Microeconomics -1 - Consumers

1 Microeconomics -1 - Consumers John Riley October 23, 2018 A1. The simple mathematics of elasticity 2 A2. The Envelope Theorem 7 B. Income and substitution effects 15 C. Application: Labor supply 24 D. Determinants of demand 28 E. Measuring consumer gains and losses 38 Technical Notes* 46 1. Equivalent Variation 2. Mathematics of income and substitution effects 3. Superlevel sets of the aggregated utility function 58 slides *Not examinable. Will be omitted from the lectures. Microeconomics -2 - Consumers John Riley October 23, 2018 A1.

2 Elasticity Consider the figure opposite. A very useful measure of the sensitivity of y with respect to z is the proportional rate of change of y with respect to z . This is called the arc elasticity Arc elasticity = yzyyzyzz * Arc elasticity Microeconomics -3 - Consumers John Riley October 23, 2018 A1. Elasticity Consider the figure opposite. A very useful measure of the sensitivity of y with respect to z is the proportional rate of change of y with respect to z . This is called the arc elasticity Arc elasticity = yzyyzyzz Consider two countries that measure both y and z in different units. Y ay and Z bz . Then Y a y and Z b z It follows that the arc elasticity is the same.

3 ( ) ()( ) ()Z Ybz a yz yY Zay b zy z Arc elasticity Microeconomics -4 - Consumers John Riley October 23, 2018 (Point) Elasticity In theoretical analysis it is helpful to take the limit and define the (point) elasticity Elasticity =( , ) limz y z dyyzy zy dz ()()zf zfz * Point elasticity Microeconomics -5 - Consumers John Riley October 23, 2018 (Point) Elasticity In theoretical analysis it is helpful to take the limit and define the (point) elasticity Elasticity =( , ) limz y z dyyzy zy dz ()()zf zfz Note that 1lnddyydzy dz.

4 Therefore ( , )lnz dydy zzyy dzdz . Using this formula we can derive the following proposition Elasticity of products and ratios The elasticity of a product is the sum of the elasticities. ( , )( , )( , )xy zx zy z The elasticity of a ratio is the difference in elasticities ( , )( , )( , )xzx zy zy Point elasticity Microeconomics -6 - Consumers John Riley October 23, 2018 Derivation of the sum rule ( , )lnz dydy zzyy dzdz Consider the elasticity of a product. ( , )lndxy zzxydz [lnln ]dzxydz lnlnddzxydzdz ( , )( , )x zy z Group exercises: Group O: Linear demand ,app a bq qb Group E: Log linear demand , lnlnlnbq apqa b p Microeconomics -7 - Consumers John Riley October 23, 2018 A2.

5 The Envelope Theorem Consider the following constrained maximization problem with a parameter p in the function to be maximized. Let x be the solution when the parameter is p . Let ()xp be the solution for all p . Let ()Fp be the maximized value. ( ){ ( , ) | ( )}xF pMax f x p g x b . Simple Example: Profit maximization ( ){( )}qF pMax pq C q To determine the rate at which ( )( ( ), )F pf x p p varies with p is appears that it is necessary to first solve for the maximizer ()xp and then substitute this into ( , )f x p. However this intuition is incorrect. The answer is much simpler. On the margin only the direct effect is a non-zero effect. Microeconomics -8 - Consumers John Riley October 23, 2018 Envelope theorem 0( ){ ( , ) | ( )}xF pMax f x p g x b.

6 ( ( ), )dFfx p pdpp Informal proof: Let ()x x p be the solution when the price is p . Suppose that the decision-maker is na ve and does not change output as the parameter changes. The na ve payoff is ( , )( ( ), )f x pf x p p . We compare ()Fp and ( , )f x p . * Microeconomics -9 - Consumers John Riley October 23, 2018 Envelope theorem 0( ){ ( , ) | ( )}xF pMax f x p g x b . ( ( ), )dFfx p pdpp Informal proof: Let ()x x p be the solution when the price is p . Suppose that the decision-maker is na ve and does not change output as the parameter changes. The na ve payoff is ( , )( ( ), )f x pf x p p . Note that (i) ( )( ( ), )F pf x p p Since ()xp is optimal, for all p (ii) ( )( ( ), )( ( ), )F pf x p pf x p p.

7 Assuming that the functions are differentiable, the graphs of the two functions must be as depicted. It follows that the graphs must be tangential at p. ( )( , )fF px pp Microeconomics -10 - Consumers John Riley October 23, 2018 Intuition for the simplest case with no constraint ( ){ ( , )}xF pMax f x p There are two effects 1. Direct effect Parameter change p ( , )f x p rises to ( ,)f x pp * Microeconomics -11 - Consumers John Riley October 23, 2018 Intuition for the simplest case with no constraint ( ){ ( , )}xF pMax f x p There are two effects 1.

8 Direct effect Parameter change p ( , )f x p rises to ( ,)f x pp 2. Indirect effect Decision variable change x . For small change in price the The graph of ( , )f x p has a slope which is close to zero. This effect disappears in the limit. Only the direct effect is a first order effect . Microeconomics -12 - Consumers John Riley October 23, 2018 A more general result (Not examinable) ( ){ ( , )}xXF pMax f x p . Note that x is constrained to belong to some unspecified set. Proposition: If ()xp is a continuous function then ( )( ( ), )fF px p pp (i) Since 0()xp is the optimizer when 0pp , it follows that 01000( ( ),)( ( ),)()f x p pf x pFpp . Therefore 1010111001( ( ),)( )( )( ( ), )()(,((),))F pF pf x p pf xpf x p pfxppp ** Microeconomics -13 - Consumers John Riley October 23, 2018 A more general result (Not examinable) ( ){ ( , )}xXF pMax f x p.

9 Note that x is constrained to belong to some unspecified set. Proposition: If ()xp is a continuous function then ( )( ( ), )fF px p pp . (i) Since 0()xp is the optimizer when 0pp , it follows that 00010( )( ( ),)( ( ),)F pf x p pf x p p . Therefore 1011001110( )( )( ( ), )( ( ),)( ( ), )( ( ),)F pF pf x p pf x p pf x p pf x p p (ii) Since 1()xp is the optimizer when 1pp , it follows that 10111( ( ), )( ( ), )()f x p pf x pFpp . Therefore 0100111000( )( )( ( ),)( ( ),)(( ( ),,)( ))fxF pF pf x p pffxpx p pppp * Microeconomics -14 - Consumers John Riley October 23, 2018 A more general result (Not examinable) ( ){ ( , )}xXF pMax f x p.

10 Note that x is constrained to belong to some unspecified set. Proposition: If ()xp is a continuous function then ( )( ( ), )fF px p pp . (i) Since 0()xp is the optimizer when 0pp , it follows that 00010( )( ( ),)( ( ),)F pf x p pf x p p . Therefore 1011001110( )( )( ( ), )( ( ),)( ( ), )( ( ),)F pF pf x p pf x p pf x p pf x p p (ii) Since 1()xp is the optimizer when 1pp , it follows that 11101( )( ( ), )( ( ), )F pf x p pf x p p . Therefore 1011000100( )( )( ( ), )( ( ),)( ( ), )( ( ),)F pF pf x p pf x p pf x p pf x p p Together, these inequalities imply that 0100101110101010( ( ), )( ( ), )( )( )( ( ), )( ( ), )f x p pf x p pF pF pf x p pf x p ppppppp . Note that as 10pp , the lower and upper bounds both approach 00( ( ),)fx p pp . Thus the derivative 000( )( ( ),)fF px p pp.