Lecture 1: The market and consumer theory. Intermediate microeconomics Jonas Vlachos Stockholms universitet

Transcription

1 Lecture 1: The market and consumer theory Intermediate microeconomics Jonas Vlachos Stockholms universitet 1

2 The market Demand Supply Equilibrium Comparative statics Elasticities 2

3 Demand Demand function. Mathematical relation between quanitity demanded (Q d ), own price (p) and other factors For example Where p is own price, p b and p c are prices of substitutes, and Y is income 3

4 Demand curve: own price change Holding everything but own price equal Movement along the demand curve dq/dp=-20 (million kg/$) Since p is on y-axis, slope=1/(dq/dp)=dp/dq=-0.05 ($/million kg) 4

5 Demand curve: changes in other prices or income Increases in substitute goods or income shifts the demand curve At any given price of pork, demand for pork increases 5

6 Supply Supply function: mathematical relation between quantity supplied (Q S ), own price (p) and other factors (for example input prices) For example Where p is own price and p h is input price 6

7 Supply curve: own price change Holding everything but own price equal Movement along the supply curve dq/dp=40 (million kg/$) Since p is on y-axis, slope=1/(dq/dp)=dp/dq=0.025 ($/million kg) 7

8 Supply curve: increase in input price Shifts the supply curve At any given pork price, less is supplied 8

9 Market equilibrium: demand=supply Q d = p = Q S = 88+40p => Q = 220 Since Q = 220, p = 3.3 9

10 Comparative statics: large shocks What happens when some shock affects the demand or supply of a good? For example: large increase in input price, from 1.5 to 1.75 Previous: Q S = 88+40p. Now: Q S = 73+40p Q d Q s p p p $3.55 Q d Q s

11 Comparative statics: small shocks Demand is a function of price, holding everything else constant: Q = D(p) Supply is a function of price and some exogenous factor a. (exogenous here means outside the control of firms ) Q = S(p,a) Prices will indirectly depend on a, so in equilibrium: D(p(a)) = S(p(a),a) 11

12 Comparative statics: small shocks To analyze the impact of a small change in a (for example input price), use the chain rule when differentiating the equilibrium condition with respect to a <0 Rearrange: >0 (equilibrium prices increase when a increases) <0 >0 12

13 Elasticities How responsive is a variable to a change in another variable? Price elasticity of demand: Price elasticity of supply: 13

14 Elasticity: example with linear demand Linear demand: Q d = a-bp => dq/dp=-b ε = dq dp p Q = b p Q 14

15 Constant elasticity demand curves (Q = Ap ε ) 15

16 Constant elasticity supply curves (Q = Bp η ) 16

17 Consumer theory The demand function builds on assumptions concerning consumer preferences Consumers also face constraints Also assume that consumers try to do as well as they can, that is maximize their well-being 17

18 Preferences: notation Consumers have preferences over bundles of goods and services Strict preference. E.g. a b (a is better than b) Weak preference. Bundle a is at least as good as b ~ Indifference. Bundle a exactly as good as b 18

19 Preferences: basic assumptions Completeness: all bundles can be ranked a consumer can rank them so that either a b, b a, or a ~ b Transitivity: Consumers rankings are logically consistent in the sense that if a b and b c, then a c. 19

20 Well behaved preferences Non-satiation (monotonicity): More is better Convexity: Averages are prefered to extremes 20

21 From preferences to indifference curves Consider combinations of two possible goods. The bundles that make a consumer equally happy make up indifference curves 21

22 Indifference curves The preference assumptions imply certain properties for indifference curves Every bundle is on an indifference curve (completeness) Indifference curves further from the origin are better (nonsatiation) Indifference curves cannot cross (transitivity) Indifference curves cannot slope upwards (non-satiation) Indifference curves cannot be thick (non-satiation) 22

23 Indifferences curves cannot cross Y ~ Z Z ~ X But Y ~ / X Transitivity violated 23

24 From preferences to utility and utility functions Utility is an analytical tool to describe preferences Ordinal: utility only describes how bundles are ranked relative to each other Not a cardinal measure that tells us by how much bundles are preferred A utility function assignes a larger value to the more prefered bundle, but units to not matter If x y, then u( ) is a utility function if u(x) > u(y) A utility function can be transformed into another utility function such that the preference ordering is maintained Positive monotonic transformation 24

25 Compare to temperatures Celsius Fahrenheit Kelvin Describe the same temperature relation, but units differ 25

26 Marginal rate of substitution (MRS) MRS is the amount of one good that an individual is willing to give up for another good for any given utility level: MRS = dq 2 /dq 1 Marginal utility is the increase in utility that a consumer gets from consuming the last unit of a good, holding other consumption constant: δu δq 1 = U 1 26

27 Different preferences => different utility functions => different MRS:s We often assume that goods are imperfect substitues, but that is not necessary 27

28 Indifference curves Perfect subsitutues and perfect complements are extremes, many different standard, convex, utility functions Cobb-Douglas never hit the axis Quasi-linear hit one of the axis 28

29 Some commonly used utility functions 29

30 The budget constraint: basics Consumers maximize utility, subject to constraints Given prices p 1, p 2, and income Y, the budget line is If p 1 = $1 p 2 = $2 and Y = $50, the budget line is: 30

31 The budget constraint: MRT The marginal rate of transformation, MRT, tells how the market allows a consumer to trade ( transform ) one good for another 31

32 Constrained consumer choice Maximize utility subject to the budget constraint Given standard indifference curves, there is an interior optimum. The highest feasible indifference curve. 32

33 Maximizing utility using calculus: I One way to solve consumer maximization is to directly assume an interior solution and thus MRS=MRT (here Cobb-Douglas utility) max X,Z U(X, Z) = Xα Z 1 α subject to Y = P X X + P Z Z MU X = δu(x, Z) δx = αx α 1 Z 1 α MU Z = δu(x, Z) δz MRS XZ = MU X MU Z = = (1 α)x α Z α αxα 1 Z 1 α (1 α)x α Z α = α 1 α Z X = P X P Z = MRT 33

34 Maximizing utility using calculus: I (cont) From this we see that Z = (1 α)p X α P Z X, where (1 α)p X α P Z is a constant. We now have the optimal allocation between Z and X, but how much we consume also depends on income Y. Insert expression for Z in the budget constraint Y = P X X + P Z (1 α)p X X = P α P X X α Z α = P X X α α + 1 α α = P X α X X = αy P X (optimal quantity of X). Continuing for Z, we get Z = (1 α)p X α P Z X = (1 α)p X α P Z αy P X = 1 α Y P Z (optimum quatity of Z) 34

35 Maximizing utility using calculus: II The Lagrangian method: optimization with constraints General: maximize f(x) + λg(x) Maximize U q 1, q 2 subject to Y = p 1 q 1 + p 2 q 2 Chose optimal values of q 1, q 2, and λ Think of λ as the cost of violating the constraint 35

36 Maximizing utility using calculus: II Find first order conditions: (1) (2) (3) Equating (1) and (2) conditions yields λ is the marginal utility of income ( shadow value of income) The value of relaxing the constraint 36

37 Special case 1: Perfect complements 37

38 Special case 2: Perfect substitutes If the marginal rate of substitution is not equal to the marginal rate of transformation, you will only chose the relatively cheap good. Corner solution 38

39 Special case 3: Quasilinear preferences Under quasilinear preferences, the price of one good may be so high that you do not consume any of it. Corner solution 39

40 Expenditure minimization An alternative approach to consumer optimization is to consider the lowest cost at which you can achieve a certain level of utility. I.e. to mimimize expenditures The solution gives you the expenditure function, the minimum expendtures needed to achive a certain utility level 40

41 Are the basic assumptions correct? Behavioural economics analyses the basic rationality assumptions. Numerous deviations recorded Endowment effects: what people actually own affects their preferences Salience: people do not (fully) incorporate costs such as taxes when making economic decisions Transitivity: basically seems to hold But, we are working with models The economic importance of such deviations is not fully understood 41