Prof. Wolfram Elsner Faculty of Business Studies and Economics iino Institute of Institutional and Innovation Economics Introductory Microeconomics The Ideal Neoclassical Market and General Equilibrium
Readings for this lecture Mandatory reading this time: The Ideal Neoclassical Market and General Equilibrium, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 97-128. The lecture and the slides are complements, not substitutes An additional reading list can be found at the companion website 2
Introduction neoclassical method What is a neoclassical method? To develop a pure theory Axiomatic approach Taking as little as possible as given To construct an optimal equilibrium model In a price-quantity world Neoclassical economics typically ends up in price-quantity spaces employing the mathematics of maximization under restrictions (Lagrange algorithm). Economism in that sense is the idea of a pure economy, isolated from the rest of society and from the natural environment. 3
Introduction - model What is a model? Imaginary Analogy Metaphor Relating variables to one another Diverging in the number of variables, agents and relations Perfect competition and perfect information are assumed at least as a benchmark to which more specific models with imperfect competition or imperfect information are compared 4
General Equilibrium Theory (GET) as the study of all interdependent ideal partial markets Comprehensive formulation of neoclassical economics Simultaneous equilibrium in all partial markets Neoclassical paradigm cannot be tested and rejected straightforwardly data are always selected Stylized facts 5
GET Walrasian Economics Fictitious auctioneer (computes market-clearing prices) No direct interdependence (only indirect via the market ) Walras was the first to emphasize the interdependence of individual partial markets in his Elements of pure economics (1874). Neoclassical or marginalist revolution Marginal utility (and marginal productivity in production) Scarcity (focus on the allocation of scarce resources) 6
Overview Optimization problems Consumer theory Production theory Utility/ profit maximization problems Cost minimization problems Partial and general equilibrium Limitiations Theory of the second best, Markets of lemons 7
Perfect Information and Perfect Competition All agents (firms and households) have all relevant information Past, present and future Reflected in prices At no costs Preferences and production technologies are exogenous given Technology can be accessed and employed by everyone without problems 8
Competitive Equilibrium No endogenous mechanism that causes a change in prices Exogenous change in preferences or technology will result in a shift of the equilibrium Such an equilibrium situation, as the first welfare theorem shows, is Pareto-optimal 9
Preferences Descartes division of the world into an inner subjective domain and an outer objective domain The mind as having values and being subjective The individual with reference to reason and consciousness Cogito ergo sum I think, therefore I am 10
Cartesian dualism The Cartesian dualism is the root of the neoclassical conception of the individual Focus shifted from individual toward her individualistic choice No explicit consideration of interaction among individuals No explicit consideration of the institutional environment Individual decision making Completely detached from her social structure 11
A rational preference relation We assume that the individual has a rational preference relation over the set of all possible choices. Let x, y, z be mutually exclusive alternatives in the set of possible choices X A rational preference relation is characterized by Completeness: for x, y X, x y (read: or) y x or both Transitivity: for x, y, z є X, x y (read: and) y z x z (Reflexivity: x ~ x (read: for all) x є X) 12
Lexicographic preferences An agent with lexicographic preferences will choose the bundle that offers the largest amount of the first good x 1 no matter how much of the other good is in the bundle. (x 1, x 2 ) (x 1, x 2) if x 1 x 1 Rational, but cannot be represented by a continuous function 13
Notation I Symbol Meaning p = (p 1, p 2,, p N ) Price vector R N, R N + Real numbers, positive real numbers (size of commodity space and consumption set) x i = (x 1 i, x 2 i,, x N i ) є R + N Commodity bundle or demand individual I x = (x 1, x 2,, x I ) є R + IN ω i = (ω 1 i, ω 2 i,, ω N i ) Allocation Endowment individual I ω = I ω i, ω = p ω Aggregate endowment, wealth i=1 14
Notation II Symbol Meaning x, x n i, p Equilibrium allocation, equilibrium demand agent i good n, equlibrium price vector u, v( ) Direct and indirect utility functions x, h( ) Marshallian and Hicksian demand e( ) Expenditure function λ, Λ( ) Lagrange multiplier, Lagrangian function K, L, r, w Production factors (capital, labor), Factors productivities (unit interest and unit wage) y(k, L) Production function 15
Lagrangian method Unrestricted utility/profit-functions of this type have no maxima but individuals usually are restricted at least by a limited budget Hence, neoclassical microeconomics needs a method for optimization under restrictions The Lagrangian method is a method to find local maxima and minima (originally from classical mechanics, developed by Joseph- Louis Lagrange in correspondence with Leonhard Euler, 1755) Restrictions are exogenously imposed Note that the method requires the solution to be on every one of the restrictions (i.e. all restrictions are binding) 16
Utility function Continuous Strictly increasing in every argument u/ x n > 0 At least differentiable twice on the interior of R N + Strictly concave 2 u/ 2 x n < 0 Satisfies lim u/ x n = + xn 0 17
Marginal utility Marginal utility is positive, but decreasing. Quantities and prices will be strictly positive in equilibrium. This leads to a budget set B i = x i R + N : px i pω i that typically takes the form of a linear decreasing budget restriction 18
The Lagrange Multiplier Approach The Lagrangian function is constructed as follows: max x 1,x 2,λ Λ = u x 1, x 2 λ(p 1 x 1 + p 2 x 2 ω) Assuming the utility function is concave, first order conditions are necessary and sufficient conditions a system of three equations: (1) Λ = u λp x 1 x 1 = 0; (2) 1 Λ = u λp x 2 x 2 = 0; 2 19
The Lagrange Multiplier Approach (3) Λ λ = p 1x 1 p 2 x 2 + ω = 0 Dividing the first and the second equation At the optimum, the ratio of prices equals the ratio of marginal costs: u/ x 1 u/ x 2 = λp 1 λp 2 = p 1 p 2 20
Walrasian or Marshallian demand function The solution to the UMP (utility maximization problem) is a bundle of commodities as a function of prices and wealth: x i = x i (p, ω) Using the Lagrange method, we maximize: Λ = u x λ(px ω) First order conditions (FOC): u x n = λp n n N; px = ω 21
Equilibrium conditions for consumers Marginal utilities are equalized formally: u/ x n1 p n1 = u/ x n2 p n2 u/ x n1 u/ x n2 = p n1 p n2 = MRS The FOCs of the UMP give us N + 1 variables (x 1, x 2,, x N and λ), a system that is solvable in principle. The relations of marginal utilities (and good prices) is called the marginal rate of substitution (MRS) 22
The utility function is given by: u x 1, x 2 = α ln x 1 + (1 α) ln x 2 subject to p 1 x 1 + p 2 x 2 ω. We set up the Lagrange function max α ln x 1 + 1 α ln x 2 + λ(ω p 1 x 1 p 2 x 2 ) x 1,x 2,λ An Example and derive the FOCs: (1) α x 1 = λp 1 (2) 1 α x 2 = λp 2 (3) p 1 x 1 + p 2 x 2 = ω 23
We combine the first two to get: p 1 x 1 = α 1 α p 2x 2 And substitute into the third, p 1 x 1 = ω p 2 x 2 We obtain: α 1 α p 2x 2 = ω p 2 x 2 An Example This equations can be solved for x 2 as a function of prices and wealth: x 2 p 1, p 2, ω = 1 α ω p 2 24
Inserting and solving for demand of x 1, we get: x 1 p 1, p 2, ω = α ω p 1 The last two equations are the individual s Walrasian or Marshallian demand functions, telling us, how much of each good the individual will demand as a function of prices and wealth. An Example 25
The consumer s utility at the optimum is given by the indirect utility function: v p, ω = u x 1, x 2 = α ln α ω p 1 + (1 α) ln (1 α) ω p 2 An indirect utility function gives utility as a function of prices and income. An Example The direct utility function gives utility as a function of goods consumed 26
x1 Indifference curves Tangential point optimum consumption bundle x1, x2 budget line Marshallian demand x2 27
Marshallian Demand, Hicksian Demand The dual problem to utility maximization is a minimization of expenditure for reaching a specific utility level u. The general form: min e(p i, x i ) subject to u x u Given an expenditure function e(p, u) that shows the minimum expenditure required for reaching a certain utility level, the relation between Hicksian and Marshallian demand is given by: h p, u = x p, e p, u 28
Slutsky Equation Given the indirect utility function just derived before, we can also state: x p, ω = h(p, v(p, ω)) The two kinds of demand can be related by what is termed the Slutsky equation. Split changes in uncompensated demand Substitution of the (after the price changes) relatively cheaper goods for relatively more expensive ones, the substitution effect Change in the consumer s purchasing power following the price changes, the income effect 29
Slutsky Equation x i p, ω p j = h p, u p j Substitution Effect x i p, ω ω x j (p, ω) Income Effect Using Shephard s Lemma e p,u p j = h j (p, u) the derivation follows with the definition of the indirect utility above (x j p, ω = h j (p, v(p, ω))) from the total differential of the Hicksian demand function (with du = 0): dh i p, u = x i(p,e p,u ) dp p j + x i(p,e p,u ) e p,u dp j e(p,u) p j + x i( ) e j e( ) u du 30
x1 New optimum Hicksian demand (compensated) x2 IE and SE in demand changes IE SE 31
The Production Function As an example for firm s production possibilities is often used the Cobb-Douglas technology, which is defined by the following production function for α (0,1) and two inputs (capital K, labor L): y = F K, L = AK α L 1 α 32
Cobb-Douglas production function 33
Marginal rate of technical substitution (MRTS) If we assume, that the production function is differentiable, the MRTS tells us at which rate one input can be exchanged for another one without altering the quantity of output. Setting the equal to zero and sticking to the Cobb-Douglas production function for the derivation, with A = 1: dy = α L K 1 α dk + 1 α K L α dl = 0 Resolving for dl dk gives dl = α dk L K 1 α 1 α K L α = α 1 α L K 1 α+α 34
MRTS The absolute value of the scope of the slope of the isoquant: MRTS KL = dl dk = α 1 α L K or sometimes also given in terms of the absolute value, MRTS KL = dl = α L dk 1 α K Formally, isoquant S(y) and input requirement set I(y) are defined as: S y = { L, K : F K, L = y} I y = { L, K : F K, L y} 35
Isoquants and input sets α=0,5 36
Law of diminishing returns The law of diminishing returns states, that if all but one input is fixed, the increase in output from an increase in the variable input decline. For the production function, this implies: Strictly increasing in both dimensions F K > 0 and F L > 0 Concave in both dimensions 2 F 2 K < 0 and 2 F 2 L < 0 (Note that as the function is also twice statically differentiable and its slope becomes infinite near zero (K = 0 or L = 0) it mathematically resembles the neoclassical utility function) 37
Cost Minimization and Cost Functions Starting with the firm s cost-minimization problem: min K,L rk + wl subject to F K, L y we construct the Lagrangian Λ K, L, λ = rk + wl + λ(f K, L y) To figure out which point on the isoquant is cost minimizing, we take the FOCs and set them to zero: Λ F Λ F (1) = r + λ = 0 (2) = w + λ = 0 K K L L (3) F K, L y = 0 38
Cost Minimization and Cost Functions The first two FOCs can conveniently be rearranged to r = λ F and w=λ F K L Combining the two equations we get the firm s optimality condition for the relative amounts of inputs used: r w = F K F L = MRTS KL At the optimum, the relative price of inputs has to equal the MRTS. 39
Input requirement set, isoquant, relative prices 40
Cost Minimization and Cost Functions Deriving the firm s cost function with using the Cobb-Douglas function (α = 0.5): C r, w, y = min K,L rk + wl subject to y K0.5 L 0.5 = KL The constraint must be binding, y K 0.5 L 0.5. Solving for K yields K = y2 L which can be substituted into the cost function: C r, w, y = min K,L r y2 L + wl 41
Cost Minimization and Cost Functions Taking the FOC for labor we have L r, w, y = r w 0.5 y. The conditional demand for capital is derived in the same way; we obtain K r, w, y = w r 0.5 y. and can rewrite the cost function as C r, w, y =wl r, w, y + K r, w, y = 2r 0.5 w 0.5 y. 42
Profit Maximization Using the cost function, we now turn to the firm s profit maximization problem. A firm s profits are defined as revenue minus cost: Π y, L, K, w, r, p 0 = p 0 y (wl + rk) The firm has now the task to choose the level of output. Substituting the cost function for (wl + rk) and taking input prices (w and r) as given: max y 0 Π = p 0y C(r, w, y) 43
Profit Maximization The Cobb-Douglas production function exhibits constant returns to scale since: F ck, cl = A ck 0.5 cl 0.5 = cf(k, L) This means that the cost function is linear and average costs as well as marginal costs are constant. C r, w, y AC = y C r, w, y MC = y 44
Existence of Equilibrium: Constant Returns to Scale 45
Existence of Equilibrium: Decreasing Returns to Scale 46
Existence of Equilibrium: Increasing Returns to Scale 47
Partial Equilibrium: Stylized Demand and Supply Functions 48
Partial Equilibrium 49
General Equilibrium Instead of just looking at one market in isolation, as we did before, we look at the complete set of markets now. In equilibrium, marginal rates of substitution (MRS) and technical substitution (MRTS) are equal: MRS 12 = u x1 = p y 1 x1 = = u p 2 y x2 x2 MRTS 12
Walras Law The combination of equilibrium prices and allocations ( p, x) as Arrow-Debreu equilibrium. We assume that all markets are in equilibrium, i.e. excess demand z (demand minus supply) is equal to zero: I I z n p = x n i (p) ω n i = 0 n {1,, N} i=1 i=1 A corollary of this is Walras Law, stating that if N 1 markets are in equilibrium, the Nth market must also be in equilibrium 51
Walras Law If N 1 markets are in equilibrium, the Nth market must also be in equilibrium. This follows from simple accounting. If we sum up all individual budgets constraints, we see that total expenditure has to equal total receipts, i.e., p I i=1 x i p = For excess demand functions, Walras Law implies that they sum up to zero: N n=1 p I i=1 z n p = 0 ω i 52
Walras Law For the analysis of general equilibrium, this result proves useful since it implies that if all market but one are in equilibrium, the last market also has to be in equilibrium: z n p = 0 n 1,, N 1 z N p = 0 53
Welfare Theorems For now, we just assume that there exists an equilibrium with positive prices and state some properties of this equilibrium: Theorem 1 (first welfare theorem): A competitive equilibrium allocation is an efficient allocation. Theorem 2 (second welfare theorem): Every efficient allocation can be transformed into a competitive equilibrium allocation by appropriate transfers 54
Conditions for Walras common equilibrium of Sonnensch./Mantel/Deb. Existence Convexity of preferences (concave utility function) Convex production set Uniqueness Either the initial allocation is already Pareto-efficient Or all goods are substitutes (positive cross price elasticities) Then the excess demand curve crosses the x-axis only once Stability Conditions for uniqueness are sufficient for global stability If excess demand becomes zero only once, this equilibrium will be stable 55
Sonnenschein-Mantel-Debreu The conditions that have to be fulfilled for an existing, unique, and stable equilibrium are called SMD conditions. The GET is a special exception, not the rule for the condition of a perfect rational agent. For all other cases the system could react differently, i.e., multiple equilibria, no equilibrium, or instable equilibria etc. 56
The General Theory of the Second Best To which degree the general equilibrium model can serve to inform relevant policy decisions in the real world? Some distortion is present in the setup Optimal conditions in all other relevant areas will not guarantee a second best result (i.e. there will not be an efficient allocation even between the undistorted elements) as the theory of the second best shows Some first-best conditions are violated, there may be numerous second best results that can be reached 57
The General Theory of the Second Best We have a function F(x 1,, x n ) of n variables x i that is to be maximized (minimized) and a constraint L x 1, x n = 0 that has to be taken into account in that operation. The optimization gives the FOCs (with F i as F i x i etc.) F i λl i = 0 i = 1,, n and, from these, the relative conditions for optimality, F i F n = L i L n i = 1,, n. 58
The General Theory of the Second Best In Lipsey and Lancaster s theory of the second best, this first best solution is excluded by an additional constraint of the form: F 1 = k L 1 with k 1. F n L n This additional constraint changes the optimization problem to: min F λ 1 L λ 2 F 1 F n k L 1 L n (or equivalently for maximization problems). λ 1 and λ 1 will generally be different from λ. 59
The General Theory of the Second Best The FOCs here are: F i λ 1 L i λ 2 F n F 1i F 1 F ni F n 2 k L nl 1i L 1 L ni L n 2 = 0 Denoting F nf 1i F 1 F ni F n 2 = Q i and L nl 1i L 1 L ni L n 2 F i F n = L i + L n + λ 2 λ 1 Q i kr i λ 2 λ 1 Q n kr n = R i we can write: The Pareto optimum conditions (achieved in the first best result above) can therefore only be attained if λ 2 = 0 (i.e. if the additional condition is irrelevant). 60
Asymmetric Information The Markets for Lemons Differences in the information (on quality) available to sellers and potential buyers of a good. Sellers will exploit their information advantage and sell lowquality goods at market price Buyers use some statistics from observed frequencies of good and bad products (lemons) to form an idea of expected quality The example will show that these information sets directly lead to the collapse of the market (and without a market, no efficient or Pareto optimal allocation of goods can be established). 61
Asymmetric Information The Markets for Lemons Quality q of goods is distributed uniformly; sellers will not sell under value, i.e. the seller price is p s q for each seller. Buyers are willing to pay a higher price kq (with 1 < k < 2) over their expected quality; since the average value of the goods is 1 k 2 i q i, the buyer s reservation prices will be p d q i 2 i Since k < 2 there are sellers with quality q > i k 2 q i who will exit the market which leads to a further fall in average quality and sends the market into a downward spiral 62
Asymmetric Information The Markets for Lemons Assume the following situation: Two groups of agents, 1 and 2. Distinguished by the utility they gain from consumption of a specific good q i is the indicator of the quality of a particular unit of the good M being a bundle of the rest of goods U 1 = M + n n 3 i=1 q i and U 2 = M + q i=1 2 i 63
Asymmetric Information The Markets for Lemons Spending one more unit on bundle M increases utility by one unit du i dm = 1 Spending on the second good, the utility effect depends on the quality q > p : The purchase is worthwhile of type 1 agents 3q 2 > p : worthwhile for type 2 agents 64
Asymmetric Information The Markets for Lemons Let Y 1 and Y 2 denote the income of all types 1 and 2 agents, respectively. Then, demand D for the good from type 1 agents is of the form: D 1 = Y 1 p D 1 = 0 if q p otherwise For type 2 agents, we have the following analogous expressions D 2 = Y 2 p D 2 = 0 if 3 2 q p otherwise 65
Asymmetric Information The Markets for Lemons Accordingly, total demand D is: D = Y 1+Y 2 p if q p D = Y 2 p if 3 q p q 2 D = 0 otherwise (i. e. if p > 3 q) 2 With uniform quality distribution, the average quality at any price p would be q = p ; therefore the demand would be D = 0. 2 No trade will never take place, even though at any price there would be someone willing to pay the asked price if quality could be assured 66
Conclusion & Limitations of GET This chapter provided a short introduction to neoclassical microeconomics, optimization methods, and GET Some critique (SMD) and two specific limitations (second best and markets for lemons) were included Further extensions (intertemporal optimization, growth models) and limitations (imperfect information) are introduced in the chapter (but are not included in this presentation) Chapters 6 and 7 of the textbook extend this and go more into detail on the critique of GET and on oligopoly models. 67
Readings for the next lecture Compulsory reading: Critiques of the Neoclassical Perfect Market Economy and Alternative Price Theories, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 129-155. For further readings visit the companion website 68