1. Suppose preferences are represented by the Cobb-Douglas utility function, u(x1,x2) = Ax 1 a x 2

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1 Additional questions for chapter 7 1. Suppose preferences are represented by the Cobb-Douglas utility function ux1x2 = Ax 1 a x 2 1-a 0 < a < 1 &A > 0. Assuming an interior solution solve for the Marshallian demand functions. 2. Consider the utility function z = lnax 1 a x 2 1-a solve for the Marshallian demand functions.

2 3. Consider the CES utility function ux 1 x 2 = x 1 p + x 2 P 1/p a. Set up the maximization problem hint: y p1x1 p2x2 0 b. Use the Lagrangian to derive the Marshallian demand functions c. Use the Marshalian demand function to find the indirect utility function. d. Verify Roy s identity applies e. Set up the expenditure minimization problem f. Use the Lagrangian to find the Hicksian demands g. Sho the relation beteen the indirect utility functions and expenditure functions h. Sho the duality beteen the Marshallian and Hicksian Demand functions. Anser: See class

3 Additional questions chapter 8 - solutions 1. Linear Expenditure System: implications of 0-homogeneity and ealth elasticity. Consider the demand function here each component function is respectively defined by: L x l p = b l + γ p jb j l l = 1... L ith the fixed parameters b j γ j j = 1... L. p l 1.1 Sho that the expenditure on each commodity i.e. p l x l is a linear function of the price vector and ealth p. L p l x l = p l x l p = p l b l + γ p jb j l p l L = p l b l + γ l γ l p j b j = γ l + b l p l γ l b l p l γ l b j p j = γ l + 1 γ l b l p l j l γ l b j p j j l This is a linear function ith coefficients γ l on 1 γ l b l on p l and γ l b j on all other p j. 1.2 What happens to commodity expenditure if all the b j parameters are zero? Then expenditure for each l th good is then γ l. Thus expenditure for each good is a fixed proportion of ealth independent of any variation in prices hich ould seem to very unrealistic. The original demand function generalizes Stone-Geary preference and ith the b j = 0 j restriction e have a generalization of Cobb-Douglas. With actual Stone-Geary and Cobb-Douglas preference e have L γ j = 1. Thus Cobb-Douglas requires the expenditure for each good to be a fixed share of ealth. In this context it is noteorthy that Cobb-Douglas is by far the most popular preference relation in the literature. 1.3 Do e need to impose conditions on the parameters b l and γ l l = 1... L to guarantee that x satisfies the condition of 0-homogeneity? No because: x i tp t = b i + γ i tp i t = b i + γ i p i = t 0 x i p L tp j b j = b i + γ i tp i t L p j b j = x i p L p j b j Hence the system of demand functions is HOD-0 for all b i and γ i for i = 1... L. 1

4 1.4 Under hat conditions on the parameters b l and γ l l = 1... L is good l a normal good? A luxury? A necessity? Explain. It might be more informative if e consider this question relative to some fixed price vector p. First calculate the ealth elasticity ε i : ε i = x i p x i p = γ i x i p hich is ell defined as long as x i p > 0. Hence ε i > 0 γ i > 0 hich is the condition for good i to be normal. Unless stated otherise e alays assume that prices are [strictly] positive. Good i is a luxury good if ε i > 1 and a necessity if ε i < 1. Therefore good i is a luxury hen: γ i x i p > 1 γ i > x i p γ i > b i γ L i p j b j 0 > b i γ L i p j b j L b i < γ i p j b j. + γ i Thus b i must be small enough and γ i must be large enough so that the product b i is less than γ i times the sum of all such products for all L goods. This is most easily accomplished by adjusting γ i for a given set of the b parameters {b j } L. On the other hand good i is a necessity hen: L b i > γ i p j b j. This ill generally require a relatively large value of b i and a relatively small value of γ i. 2

5 2. Obtaining a simple Slutsky matrix. For the demand function defined by the to component functions L = 2 x 1 p 1 p 2 = [ + 5p 1 / 3p 1 ] 5 and x 2 p 1 p 2 = 2 + 5p 1 / 3p 2 provide the complete Slutsky matrix. Sho your ork and simplify the elements. With L = 2 the Slutsky matrix is ith representative element Sp = s11 p s 12 p s 21 p s 22 p s ij p = x i p + x j p x i p. p j With this demand function it may be easier if e first calculate the individual partial derivatives: x 1 p = 5 3p p 1 p 1 3p 1 2 = 15p p 1 9 p 1 2 = 3 9 p 1 2 = 3 p 1 2 x 1 p = 0 p 2 x 1 1 p = 3p 1 x 2 p = 10 p 1 3p 2 x 2 p = 2 + 5p 1 p 2 3 p 2 2 x 2 2 p = 3p 2 We then have: s 11 p = x 1 p + x 1 p x 1 p p 1 + = 3 p p p 1 = p 1 p 1 = 2 + 5p 1 9 p 1 2 s 12 p = x 1 p + x 2 p x 1 p p 2 = p 1 1 = 2 + 5p 1 3p 2 3p 1 3p 1 = 3p p p 1 5 3p 1 = 2 9p p 1 3

6 s 21 p = x 2 p + x 1 p x 2 p p 1 = p p 2 3p 1 = 2 + 5p 1 3p 2 = 10 3p p p 2 = p 1 s 22 p = x 2 p + x 2 p x 2 p p 2 = 2 + 5p 1 3 p p 1 2 = 2 + 5p 1 3p 2 3p 2 3 p p 1 9 p 2 2 = 6 + 5p 1 9 p p 1 9 p 2 2 = 2 + 5p 1 9 p 2 2. Therefore the Slutsky matrix is Sp = 2 + 5p 1 9 p p p p 1 9 p 2 2. As expected ith L = 2 it is symmetric. Some students factored the matrix to obtain a solution of the form 1 p p 1 p 2 Sp = 2 + 5p p 1 p 2 1 p 2 2 While technically correct this breaks up the expression of each of the individual substitution elements and therefore should be avoided.. 4