12 Chapter Highlights 1)can use any monotonic transformation of this as well, such as b)perfect complements- what matters is the minimum of the left and right shoes you have, so u(a1, r2)=minfal, r2) works c)quasilinear preferences - indifference curves are vertically parallel Figure 4.4 1)utility function has form u(a1, I2)=v(a1)+a2 Cobb-Dougl glas preferences. Figure 4.5 1)utility has form u(r1, I2)=zfrs 2)convenient to take transformation f(u)=ut and write erg +e 3 or rfa2, where a=b/(b+c) E Marginal utility extra utility from some extra consumption of one of the goods, holding the fixed 2. this is a derivative, but a special kind of derivative a partial derivative 3. this just means that you look at the derivative of u(a1, r2) keeping r2 fixed treating it like a constant a)if u(a1, I2)=51+a2, then MU1=au/ax1=1 b)if u( rar2, then MU1 5. note that marginal utility depends on which utility function you choose to represent preferences a) if you multiply utility times 2, you multiply marginal utility times 2 b)thus it is not an operational concept c)however, MU is closely related to MRS, which is an operational co concept 6. relationship between MU and MRS a)u(a1, a2)=k, where k is a constant, describes an indifference curve b)we want to measure slope of indifference curve, the MRS c)so consider a change(dr1, dr2) that keeps utility constant. Then MU1dx1 MU2dx2=0 dx1+-dx2=0 MUI e)so we can compute MRS from knowing the utility function F Example 1. take a bus or take a car to work? 2. let I1 be the time of taking a car, yi be the time of taking a bus. Let az be cost of car. etc 3. suppose utility function takes linear form U(I,., In)=B1I1+.+BnIn 4. we can observe a number of choices and use statistical techniques to estimate the parameters Bi that best describe choice 5. one study that did this could forecast the actual choice over 93% of the 6. once we have the utility function we can do many things with it a)calculate the marginal rate of substitution between two characteristics 1)how much money would the average consumer give up in order to get a shorter travel time? b)forecast consumer response to proposed changes c)estimate whether proposed change is worthwhile in a benefit-cost sense12 Chapter Highlights 1) can use any monotonic transformation of this as well, such as log (x1 + x2) b) perfect complements — what matters is the minimum of the left and right shoes you have, so u(x1, x2) = min{x1, x2} works c) quasilinear preferences — indifference curves are vertically parallel. Figure 4.4. 1) utility function has form u(x1, x2) = v(x1) + x2 d) Cobb-Douglas preferences. Figure 4.5. 1) utility has form u(x1, x2) = xb 1xc 2 2) convenient to take transformation f(u) = u 1 b+c and write x b b+c 1 x c b+c 2 3) or xa 1x1−a 2 , where a = b/(b + c) E. Marginal utility 1. extra utility from some extra consumption of one of the goods, holding the other good fixed 2. this is a derivative, but a special kind of derivative — a partial derivative 3. this just means that you look at the derivative of u(x1, x2) keeping x2 fixed — treating it like a constant 4. examples a) if u(x1, x2) = x1 + x2, then MU1 = ∂u/∂x1 = 1 b) if u(x1, x2) = xa 1x1−a 2 , then MU1 = ∂u/∂x1 = axa−1 1 x1−a 2 5. note that marginal utility depends on which utility function you choose to represent preferences a) if you multiply utility times 2, you multiply marginal utility times 2 b) thus it is not an operational concept c) however, MU is closely related to MRS, which is an operational concept 6. relationship between MU and MRS a) u(x1, x2) = k, where k is a constant, describes an indifference curve b) we want to measure slope of indifference curve, the MRS c) so consider a change (dx1, dx2) that keeps utility constant. Then MU1dx1 + MU2dx2 = 0 ∂u ∂x1 dx1 + ∂u ∂x2 dx2 = 0 d) hence dx2 dx1 = −MU1 MU2 e) so we can compute MRS from knowing the utility function F. Example 1. take a bus or take a car to work? 2. let x1 be the time of taking a car, y1 be the time of taking a bus. Let x2 be cost of car, etc. 3. suppose utility function takes linear form U(x1,...,xn) = β1x1+...+βnxn 4. we can observe a number of choices and use statistical techniques to estimate the parameters βi that best describe choices 5. one study that did this could forecast the actual choice over 93% of the time 6. once we have the utility function we can do many things with it: a) calculate the marginal rate of substitution between two characteristics 1) how much money would the average consumer give up in order to get a shorter travel time? b) forecast consumer response to proposed changes c) estimate whether proposed change is worthwhile in a benefit-cost sense