Properties of Demand Functions Comparative statics analysis of ordinary demand functions -- the study of how ordinary demands x 1 *(p 1,p 2,m) and x 2 *(p 1,p 2,m) change as prices p 1, p 2 and income m change. Own-Price changes: How does x 1 *(p 1,p 2,m) change as p 1 changes, holding p 2 and m constant?

x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 Fixed p 2 and m.

Own-Price Changes The curve containing all the utility- maximizing bundles traced out as p 1 changes, with p 2 and m constant, is the p 1 - price offer curve. The plot of the x 1 -coordinate of the p 1 - price offer curve against p 1 is the ordinary demand curve for commodity 1.

Own-Price Changes: Cobb-Douglas Take Then the ordinary demand functions for commodities 1 and 2 are What does the demand curve look like? What does a p 1 price-offer curve look like?

x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 is Fixed p 2 and m.

Own-Price Changes: Perfect complements Take Then the ordinary demand functions for commodities 1 and 2 are What does the demand curve look like? What is the p 1 price-offer curve?

p1p1 x1*x1* Ordinary demand curve for commodity 1 is Fixed p 2 and m. Own-Price Changes x1x1 x2x2 p1’p1’ p 1 ’’ p 1 ’’’ m/p 2

Own-Price Changes: Perfect Substitutes Take Then the ordinary demand functions for commodities 1 and 2 are What does the demand curve look like? What is the p 1 price-offer curve?

Own-Price Changes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and m. p1’p1’ p 2 = p 1 ’’ p 1 ’’’ p 1 price offer curve Ordinary demand curve for commodity 1

Income Changes How does the value of x 1 *(p 1,p 2,m) change as m changes, holding both p 1 and p 2 constant? A plot of quantity demanded against income is called an Engel curve. A plot of bundles chosen as we vary income is called the income-offer curve. Draw these two curves for Perfect complements, Cobb-Douglas, and Perfect Substitutes.

Income Changes In every example so far the income offer curves have all been straight lines for the origin? Q: Is this true in general? A: No. Income offer curves are straight lines only if the consumer’s preferences are homothetic.

Homotheticity A consumer’s preferences are homothetic if and only if for every k > 0. That is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin.  (x 1,x 2 ) (y 1,y 2 ) (kx 1,kx 2 ) (ky 1,ky 2 ) 

Why the connection? Take a budget and choice (x1*,x2*). Double the budget so new budget is 2m. Notice if (x1,x2) were in the old budget, (2 x1, 2 x2) is in the new budget. Homotheticity implies that if (x1*,x2*)> (x1,x2) then (2x1*,2x2*)>(2x1,2x2), thus new choice is (2x1*,2x2*). Notice if you draw a line from origin to (x1*,x2*) it goes through (2x1*,2x2*). Since these are the choices, the MRS must be the same at both points to be tangent to the ICs.

Homogeneous Utility Functions U(k x1,k x2)= g(k) u(x1,x2) and g(k)>0. This implies homotheticity! U(x1,x2)>U(y1,y2) => g(k) U(x1,x2)> g(k) U(y1,y2) => U(k x1,k x2)>U(k y1,k y2) Does Cobb-Douglas, Perfect Substitutes, Perfect Complements satisfy this?

Income Effects -- A Nonhomothetic Example Quasilinear preferences are not homothetic. For example, U(1,0)>U(0,3/4) but when k=4, we have U(4,0)<U(0,3)!

Income Effects A good for which quantity demanded rises with income is called normal. Therefore a normal good’s Engel curve is positively sloped. A good for which quantity demanded falls as income increases is called income inferior. Therefore an income inferior good’s Engel curve is negatively sloped. A luxury good is where the proportion of income spent on it goes up with income. The opposite of a luxury good is a necessary good. Is a necessary good always inferior? Is a inferior good always necessary? Can we have a necessary good w/o a luxury good? Vice-versa? Can we have an inferior good w/o a luxury good? Vice-versa?

Ordinary and Giffen goods. A good is called ordinary if the quantity demanded of it always increases as its own price decreases. If, for some values of its own price, the quantity demanded of a good rises as its own-price increases then the good is called Giffen. Example of a Giffen good – U(x 1,x 2 )=-e -x1 -e -x2

Cross-Price Effects If an increase in p 2 –increases demand for commodity 1 then commodity 1 is a gross substitute for commodity 2. – reduces demand for commodity 1 then commodity 1 is a gross complement for commodity 2. What happens with Cobb-Douglas preferences?