Introduction to Economics I: Consumer Theory

Transcription

1 Introduction to Economics I: Consumer Theory Leslie Reinhorn Durham University Business School October 2014

2 What is Economics? Typical De nitions: "Economics is the social science that deals with the production, distribution, and consumption of goods and services." "Economics is the study of the allocation of scarce resources." Economics aims to explain how economic agents make decisions and interact with each other how economies work and function as a whole Two methodological foundations Rationality (econ agents are smart and consistent) Utilitarianism (utilities are measurable & comparable; maximisation)

3 Divide of Economics Microeconomics vs. Macroeconomics micro: behaviour of basic economic unit (consumers and rms, buyers and sellers); macro: behaviour of economic aggregate (unemployment, in ation, scal policy, etc.) Positive vs. Normative Economics positive: describing "what is", neutral to value judgement normative: "what ought to be", subject to value judgement Rational and Behavioral Economics economic agents assumed to be rational agents may be irrational and inconsistent in their decisions Classi ed according to the subjects of study nancial economics; public economics; labour economics; international economics; etc.

4 Basic Framework of Economic Analysis How economic agents respond to environments consumer facing prices a player s strategy depends on other players strategies Example: increasing price! lower demand by consumers Environments adjust until equilibrium is reached equilibrium prices equilibrium strategies Example: market equilibrium prices (demand=supply)

5 Consumer Theory: choice under certainty General Optimization Problems Budget Constraints Utility and Indi erence Curve Utility Maximization Problem Demand Function, Comparative Statics Consumer Surplus

6 The Optimization Problems Typically, an optimization problem has three key components: 1 The Objects of Choice consumers: consumption bundles rms: outputs, inputs 2 The Objective Function 3 Constraints consumers: to maximize utility rms: to maximize pro t consumers: prices, incomes rms: technology (production function), demands

7 Consumption Bundles and Budget Constraints Consumption Bundles (x 1, x 2 ) objects of choice: (x 1, x 2 ) 2 R 2 +, the amount of good 1 and 2 that the consumer wants to buy Budget Constraints The environment: 3 parameters, (p 1, p 2, M), which are out of control constraints: Budget Constraints p 1 x 1 + p 2 x 2 M Budget Line: p 1 x 1 + p 2 x 2 = M Budget Set: B = (x 1, x 2 ) 2 R 2 + : p 1x 1 + p 2 x 2 M

8 Budget Set Fig 1.1 Budget Set Fig 1.2 increasing p 1

9 Utility Functions: MU and MRS Utility function, u (x 1, x 2 ), represents a consumer s preference,. Marginal Utility (MU) (x 1, x 2 ) (y 1, y 2 ) () u (x 1, x 2 ) > u (y 1, y 2 ) measure the rate at which utility increases with the increase of the amount of a particular good math de nition: partial derivative MU 1 (x 1, x 2 ) = u (x 1, x 2 ) x 1 Marginal Rate of Substitution (MRS) MRS: the slope of an indi erence curve math: take total derivative of the utility function du (x 1, x 2 ) = u(x 1,x 2 ) x 1 dx 1 + u(x 1,x 2 ) x 2 dx 2. Let du = 0, we get MRS = dx 2 dx 1 = u(x 1,x 2 ) x 1 = MU 1 u(x 1,x 2 ) MU 2 x 2

10 Consumer s Problem: Utility Maximization The utility maximization problem is thus (P) max (x1,x 2 ) u (x 1, x 2 ) s.t. p 1 x 1 + p 2 dx 2 M The tangent condition for interior solution the indi erence curve is tangent to the budget line, p 1 p 2 = MRS

11 An Example: Cobb-Douglas Utility Example The problem, α > 0, β > 0, α + β 1 (P) max (x1,x 2 ) u (x 1, x 2 ) = x1 αx β 2 s.t. p 1 x 1 + p 2 dx 2 M Marginal Rate of Substitution Tangent condition is therefore MRS = MU 1 = αx α 1 1 x β 2 MU 2 βx1 αx β 1 2 = αx 2 βx 1 p 1 p 2 = αx 2 βx 1

12 Demand Function Solutions to the utility maximization problem (P) max (x1,x 2 ) u (x 1, x 2 ) s.t. p 1 x 1 + p 2 dx 2 M Example are the demand functions: x 1 (p 1, p 2, M) and x 2 (p 1, p 2, M) the parameters, (p 1, p 2, M), de ne the budget constraint which is out of control comparative statics: how x1 and x 2 change with the parameters, (p 1, p 2, M)? In the case of C-D utility, u (x 1, x 2 ) = x1 αx β 2, we get the demand functions are x1 (p 1, p 2, M) = α M x2 (p 1, p 2, M) = β M α + β p 1 α + β p 2

13 Consumer Surplus Inverse demand function: p 1 (x 1 ; p 2, M), showing the unit price that you re willing to pay at the amount of x 1 Consumer Surplus at the price p is thus CS = Z x 0 p 1 (x 1 ; p 2, M) dx 1 p x

14 Consumer Theory: Choice under Uncertainty Choices between lotteries; Expected utility theory; Risk aversion An Example of Insurance

15 Introduction In the lectures so far, we have considered the cases where agents are fully sure about the outcomes of their choices. However, in countless cases, the outcomes of our choices are not certain i.e. 1) when you buy a share of Facebook, it s stock price is not certain; i.e. 2) when buying a lottery, the outcomes are not certain; etc.; in these cases, though you re fully sure what you re buying, but the outcomes which you may care about are uncertain

16 Choice under Uncertainty: Example I Example Flipping a Coin: If it comes down as heads, you get 10; If it comes down as tails, you get 0. And it is a fair coin, with each outcome with a probability of 50%. Suppose you re charged x for playing this game. So for what price x would you like to play this game?

17 Choice under Uncertainty: Example I Example (cont d) Actually, your choice is between the following two options Not play and get 0; Play: for sure you pay x, and with 50% chance of getting 10 Is expected value of outcomes a good criterion for decision? Option 1: 0 Option 2: x + 5 So if x 5, choose to play the game. Is this a sensible decision rule?

18 Choice under Uncertainty: Example II Example Suppose you re a pauper, living homelessly on streets. You fortunately get a lottery: with 50% chance of getting nothing, while 50% chance of getting 1 million! A man comes up to you, and o ers to buy your lottery for 499, 999 for sure. Would you take his o er? According to expected value, you will decline. But this seems ridiculous, isn t it? You can get 499, 999 for sure!

19 Choice under Uncertainty: Example II Solution? Perhaps, Valuation depends on the happiness with each possible outcome; Utility increase of getting extra incomes decreases as we become richer u ( 0) = 0 u ( 499, 999) = 100 u ( 1, 000, 000) = 110 We d use expected utility, rather than expected value, here 1 2 u ( 0) + 1 u ( 1, 000, 000) < u ( 499, 999) 2

20 Expected Utility Theory Formally, let s de ne an object with uncertain outcomes as a lottery De nition A simple lottery L = (p 1, p 2,, p n ), with p i 0 and n i=1 p i = 1, is a collection of probabilities for the possible outcomes are X = x 1, x 2,, x n, where p i is the probability for x i Preference over lotteries: Expected Utility Eu (L) = p 1 u (x 1 ) + p 2 u (x 2 ) + p n u (x n ) where u () is called vnm utility function.

21 Risk Aversion De nitions An agent is risk-averse i, he never prefers a random variable X to its mean EX, that is u (EX ) Eu (X ) An agent is risk-averse i his vnm utility, u (), is concave. [Jensen s Inequality: For any random variable X, Ef (X ) f (EX ) i f is concave.] Example Risk-lover if u (EX ) Eu (X ) Risk-neutral u (EX ) = Eu (X ): ) u linear means risk-neutrality Consider a lottery costing $2, with Pr =.5 you get $10, and with Pr =.5 you lose $6. Will you play it?

22 Insurance: an Example Initial State There are two states of the Nature: Good with probability p g, and Bad with probability p b = 1 p g The consumer has initial income W, and with probability p b, he will su er a loss of L, L < W The contingent consumption plan is thus c g = W c b = W L

23 Insurance γ = insurance premium = price of every 1 of insurance (to be paid by the insurance company in the Bad state only!) K (K W ) = amount of insurance purchased The contingent consumption plan under insurance is thus The budget constraint is thus c g c b = c g = W γk c b = W L γk + K W (W γk ) W L (W L γk + K ) = γ 1 γ

24 Consumer s Expected Utility Eu (C ) = p g ln c g + p b ln c b, so in optimum where MRS cg,c b = MU b MU g = p b/c b p g /c g = γ 1 γ MRS cg,c b = p bc g p g c b = p b p g W γk W L γk + K Fair premium: the insurer wins zero pro t, and therefore By substitution, we get Full insurance! γ = p b W γk W L γk + K = 1 The consumption in both states are the same! Theorem A risk-averse consumer facing a fair premium will always fully insure.