Economics 58 Fall 2009 Solutions to PS # 3

Economics 58 Fall 2009 Solutions to PS # 3 1. CES Utility a. MRS = ∂U/ ∂x δ −1 = ( x y ) = px /p y for utility maximization. ∂U/ ∂y Hence, x/y = ( px p y )1 (δ −1) = ( p x p y ) −σ b. If δ = 0, x y = p y p x so p x x = p y y . c. Part a shows p x x p y y = ( px p y )1−σ where σ = 1 (1 − δ ) . Hence, for σ < 1 the relative share of income devoted to good x is positively correlated with its relative price. This is a sign of low substitutability. For σ > 1 the relative share of income devoted to good x is negatively correlated with its relative price – a sign of high substitutability. d. The algebra is a bit tricky here, but worth doing once. Let’s solve for indirect utility −σ −σ ⎛p ⎞ x ⎛⎜ p x ⎞⎟ = or x = y ⎜ x ⎟ ⎜p ⎟ y ⎜⎝ p y ⎟⎠ ⎝ y⎠ Substituting into the budget constraint yields −σ ⎛p ⎞ I = p x y⎜ x ⎟ + p y y ⎜p ⎟ ⎝ y⎠ Similarly, Ip −σ x = 1−σ x 1−σ px + p y or y= Ip −y σ p 1x−σ + p1y−σ ⎛ p x−σ ⎜ Hence, δU = x + y = I ⎜ p1−σ + p1−σ y ⎝ x δ δ δ δ ⎞ ⎛ p −σ ⎟ + I δ ⎜ 1−σ y 1−σ ⎟ ⎜p +p y ⎠ ⎝ x ⎞ ⎛ 1 ⎟ or Now − δσ = 1 − σ so δU = I δ ⎜ 1−σ 1 − σ δ − 1 ⎜(p + p ) ⎟ x y ⎠ ⎝ ⎞ ⎟ ⎟ ⎠ δ V ' = I ( p1x−σ + p1y−σ ) −1 1−σ where V ' = (δU ) 1 δ This is the indirect utility function. Clearly it is homogeneous of degree zero in income and prices. Inverting the expression yields the expenditure function: E = I = V ' ( p1x−σ + p1y−σ ) 1 (1−σ ) Clearly this is homogeneous of degree one in the prices. Note that the odd form for V’ here suggests the use of the CES form given in Problem 4.13 in applications involving these functions. 2. CES Indirect Utility and Expenditure Functions a. See prior problem b. Scale all variables by t and the function is unchanged. c. The partial derivative of V w.r.t. I is positive as the prices are positive. d. Again, partial derivatives of V w.r.t. the prices are both negative: for example, ∂V ∂p x = − Ip xr −1 ( p xr + p ry ) − (1+ r ) / r < 0 . e. Simply reversing the positions of V and I in the indirect utility function yields E = V ( p xr + p ry )1 / r . f. Multiplying prices by any factor t multiplies expenditures by t. g. For example, ∂E / ∂p x = Vp xr −1 ( p xr + p ry ) (1− r ) / r > 0 . h. ∂ 2 E / ∂p x2 = (1 − r ) ⋅ Vp x2 r −2 K (1−2 r ) / r + ( r − 1)Vp xr −2 K (1−r ) / r where K = ( p xr + p ry ) . Division of this expression by Vp xr −2 K (1−r ) / r yields ( r − 1)(1 − k ) < 0 where k = p xr / K < 1 . 3. a. Utility maximization requires pb = 2j and the budget constraint is .05pb +.1j = 3. Substitution gives pb = 30, j = 15 b. If pj = $.15 substitution now yields j = 12, pb = 24. c. To continue buying j = 15, pb = 30, David would need to buy 3 more ounces of jelly and 6 more ounces of peanut butter. This would require an increase in income of: 3(.15) + 6(.05) = .75. d. 4. e. Since David N. uses only peanut better and jelly to make sandwiches (in fixed proportions), and because bread is free, it is just as though he buys the good ‘sandwiches’, where psandwich = 2ppb + pj. In part a, ps = .20, qs = 15; In part b, ps = .25, qs = 12; In general, 3 so the demand curve for sandwiches is a hyperbola. qs = ps f. There is no substitution effect due to the fixed proportion. A change in price results in only an income effect. Hausman’s terminology refers to the terms in the Taylor expansion of the expenditure function. For the introduction of new goods, price falls from p* to p1 and the reduction in necessary expenditure is represented by the first order term in the Taylor expansion (notice that this happens because the envelope theorem ∂E shows that = x c (which is denoted by hu in the paper. Substitution bias ∂p x occurs because people change what they buy in response to changing prices. Such reactions to changing prices are captured by the second order term in the ∂x c ∂ 2 E Taylor expansion – that is by the term in = . ∂p x ∂p x2 Consumer Surplus is given by the area under the Hicksian demand curve in Figure 1. When price falls from p2 to p1, the total gain in consumer surplus is given by the rectangular area plus the shaded triangle. The gain in Consumer surplus from the substitution effect is given only by the area of the shaded triangle. Hence, Hausman argues, focusing on substitution bias misses a lot – especially when it comes to new goods.