Lecture 3 – Consumer choice (part 2) 1. Walrasian demand and indirect utility function: Roy’s identity 2. Comparative statics 3. Slutzky decomposition 4. Welfare: consumer surplus 1. Walrasian demand and indirect utility function: Roy’s identity • In the previous lecture we have shown that the Hicksian demand is the derivative vector of the expenditure function with respect to prices (Shephard lemma) • Somehow in analogous way we can show the following relationship between indirect utility function and walrasian demand: Roy’s identity: Suppose that u(.) is a continuous utility function and suppose that the indirect utility function v(p,m) is differentiable at (p*, w*)>>0, then for every l = 1.....L the following holds: ∂v ( p*, w*) / ∂p l xl ( p*, w*) = − ∂v ( p*, w*) / ∂w In “words”, the walrasian demand is equal to the derivative of the indirect utility function with respect to prices normalized by the marginal utility of wealth. Proof: Consider u*=v(p*, w*), where w*= e(p*,u*). Because this identity holds for all p, diffentiating v(.) with respect to prices, and evaluating at p=p* we obtain: ∂v( p*,e( p*,u*) ∂v( p*,e( p*,u*) ∂e( p*,u*) + =0 ∂p ∂e( p*,u*) ∂p Since by the Shepard lemma we know that the derivative of the expenditure function with respect to the price p is equal to the Hicksian h(p, u) demand, and remembering that that w= e(p,u), substituting in the above expression we get ∂v( p*,e( p*,u*) ∂v( p*,e( p*,u*) + h( p*,u*) = 0 ∂p ∂w Since, w*=e(p*,u*) and from the last lecture we know that a) if the quantity x* is optimal in the expenditure minimisation problem where u=u*, then x* is optimal in the utility maximisation problem when w*=p*x*. Moreover, the maximised utility level is u b) If x* is optimal in the utility maximisation problem, then x* is optimal in the expenditure minimisation problem where the required level of utility is u(x*) Then from a) and b), h(p*,u*)=x(p*,w*), which implies: ∂v( p*,e( p*,u*) ∂v( p*,e( p*,u*) + x( p*, w*) = 0 ∂p ∂w and re-arranging: ∂v( p*,e( p*,u*) ∂p x( p*, w*) = − ∂v( p*,e( p*,u*) ∂w 2. Comparative statics Given the UMP and EMP we have derived: - walrasian demand, indirect utility function and ROY identity; - hicksian demand, expenditure function and SHEPARD lemma; Now, we should be able to analyze how the consumption changes if we change price and income, i.e. we should be able to make comparative statics exercises! • Income changes - - Engel curve DRAW A GRAPH! Positive effect: normal goods Negative effect: inferior goods • Price changes - - price offer good DRAW A GRAPH! Negative effect: ordinary good Positive effect: Giffen good 3. Slutzky decomposition The price effect can be decomposed into two parts - Income effect - Substitution effect Exercice: SHOW GRAPHICALLY THESE TWO EFFECTS In analytical terms, these two effects can be precisely measured using the Slutzky decomposition. Formally, for all (p,w) and u=v(p,w) the following holds: ∂hl ( p, u ) ∂xl ( p, w) ∂xl ( p, w) = + .xk ( p, w) ∂pk ∂pk ∂w (1) Proof: When the consumer solves the UMP or the EMP we know that we know that for all (p,u), h(p,u)=x(p, e(p,u)) where w=e(p, u). Hence, differentiating the hicksian demand for good l with respect to the price pk and evaluating the derivative at (p*, u*) we obtain: ∂hl ( p*, u *) ∂xl ( p*, e( p*, u*)) ∂xl ( p*, e( p*, u*)) ∂e( p*, u*) = + ∂pk ∂pk ∂e(.) ∂pk Remembering that the derivative of the expenditure function with respect to the price of a good corresponds to the hicksian demand for that good, and that w=e(p,u), we obtain: ∂hl ( p*, u *) ∂xl ( p*, w *) ∂xl ( p*, w *) = + hk ( p*, u*) ∂pk ∂pk ∂w Remembering that for all (p,u), h(p,u)=x(p, e(p,u)), we prove that: ∂hl ( p*, u *) ∂xl ( p*, w *) ∂xl ( p*, w *) = + xk ( p*, w*) ∂pk ∂pk ∂w If you consider all the possible price change at the same time, you can write the slutzky substitution matrix where each term is given by the equation (1)… Exercice: suppose there are only 3 goods and 3 prices. Write the slutzky substitution matrix! Property: when the demand generated from preference maximization, the slutzky substitution matrix is symmetric and negative definite. Re-arranging the terms of equation (1) ∂xl ( p, w) ∂hl ( p, u ) ∂xl ( p, w) = − .xk ( p, w) ∂pk ∂pk ∂w substitution effect: ∂hl ( p, u ) ∂pk income effect: ∂xl ( p, w) .xk ( p, w) ∂w Question: when do you have a Giffen good? Consumer Welfare • Positive analysis: what are the actual consumer choices given preferences and constraints? • Normative analysis: when consumer choices change, how does consumer well-being change? So far we just investigated consumer choice (positive analysis), now we would like to evaluate the welfare implication of these choices. We have already done comparative static exercises and we know that if prices and/or income change, then the consumer optimal choice change Hence, given that the consumer optimal choice changes, we would like to know how the well-being of the consumer changes. The type of question we would like to answer is: is the consumer better off or worse off as a consequence of a price and/or income change? Measuring consumer’s welfare The first problem we have to solve in order to evaluate the change of consumer’s welfare is to find a MEASURE of consumer welfare. • Which MEASURE do you suggest? Utility? Suppose we use utility. Then, the obvious way to measure the well-being of the consumer at two different consumption choices is the indirect utility function Let (p*,w*) be a given price-income pair and (p’,m’) be another price-income pair, then the difference in utility associated to the consumer choices for these two price- income pairs is simply: V(p*,w*)-V(p’,m’) However, remember that utility is a pure ORDINAL concept! Problem: suppose that initially the price and income level in an economy are (p*,w*). Suppose that following an exogenous price shock, the level of prices increases. • What will happen to consumer welfare? • Suppose that the government wanted to keep the consumer welfare constant. What should the government do? To answer this type of questions, it would be useful to have a cardinal measure of welfare change. For example, a measure in terms of income. Construction of a money-metric indirect utility function A money metric utility function tells you the wealth required to reach a level of utility V(p,w), when the price ∧ level is any arbitrary vector p Formally: ∧ e( p, v( p, w) This function gives you the income necessary to achieve ∧ the utility v(p,m) when the price level is p ∧ This measure can be constructed for any p . But in particular, since we are interested in the change of welfare associated to a particular price change, we can write this measure either for the initial price vector or for the final price vector. o 1 Let p and p be the initial and final price vector Let u = v( p , w) and u = v( p , w) o o 1 1 Then we can write two measure of consumer welfare: • EQUIVALENT VARIATION (EV) EV ( p o , p1 , w) = e( p o , u1 ) − e( p o , u o ) The equivalent variation tells you the amount of money you need to give the consumer so that he is indifferent between staying in the status quo or accepting the price change. o 1 Note: the term e( p , u ) gives you the income necessary to 1 guarantee the utility level u at the original prices. Loosely speaking, this tells you how much money you should give the consumer so as to bring him to the new level of utility without changing prices. • COMPENSATING VARIATION (CV) CV ( p o , p1 , w) = e( p1 , u1 ) − e( p1 , u o ) The compensating variation tells you the amount of money you should give the consumer to bring him at the level of utility he was enjoying before the price change 1 o Note: the term e( p , u ) gives you the money necessary to guarantee the initial level of utility at the new prices. Question: Can you represent graphically the EV and CV? Consumer surplus and money-metric utility measures Another common measure of the consumer welfare is the consumer surplus, i.e. the integral of the marshallian demand. DRAW A MARSHALLIAN DEMAND AND SHOW THE CONSUMER SURPLUS! QUESTION: how do consumer surplus, EV and CV relate? HINT: • Consider the definition of EV and CV • Apply Shepard’s lemma RESULT? We can re-write the EV and CV in terms of hicksian demand: po EV ( p , p , w) = e( p , u ) − e( p , u ) = ∫ 1 h( p, u1 )dp o 1 o 1 o o p po CV ( p , p , w) = e( p , u ) − e( p , u ) = ∫ 1 h( p, u o )dp o 1 1 1 1 o p Now remember that from Slutzky decomposition we know the slope of hicksian and walrasian demand…. • DRAW ON THE SAME GRAPH HICKSIAN AND WALRASIAN • SHOW CONSUMER SURPLUS, EV, CV Consumer welfare measures and income effect • Remember from slutzky decomposition that the difference in slope between hicksian and walrasian demand function is due to the income effect • What happens if income effect is equal to zero? • Quasi-linear preferences are a special case where the income effect, at least for “high” income level is equal to zero Quasi-linear function U ( x1 , x 2 ) = x1 + u ( x 2 ) where u ( x 2 ) is a strictly concave function Property: when the income effect is equal to zero we have that Consumer Surplus=EV=CV EXERCICE: show if this property holds for a quasi-linear utility function.