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Unformatted text preview: EE263 S. Lall 2011.04.11.04 Problem Session 2 Solutions 1. Price elasticity of demand. 43010 The demand for n different goods is a function of their prices: q = f ( p ) , where p is the price vector, q is the demand vector, and f : R n → R n is the demand function. The current price and demand are denoted p ∗ and q ∗ , respectively. Now suppose there is a small price change δp , so p = p ∗ + δp . This induces a change in demand, to q ≈ q ∗ + δq , where δq ≈ Df ( p ∗ ) δp, where Df is the derivative or Jacobian of f , with entries Df ( p ∗ ) ij = ∂f i ∂p j ( p ∗ ) . This is usually rewritten in term of the elasticity matrix E , with entries E ij = ∂f i ∂p j ( p ∗ ) 1 /q ∗ i 1 /p ∗ j , so E ij gives the relative change in demand for good i per relative change in price j . Defining the vector y of relative demand changes, and the vector x of relative price changes, y i = δq i q ∗ i , x j = δp j p ∗ j , we have the linear model y = Ex . Here are the questions: (a) What is a reasonable assumption about the diagonal elements E ii of the elasticity matrix? (b) Goods i and j are called substitutes if they provide a similar service or other sat- isfaction ( e.g. , train tickets and bus tickets, cake and pie, etc.). They are called complements if they tend to be used together ( e.g. , automobiles and gasoline, left and right shoes, etc.). For each of these two generic situations, what can you say about E ij and E ji ? (c) Suppose the price elasticity of demand matrix for two goods is E = bracketleftbigg − 1 − 1 − 1 − 1 bracketrightbigg . Describe the nullspace of E , and give an interpretation (in one or two sentences). What kind of goods could have such an elasticity matrix? Solution. (a) The i th diagonal entry of E relates y i to x i , i.e. , the demand for the i th good to its price. When the price of a product is increased, and all other prices are held constant, the demand for that product can be expected to decrease. Hence, the diagonal elements of E should be negative. (Whether any good with a positive elasticity exists at all is a debated question, but most economists’ answer is no.) (b) The entry E ij relates the demand for good i to the price of good j . A price increase in good j leads to a decreased demand for that good. If good i is a substitute , it also leads to an increased demand for good i , as some of the consumption switches to 1 EE263 S. Lall 2011.04.11.04 what now seems a more attractive price. Hence, E ij is positive. The same argument tells us that E ji is positive when goods i and j are substitutes. If the goods are complements , the converse is true. Since the consumption of good i is associated with comsumption of good j , a decreased demand for good i follows from the decreased demand for good j . Hence, E ij (and E ji ) is negative....
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This note was uploaded on 07/28/2011 for the course EE 263 taught by Professor Boyd,s during the Summer '08 term at Stanford.
- Summer '08