam Jacobsson, Department of Economics 39
At the optimal solution the following is true for the MRS: MRS = u x F, x M x F u x F, x M x M = p 1 p 2 Same as under the utility maximization problem! The expenditure function is defined as: E p 1, p 2, uš : = p 1 x F + p 2 x M 2017-01-18 Adam Jacobsson, Department of Economics 40
A perspective on preferences Preferences indicate what individuals like and utility functions are used to represent different kinds of preferences. Common assumption in (traditional) economic analysis: Homo oeconomicus, i.e. the individual wants to maximise his/her own material well being. Is this always so? 2017-01-18 Adam Jacobsson, Department of Economics 41
Not necessarily! Individuals can also be motivated by altruism, a desire for justice, fairness etc (there is evidence for this!). Does this mean we should throw away (traditional) economic models? No! But we should be aware of the assumptions we make and how these assumptions match the problem we are analysing. Lets look at a recent article: 2017-01-18 Adam Jacobsson, Department of Economics 42
Evolution and Kantian morality by Ingela Alger and Jörgen W. Weibull, Games and Economic Behaviour 98, (2016), pp 56-67. (Voluntary reading. This is an advanced scientific text, focus on sections 1, 6 & 7.) For an easy-to-read article about the authors reaserach by Thomas Lerner, see Dagens Nyheter, 161228. What kind of preferences should one expect evolution to favour?, p56. Selfish preferences or, perhaps, Kantian preferences? Kantian imperative: Act only according to that maxim whereby you can, at the same time, will that it should become a universal law. 2017-01-18 Adam Jacobsson, Department of Economics 43
Evolutionary game theoretical model. Individuals can play either selfish or more Kantian strategies. Individuals randomly and repeatedly interact with other individuals (they have a tendency to be matched with similar individuals). The authors show that individuals who play more or less Kantian strategies tend to better than the more selfish individuals. This means that the proportion of Kantian individuals in the population remains large over time. That is, evolution favours Kantian behaviour! Homo moralis! 2017-01-18 Adam Jacobsson, Department of Economics 44