1. Chapter 6 Demand

2. Max U subject to I=p1*x1+p2*x2
X1=x1(p1, p2, m)
X2=x2(p1, p2, m)
Proportional change in p1, p2, and m leads to no change in demand for X1 and X2. To see this, assume that prices and income increase from (p1, p2, m) to (t*p1, t*p2, t*m). This creates no effect on the marginal conditions: MRS=p1/p2 and m=p1*x1+p2*x2.

3. Effect of Income on X1 and X2
Consider two scenarios: (1) higher income leads to higher consumption of X1 (see Figure 6.1); (2) higher income leads to lower consumption of X1 (see Figure 6.2).

4. 06.01

5. 06.02

6. Normal vs Inferior Goods
Xi is normal if higher Income leads to higher consumption of xi (dxi / dm>0)
Xi is inferior if higher Income leads to lower consumption of xi (dxi / dm<0)
The relationship between m and xi yields Engel curve (see the next few diagrams: Figures )

7. 06.03

8. Perfect substitutes: Max a*x1+b*x2 subject to m=p1*x1+p2*x2
The demand for x1: x1=m/p1. Thus, dx1/dm=1/p1. (assuming that a/b>p1/p2)
The demand for x2: x2=0.

9. 06.04

10. Perfect complements: Max min[x1, x2] subject to m=p1*x1+p2*x2
The demand for x1: x1=m/(p1+p2). Thus, dx1/dm=1/(p1+p2).
The demand for x2: x2= m/(p1+p2). Thus, dx2/dm=1/(p1+p2).

11. 06.05

12. Cobb-Douglas: max x1^a x2^(1-a) subject to m=p1*x1+p2*x2
Demand for x1: x1=a*m/p1. The slope of Angel curve for x1 is a/p1.
Demand for x2: x2=(1-a)*m/p2. The slope of Angel curve for x2 is (1-a)/p1.

13. 06.06

14. Cobb-Douglas: max ln(x1)+x2 subject to m=p1*x1+p2*x2
Demand for x1: x1=p2/p1, which is independent of m. Thus, dx1/dm=0.
Demand for x2: x2=m/p2 -1. Thus, dx2/dm=1/p2.

15. 06.08

16. Price Effect
Change in prices affects the relative slope of budget line. Lower p1 implies lower p1/p2.
The price effect on x1 can be either positive or negative, as seen in the next few diagrams (Figures ). If lower p1 leads to higher x1, then x1 is an ordinary good. But if lower p1 leads to lower x1, then x1 is a Giffen good.
Price offer curve (price expansion path) can be derived.
Marshallian demand (with both subs and income effects included) for x1 can be obtained.

17. 06.09

18. 06.10

19. 06.11

20. 06.12

21. 06.13

22. 06.15