UNIT 4. CONSUMER BEHAVIOR J. Alberto Molina – J. I. Giménez Nadal UNIT 4. CONSUMER BEHAVIOR 4.1 Consumer equilibrium (Pindyck → 3.3, 3.5 and T.4) Graphical analysis. Analytical solution. 4.2 Individual demand function (Pindyck → 4.1) Derivation of the individual Marshallian demand Properties of the individual Marshallian demand 4.3 Individual demand curves and Engel curves (Pindyck → 4.1) Ordinary demand curves Crossed demand curves Engel curves 4.4 Price and income elasticities (Pindyck → 2.4, 4.1 and 4.3) Price elasticity of demand Crossed price elasticity Income elasticity 4.5 Classification of goods and demands (Pindyck → 2.4, 4.1 and 4.3) APPENDIX: Relation between expenditure and elasticities Unit 4 – Pg. 1 4.1 Consumer equilibrium Consumer equilibrium: • We proceed to analyze how the consumer chooses the quantity to buy of each good or service (market basket), given his/her: – Preferences – Budget constraint • We shall assume that the decision is made rationally: Select the quantities of goods to purchase in order to maximize the satisfaction from consumption given the available budget • We shall conclude that this market basket maximizes the utility function: – The chosen market basket must be the preferred combination of goods or services from all the available baskets and, particularly, – It is on the budget line since we do not consider the possibility of saving money for future consumption and due to the non‐satiation axiom Unit 4 – Pg. 2 4.1 Consumer equilibrium Graphical analysis • The equilibrium is the point where an indifference curve intersects the budget line, with this being the upper frontier of the budget set, which gives the highest utility, that is to say, where the indifference curve is tangent to the budget line q2 * q2 U3 U2 U1 * q1 q1 Unit 4 – Pg. 3 4.1 Consumer equilibrium Consumer problem • The consumer will assign limited resources (i.e., income “Y”) among different goods or services (q1, q2) in order to maximize utility: maxUUqq (12 , ) qq12, subject to: Ypqpq11 2 2 qq12, 0 • Assumptions: – There is always some positive consumption of all goods or services – The consumer spends all income • Therefore, the optimization problem can be written as follows: maxU U (q1,q2 ) q1 ,q2 s.t. Y p1q1 p2q2 Unit 4 – Pg. 4 4.1 Consumer equilibrium Mathematical resolution of the consumer problem: • Lagrange Multipliers Method (LMM): We use an auxiliary function called “ The Lagrangian function” (λ: Lagrange multiplier): L L(q1,q2,) U(q1,q2 ) [Y p1q1 p2q2 ] which is maximized: max L L(q1,q2,) q1,q2 , The First Order Conditions (F.O.C) are: L L1 U1 p1 0 U1 p1 (1) q1 L L2 U2 p2 0 U2 p2 (2) q2 L L Y p q p q 0 Y p q p q (3) 1 1 2 2 1 1 2 2 Given particular values for the exogenous variables (income and prices), we solve the system of three equations for the three endogenous variables (q1, q2, λ) Unit 4 – Pg. 5 4.1 Consumer equilibrium Mathematical resolution of the problem: • From equations (1) and (2) we derive: Equal marginal principle (EMP): The consumer must, in U U 1 2 equilibrium, obtain the same utility from the last monetary unit p1 p2 spent on either good 1 or good 2. • In equilibrium, we obtain: U1 U2 2 U1 p1 MRS1 U p p1 p2 2 2 That is to say, the consumer is in equilibrium when the following two magnitudes are equated: 2 The subjective exchange rate between goods (MRS 1): the rate at which the consumer wants to exchange one good for the other with constant utility The objective exchange rate between goods offered in the market (P1/P2) Unit 4 – Pg. 6 4.1 Consumer equilibrium Mathematical resolution of the problem: • SUMMARIZING: the consumer problem is solved as follows: Problem From F.O.C.s: Solution: U U 0 0 0 maxU U (q ,q ) 1 2 Given p1 , p2 and Y 1 2 q1 ,q2 p p 1 2 (q1*, q2*) s.t. Y p1q1 p2q2 Y p1q1 p2q2 Endogenous variables: q1 and q2 Exogenous variables: p1, p2 and Y MRS 2 U 1 p1 In equilibrium : 1 U 2 p 2 The marginal rate of substitution must be equal to the relative price Unit 4 – Pg. 7 4.1 Consumer equilibrium Mathematical resolution of the problem: p • SPECIAL CASE: Corner solution MRS 2 1 1 p2 The consumer spends the total income on just one good or service and decides not to buy any of the other good: MRS 2 p1 1 p2 Unit 4 – Pg. 8 4.1 Consumer equilibrium Economic interpretation of the Lagrange Multiplier • In equilibrium, the Lagrange Multiplier is equal to the marginal utility of income: (1) First we differentiate the utility function: dU U1dq1 U2dq2 (2) Also the budget line equation (constant dY P1dq1 P2dq2 prices): (3) We know that in equilibrium:U1 P1 and U2 P2 (4) Finally, substituting (3) into (1): dU P1dq1 P2dq2 ( P1dq1 P2dq2 ) dY And using (2) Unit 4 – Pg. 9 4.2 Individual demand function Derivation of the the Marshallian individual demand function Def: The general demand function (or Marshallian demand function) is the mathematical relationship, in equilibrium, between prices and income. and the corresponding quantity of the good: q1= q1(p1, p2, Y) Demand function of good 1 q2= q2(p1, p2, Y) Demand function of good 2 These functions are obtained from the F.O.C.s of the consumer maximization problem, considering p1, p2 and Y as parameters/variables in place of of specific numerical values. Unit 4 – Pg. 10 4.2 Individual demand function Properties of the Marshallian individual demand functions • They are functions such that: For each three values (p1, p2, Y) there is only one optimal quantity demanded of each good or service, that is to say, the optimal equilibrium basket is unique • They describe optimal behavioral relations: They incorporate the objectives and restrictions of the consumer problem The consumer tastes or preferences fully determine the functional form • They are homogeneous degree “0” functions in prices and income: If all prices and income are multiplied by the same positive constant “k”, the equilibrium remains: 0 qi(kp1, kp2, kY) = k qi(p1, p2, Y) = qi(p1, p2, Y); i = 1, 2 Unit 4 – Pg. 11 4.3 Individual demand curves and Engel curves Once we have the individual demand functions, we can define several behavioral curves according to the particular values of specific exogenous variables (p1, p2, Y): • (1) Ordinary demand curve: Shows the quantity demanded of a good as a function of its price (the income and the price of the other good are given): 0 0 q1 q1 ( p1 , p 2 ,Y ) q1 ( p1 ) Ordinary demand curve of good 1 0 0 q 2 q 2 ( p1 , p 2 , Y ) q 2 ( p 2 ) Ordinary demand curve of good 2 • (2) Crossed demand curve: Shows the quantity demanded of a good as a function of the price of the other good (the income and the price of thesamegoodare given): 0 0 q1 q1 ( p1 , p 2 ,Y ) q1 ( p 2 ) Crossed demand function of good 1 0 0 q 2 q 2 ( p1 , p 2 ,Y ) q 2 ( p1 ) Crossed demand function of good 2 • (3) Engel curve: Shows the quantity demanded of a good as a function of the consumer’s income (both prices are given): 0 0 q1 q1 ( p1 , p 2 , Y ) q1 (Y ) Engel curve of good 1 0 0 q 2 q 2 ( p1 , p 2 , Y ) q 2 (Y ) Engel curve of good 2 Unit 4 – Pg. 12 4.3 Individual demand curves and Engel curves Behavioral curves pi is variable 0 0 (1) Ordinary demand curve: (pj , Y ) qi=qi(pi) Consumer Individual demand pj is variable (2) Crossed demand curve: Equilibrium 0 0 (pi , Y ) functions qi=qi(pj) qi=qi(pi, pj, Y) Y is variable (3) Engel curve: 0 0 (pi , pj ) qi=qi(Y) Unit 4 – Pg. 13 4.3 Individual demand curves and Engel curves q2 Y/p 2 (1)Ordinary demand curve of good 1: PCC1 Price-consumption curve (PCC): Shows 1 q2 how the quantity demanded changes when prices change. First we change p1: 0 q2 2 p1 CPC1 (q1,q2 ) MRS1 ; p2 ,Y constant; p1 free p2 q1 q 0 Y/p 0 q 1 Y/p 1 P1 1 1 1 1 0 p1 Ordinary demand curve (ODC): 2 p1 ODC1 (q1, p1 ) MRS1 ; p2 ,Y constant p 1 1 p2 ODC1 q1 0 1 q1 q1 Unit 4 – Pg. 14 4.3 Individual demand curves and Engel curves q2 0 Y/p2 (2) Crossed demand function of good 1: PCC2 0 q2 Price-consumption curve of good 1 (PCC): Shows how the quantity demanded Y/p 1 2 changes when the price of good 2 changes. Therefore, we allow p to change: 1 2 q2 2 p Y/p2 2 1 CPC2 (q1,q2 ) MRS1 ; p1,Y constant; p2 free 2 p2 q2 q1 q 0 q 1 q 2 1 1 1 Y/p1 P2 CDC1 2 p2 Crossed demand curve (CDC): In this case q1 and q2 are p p 1 2 1 2 substitutes CDC1 (q1, p2 ) MRS1 ; p1,Y constant 0 p2 p2 q1 0 1 2 q1 q1 q1 Unit 4 – Pg. 15 4.3 Individual demand curves and Engel curves q 2 (3) Engel curve of good 1: NORMAL GOOD ICC Y1/p 2 Income-consumption curve (ICC): Shows how the quantity demanded changes when the consumer’s income Y0/p 2 changes: 1 q2 q 0 2 2 p1 ICC (q1,q2 ) MRS1 ; p1, p2 constant; Y free p2 0 1 0 1 q1 q1 q1 Y /p1 Y /p1 Y EC1 Engel curve (EC): 1 Y 2 p1 ↑ Y ↑ q EC1 (q1,Y) MRS1 ; p1, p2 constant 1 p Y0 Normal good 2 (increasing EC) 0 1 q1 q1 q1 Unit 4 – Pg.