Microecon: optimal choice and demand?

Question

a.Anita’s preference for nutella (#1) and books (#2) is given by the utility function u(x1,x2)= a*ln x1 +b*x2. It's an example of quasilinear prefs, where utility is linear in one good and non-linear in the other. Assume prices, income denoted by (p1, p2, y).

(i)Anita’s MRS? Interpret expression you get.

(ii)The prefs form convex set, utility function is smooth &Anita consumes positive amounts of both. Derive her ordinary demand funtions for nutella and books using the substitution method. In other words, find x1* (p1,p2,y) and x2* (p1,p2,y). (Hint: Use budget equation to substitute x2 into utility function &maximize this new utility w/respect to independent variable x1).

(iii)What does the ordinary demand of x1 depend on? x2?

(iv)Are nutella &books ordinary goods? Normal goods? Support answer w/appropriate derivatives. Interpret the derivatives

(v)Are nutella and books complements? Substitutes? Support your answer w/appropriate derivatives. Interpret the derivatives

Answer 1

Good lord economics is a silly science. All those formulas and still the real world keeps surprising economists.

Talk about LOL.

Answer 2

ii) this is how we did it in my class:

start with income function

Y=p1x1+p2x2

x2=(Y-p1x1)/p2

then substitute that into the utility function:

U=a ln x1 + b((Y-p1x1)/p2)

then take the derivative:

dU/dx1=a/x1 - (b*p1)/p2

foc: a/x1=(b*p1)/p2

x1=(a*p2)/(b*p1)

that's your demand curve for good 1

then substitute that into the x1 function

x2=(Y-p1(a*p2/b))/p2

iii) you should be able to do that now that you have the demand curve

iv) take the derivative of the demand functions with respect to the price of the good

v) take the derivative of the demand functions with respect to the price of the other good

EDIT: for part iv, to find whether it's an ordinary good, you take the derivative of the demand curve because you want to know what will happen to quantity demanded with a change in price.  if the derivative is negative, it's an ordinary good; if it's positive, it's a giffen good