Report

Prerequisites Almost essential Firm: Optimisation Consumption: Basics CONSUMER OPTIMISATION MICROECONOMICS Principles and Analysis Frank Cowell March 2012 Frank Cowell: Consumer Optimisation 1 What we’re going to do: We’ll solve the consumer's optimisation problem… …using methods that we've already introduced This enables us to re-cycle old techniques and results A tip: • Run the presentation for firm optimisation… • look for the points of comparison… • and try to find as many reinterpretations as possible March 2012 Frank Cowell: Consumer Optimisation 2 The problem Maximise consumer’s utility U(x) U assumed to satisfy the standard “shape” axioms Subject to feasibility constraint Assume consumption set X is the non-negative orthant and to the budget constraint The version with fixed money income x X n S pixi ≤ y i=1 March 2012 Frank Cowell: Consumer Optimisation 3 Overview… Consumer: Optimisation Two fundamental views of consumer optimisation Primal and Dual problems Lessons from the Firm Primal and Dual again March 2012 Frank Cowell: Consumer Optimisation 4 An obvious approach? We now have the elements of a standard constrained optimisation problem: • the constraints on the consumer • the objective function The next steps might seem obvious: • set up a standard Lagrangean • solve it • interpret the solution But the obvious approach is not always the most insightful We’re going to try something a little sneakier… March 2012 Frank Cowell: Consumer Optimisation 5 Think laterally… In microeconomics an optimisation problem can often be represented in more than one form Which form you use depends on the information you want to get from the solution This applies here The same consumer optimisation problem can be seen in two different ways I’ve used the labels “primal” and “dual” that have become standard in the literature March 2012 Frank Cowell: Consumer Optimisation 6 A five-point plan The primal problem Set out the basic consumer optimisation problem Show that the solution is equivalent to another The dual problem problem Show that this equivalent problem is identical to that of the firm The primal problem again Write down the solution Go back to the problem we first thought of… March 2012 Frank Cowell: Consumer Optimisation 7 The primal problem x2 The consumer aims to maximise utility… Subject to budget constraint Contours of objective function Defines the primal problem Solution to primal problem Constraint set max U(x) subject to n S pixi y x* i=1 x1 March 2012 But there's another way of looking at this Frank Cowell: Consumer Optimisation 8 The dual problem z2x2 Alternatively the consumer could aim to minimise cost… Subject to utility constraint q u Constraint set Defines the dual problem Solution to the problem Cost minimisation by the firm minimise n S pixi x* z* i=1 subject to U(x) u z1 x1 March 2012 But where have we seen the dual problem before? Frank Cowell: Consumer Optimisation 9 Two types of cost minimisation The similarity between the two problems is not just a curiosity We can use it to save ourselves work All the results that we had for the firm's “stage 1” problem can be used We just need to “translate” them intelligently • Swap over the symbols • Swap over the terminology • Relabel the theorems March 2012 Frank Cowell: Consumer Optimisation 10 Overview… Consumer: Optimisation Reusing results on optimisation Primal and Dual problems Lessons from the Firm Primal and Dual again March 2012 Frank Cowell: Consumer Optimisation 11 A lesson from the firm Compare cost-minimisation for the firm… …and for the consumer x2 u z2 q So their solution functions and response functions must be the same x* z* z1 March 2012 The difference is only in notation x1 Frank Cowell: Consumer Optimisation Run through formal stuff 12 Cost-minimisation: strictly quasiconcave U Minimise Lagrange multiplier n S pi xi i=1 Use the objective function …and output constraint …to build the Lagrangean + λ[u – U(x)] U(x) Because of strict quasiconcavity we have an interior solution Differentiate w.r.t. x1, …, xn and set equal to 0 … and w.r.t l Denote cost minimising values with a * A set of n+1 First-Order Conditions l* U1 (x* ) = p1 one for each good l* U2 (x* ) = p2 … … … l* Un (x* ) = pn u = U(x* ) March 2012 utility constraint Frank Cowell: Consumer Optimisation 13 If ICs can touch the axes… Minimise n Spixi i=1 + l[u – U(x)] Now there is the possibility of corner solutions A set of n+1 First-Order Conditions l*U1 (x*) p1 l*U2 (x*) p2 … … … l*Un(x*) pn u = U(x*) March 2012 Interpretation Can get “<” if optimal value of this good is 0 Frank Cowell: Consumer Optimisation 14 From the FOC If both goods i and j are purchased and MRS is defined then… Ui(x*) pi ——— = — * Uj(x ) pj MRS = price ratio “implicit” price = market price If good i could be zero then… Ui(x*) pi ——— — * Uj(x ) pj MRSji price ratio “implicit” price market price Solution March 2012 Frank Cowell: Consumer Optimisation 15 The solution… Solving the FOC, you get a cost-minimising value for each good… xi* = Hi(p, u) …for the Lagrange multiplier l* = l*(p, u) …and for the minimised value of cost itself The consumer’s cost function or expenditure function is defined as C(p, u) := min S pi xi {U(x) u} vector of goods prices March 2012 Specified utility level Frank Cowell: Consumer Optimisation 16 The cost function has the same properties as for the firm Non-decreasing in every price, increasing in at least one price Increasing in utility u Concave in p Homogeneous of degree 1 in all prices p Shephard's lemma March 2012 Frank Cowell: Consumer Optimisation 17 Other results follow Shephard's Lemma gives H is the “compensated” or demand as a function of prices conditional demand function and utility Hi(p, u) = Ci(p, u) Downward-sloping with respect Properties of the solution function determine behaviour to its own price, etc… of response functions For example rationing “Short-run” results can be used to model side constraints March 2012 Frank Cowell: Consumer Optimisation 18 Comparing firm and consumer Cost-minimisation by the firm… …and expenditure-minimisation by the consumer …are effectively identical problems So the solution and response functions are the same: Consumer Firm m n min Swizi + l[q – f(z)] min Spixi + l[u – U(x)] Solution function: C(w, q) C(p, u) Response function: zi* = Hi(w, q) xi* = Hi(p, u) Problem: z March 2012 i=1 x i=1 Frank Cowell: Consumer Optimisation 19 Overview… Consumer: Optimisation Exploiting the two approaches Primal and Dual problems Lessons from the Firm Primal and Dual again March 2012 Frank Cowell: Consumer Optimisation 20 The Primal and the Dual… There’s an attractive symmetry about the two approaches to the problem In both cases the ps are given and you choose the xs But… …constraint in the primal becomes objective in the dual… n S pixi+ l[u – U(x)] i=1 n U(x) + m y – S pi xi [ ] i=1 …and vice versa March 2012 Frank Cowell: Consumer Optimisation 21 A neat connection Compare the primal problem of the consumer… …with the dual problem x2 u x2 The two are equivalent So we can link up their solution functions and response functions x* x* x1 March 2012 x1 Run through the primal Frank Cowell: Consumer Optimisation 22 Utility maximisation Maximise Lagrange multiplier nn – Spi xi U(x) + μ y [ i=1 i=1 Use the objective function …and budget constraint …to build the Lagrangean ] If U is strictly quasiconcave we have an interior solution A set of n+1 First-Order Conditions one for each good U1(x* ) = m* p1 If U not strictly U2(x* ) = m* p2 quasiconcave then … … … replace “=” by “” budget Un(x* ) = m* pn constraint n y = S pi xi* Differentiate w.r.t. x1, …, xn and set equal to 0 … and w.r.t m Denote utility maximising values with a * Interpretation i=1 March 2012 Frank Cowell: Consumer Optimisation 23 From the FOC If both goods i and j are purchased and MRS is defined then… Ui(x*) pi ——— = — * Uj(x ) pj MRS = price ratio (same as before) “implicit” price = market price If good i could be zero then… Ui(x*) pi ——— — * Uj(x ) pj MRSji price ratio “implicit” price market price Solution March 2012 Frank Cowell: Consumer Optimisation 24 The solution… Solving the FOC, you get a utility-maximising value for each good… xi* = Di(p, y) …for the Lagrange multiplier m* = m*(p, y) …and for the maximised value of utility itself The indirect utility function is defined as V(p, y) := max U(x) {S pixi y} vector of goods prices March 2012 money income Frank Cowell: Consumer Optimisation 25 A useful connection The indirect utility function maps prices and budget into maximal utility The indirect utility function works like an "inverse" to the cost function u= V(p, y) The cost function maps prices The two solution functions have and utility into minimal budget to be consistent with each other. y = C(p, u) Therefore we have: u= V(p, C(p, u)) y = C(p, V(p, y)) March 2012 Two sides of the same coin Odd-looking identities like these can be useful Frank Cowell: Consumer Optimisation 26 The Indirect Utility Function has some familiar properties… (All of these can be established using the known properties of the cost function) Non-increasing in every price, decreasing in at least one price Increasing in income y quasi-convex in prices p Homogeneous of degree zero in (p, y) Roy's Identity March 2012 But what’s this…? Frank Cowell: Consumer Optimisation 27 Roy's Identity u = V(p, y)= V(p, C(p,u)) “function-of-afunction” rule 0 = Vi(p,C(p,u)) + Vy(p,C(p,u)) Ci(p,u) Use the definition of the optimum Differentiate w.r.t. pi Use Shephard’s Lemma Rearrange to get… 0 = Vi(p, y) + Vy(p, y) xi* So we also have… Marginal disutility of price i Vi(p, y) xi* = – ———— Vy(p, y) Marginal utility of money income Ordinary demand function xi* = –Vi(p, y)/Vy(p, y) = Di(p, y) March 2012 Frank Cowell: Consumer Optimisation 28 Utility and expenditure Utility maximisation …and expenditure-minimisation by the consumer …are effectively two aspects of the same problem So their solution and response functions are closely connected: Primal Dual [ Problem: max U(x) + μ y – x Solution function: V(p, y) Response x * = Di(p, y) function: i March 2012 n Spi xi ] i=1 n min S pixi + l[u – U(x)] x i=1 C(p, u) xi* = Hi(p, u) Frank Cowell: Consumer Optimisation 29 Summary A lot of the basic results of the consumer theory can be found without too much hard work We need two “tricks”: 1.A simple relabelling exercise: • cost minimisation is reinterpreted from output targets to utility targets 2.The primal-dual insight: • utility maximisation subject to budget is equivalent to cost minimisation subject to utility March 2012 Frank Cowell: Consumer Optimisation 30 1. Cost minimisation: two applications THE FIRM THE CONSUMER min cost of inputs min budget subject to output target subject to utility target Solution is of the form C(w,q) Solution is of the form C(p,u) March 2012 Frank Cowell: Consumer Optimisation 31 2. Consumer: equivalent approaches PRIMAL DUAL max utility min budget subject to budget constraint subject to utility constraint Solution is a function of (p,y) Solution is a function of (p,u) March 2012 Frank Cowell: Consumer Optimisation 32 Basic functional relations Utility C(p,u) Compensated i H (p,u) Review V(p, y) indirect utility Review Di(p, y) ordinary demand for input i Review Review cost (expenditure) H is also known as "Hicksian" demand demand for good I money income March 2012 Frank Cowell: Consumer Optimisation 33 What next? Examine the response of consumer demand to changes in prices and incomes Household supply of goods to the market Develop the concept of consumer welfare March 2012 Frank Cowell: Consumer Optimisation 34