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Part IIA, Paper 1 Consumer and Producer Theory Lecture 2 Direct and Indirect Utility Functions Flavio Toxvaerd Today’s Outline Indifference curves Marginal rates of substitution Marshallian demand functions Types of goods Indirect utility function Consumer surplus and welfare Some mathematical results (Envelope Thm.) Roy’s Identity Utility Function Recall from last lecture – given Axioms of Choice, continuity and local non-satiation - a consumer’s preference ordering can be represented by a utility function Simplifying Assumption: The domain consists of only two types of commodities, types 1 and 2 A specific consumption bundle, x, will be represented by a vector x = (x1,x2) and we can write the utility function as u(x1,x2) Indifference Curves Indifference curves show combinations of commodities for which utility is constant Mathematic ally, we require that du ( x1, x2 ) 0 u ( x1, x2 ) u ( x1, x2 ) dx1 dx2 0 x1 x2 u ( x1, x2 ) dx2 x1 dx1 u ( x1, x2 ) x1 Slope of Indifference Curve = Marginal Rate of Substitution Utility Maximisation max u ( x1 , x 2 ) x1 , x2 subject to p1 x1 p 2 x 2 m L u ( x1 , x2 ) [m p1 x1 p 2 x2 ] u ( x1 , x 2 ) First order conditions: p1 x1 u ( x1 , x 2 ) p 2 x 2 p1 x1 p 2 x 2 m Utility Maximisation Eliminating the Lagrange Multiplier () gives u ( x1, x2 ) MRS = Ratio of Prices x2 p2 Slope of Indifference Curve u ( x1, x2 ) p1 = Slope of Budget Line x1 and p1x1 p2 x2 m Budget Line Note, with more than two commodities have FOC u xi MUxi MU of income pi pi Utility Maximisation To solve: move along budget line, until point of tangency with the indifference curve x2 x2 A D C B x1 x1 Demand Function When the indifference curves are strictly convex the solution is unique, say x*, where x1*= x1(p1,p2,m) , x2*= x2(p1,p2,m) giving demand as a function of prices and income Marshallian Demand Functions ' Ordinary' Good : Goods dx1 ( p1 , p2 , m) 0 x2 dp1 Giffen Good : m Price expansion path p2 dx1 ( p1 , p2 , m) 0 dp1 Complements : dx2 ( p1 , p2 , m) 0 m p1 m p '1 x1 dp1 Substitutes : p1 Demand Curve dx2 ( p1 , p2 , m) 0 p’1 dp1 x1(p1,p2,m) x1(p’1,p2,m) x1 Goods 0 : Inferior Good < 0 dx1 ( p1, p2 , m) dm dx1 ( p1, p2 , m) 0 : Normal Good > 0 dm dx1 ( p1, p2 , m) x1 : Superior Good > 1 dm m IncomeElasticity of Demand : Graphically: dx1 m . Engel Curve dm x1 Practice Problem 1: Show that the Marshallian Demand Function is homogenous degree zero. (So consumers never suffer from Money Illusion) Problem 2: Show that consumers’ purchase decisions are unaffected by any monotonic transformation of the utility function. (Hint: A monotonic transformation can be represented by a strictly increasing function f(.). Use the chain rule to show that the MRS remains unaffected) Convex Indifference Curves Convex indifference curves means that the (absolute value of the) slope of the indifference curve is decreasing as x1 increases That is: Diminishing MRS Problem 3: Show that diminishing marginal utilities is neither a necessary nor a sufficient condition for convex indifference curves Example: Cobb-Douglas Utility 1 u ( x1, x2 ) x1 x2 1 Lagrangian : L x 1 x2 ( p1x1 p2 x2 m) x1 1x1 p1 2 First Order Conditions (1 ) x1 x2 p2 p1 x1 p2 x2 m Example: Cobb-Douglas Utility Eliminating gives: (1 ) x1 p2 MRS x2 p1 Substitution into the budget constraint gives the solution m x *1 x1 ( p1 , p2 , m) p1 (1 ) m x *2 x2 ( p1 , p2 , m) p2 With Cobb-Douglas Utility the consumer spends a fixed proportion of income on each commodity Indirect Utility Function It is often useful to consider the utility obtained by a consumer indirectly, as a function of prices and income rather than the quantities actually consumed Let v( p1 , p2 , m) u ( x *1 , x *2 ) u ( x1 ( p1 , p2 , m), x2 ( p1 , p2 , m)) max u ( x1 , x2 ) such that p1 x1 p2 x2 m v( p1 , p2 , m) is called the Indirect U tility Function Properties of Indirect Utility Fn Property 1: v(p,m) is non-increasing in prices (p), and non-decreasing in income (m). Proof: Diagramatically, it is clear that any increase in prices or decrease in income contracts the ‘affordable’ set of commodities – as nothing new is available to the consumer utility cannot increase Property 2: v(p,m) is homogeneous degree zero. Proof: No change in the affordable set, or in preferences Properties of Indirect Utility Fn These two are General Properties and NOT reliant on additional restrictions such as convexity of indifference curve, more is better etc. Direct and Indirect Utility We will see that direct and indirect utility functions are closely related - and that any preference ordering that can be represented by a utility function can also be represented by an indirect utility function. This means we are free to use whichever specification we please For Example: If the price of commodity 1 changes from, say, p1=a to p1=b, we may want to use the indirect utility function to measure the change in consumer welfare: (utility) v(b, p2 , m) v(a, p2 , m) Mathematical Digression The Envelope Theorem: If M (a ) max x , x g ( x1 , x2 , a ) s.t. h( x1 , x2 , a ) 0 1 2 g ( x *1 , x *2 , a ) for which the Lagrangian can be written L g ( x1 , x2 , a ) - h( x1 , x2 , a ) then M L Evaluated at the ( x *1 , x *2 , a ) maximising a a values Proof: See Varian Microeconomic Analysis, p. 502. Application 1 Marginal utility of Income As v( p1, p2 , m) max x , x u ( x1, x2 ) s.t. p1x1 p2 x2 m 0 1 2 and the Lagrangian L u ( x1, x2 ) - ( p1x1 p2 x2 m) then, by the Envelope Theorem, v L ( x *1, x *2 , p1, p2 , m) m m The marginal utility of income is given by the Lagrange multiplier Application 2 Roy’s Identity Similarly, by the Envelope Theorem, v( p1, p2 , m) L( x1 , x1 ) * * x1( p1, p2 , m) p1 p1 and so, v( p1, p2 , m) p1 Roy’s x1( p1, p2 , m) v( p1, p2 , m) Identity m Consumer Surplus and Welfare We saw earlier that, following a price change, (utility) v(b, p2 , m) v(a, p2 , m) v( p1, p2 , m) a dp1 b p1 a v( p1, p2 , m) .x1 ( p1, p2 , m) dp1 b m a x1 ( p1, p2 , m) dp1 , b v( p1, p2 , m) if constant m Consumer Surplus and Welfare a x ( p , p , m)dp b 1 1 2 1 (Consumer Surplus) So, ( Utility) (Consumer Surplus) p1 a b Marshallian Demand x1 Summary Indifference curves Marginal rates of substitution Marshallian demand functions Types of goods Indirect utility function Consumer surplus and welfare Roy’s Identity Readings Texts: Varian, Intermediate Economics (7th ed.) chapters 4, 5, 6, 14. Varian, Microeconomic Analysis, chapter 7

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posted: | 4/6/2010 |

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