1. Tutorial 2: Demand
Matthew Robson

2. Introduction There are 2 Goods; Good 1 and Good 2.
-The price of Good 2 is 1, The price of Good 1 is p.
-The income of your team is m.
-The quantities of the 2 Goods demanded by your team are q1 and q2.
-Note that p1q1 + q2 = m.
Your team should decide what your tastes concerning the two goods are. These tastes must be of one of the following three types:
Type 1: Perfect Substitutes
Type 2: Perfect Complements
Type 3: Cobb-Douglas
Once you have chosen a Type, you must choose an a. 1

3. The Game What are your demands, q1 and q2, if p is … and m is …?
Now each team has a Type and an a you must pose the following question to the opposite team:
What are your demands, q1 and q2, if p is … and m is …?
Each team must answer truthfully.
The first team to guess the Type and a of the other team correctly wins. 2

4. Type 1: Perfect Substitutes
-One Unit of Good 1 is always substitutable for, or by, any a units of Good 2
U(q1, q2) = q1 + q2/a
-0<a<infinity
Here a = 1 3

5. Type 1: Perfect Substitutes
-One Unit of Good 1 is always substitutable for, or by, any a units of Good 2
U(q1, q2) = q1 + q2/a
-Here a = 1
-p1<1 4

6. Type 1: Perfect Substitutes
-One Unit of Good 1 is always substitutable for, or by, any a units of Good 2
U(q1, q2) = q1 + q2/a
-Here a = 1
-p1>1 5

7. Type 2: Perfect Complements
-One Unit of Good 1 is always needed to be complemented by a units of Good 2
U(q1, q2) = min(q1, q2/a)
0<a<infinity
Here a = 1 6

8. Type 2: Perfect Complements
-One Unit of Good 1 is always needed to be complemented by a units of Good 2
U(q1, q2) = min(q1, q2/a)
-0<a<infinity
Here a = 1
p1 = 1 7

9. Type 3: Cobb-Douglas
-With Relative Weights a for Good 1 and (1-a) for Good 2
-U(q1, q2) = 𝑞 1 𝑎 𝑞 2 1−𝑎
-0<a<1
-Here a = 0.5 8

10. Type 3: Cobb-Douglas
-With Relative Weights a for Good 1 and (1-a) for Good 2
-U(q1, q2) = 𝑞 1 𝑎 𝑞 2 1−𝑎
-0<a<1
-Here a = 0.5
- p1 = 1 9

11. Solutions Perfect Substitutes If P1 < a  q1 = m/p1  q2 = 0
If P1 > a  q1 = 0  q2 = m
If a = p1  q1 = Any  q2 = Any  Such that p1q1 + q2 = m
Perfect Complements
q1 = m/(p1+a)
q2 = am/(p1+a)
Cobb-Douglas
q1 = am/p1
q2 = (1-a)m 10

12. Example 1
Round m p1 p2 q1 q2 1
100
-Which Type would this be? 11

13. Example 1
Round m p1 p2 q1 q2 1
100 2 5 3
33.33 4 20
-Given they are Type 1: Perfect Substitutes, what is their a? 12

14. Example 2 -Which Type would this be? Round m p1 p2 q1 q2 1 100 80 20 25 ?
-Which Type would this be? 13

15. Example 2 & 3 Round m p1 p2 q1 q2 1 100 80 20 2 5 19.05 4.76 Round m16 14

16. Example 2 & 3 -Perfect Complements, a = 0.25 -Cobb-Douglas, a = 0.8
Round m p1 p2 q1 q2 1
100 80 20 2 5
19.05
4.76
-Perfect Complements, a = 0.25
Round m p1 p2 q1 q2 1
100 80 20 2 5 16
-Cobb-Douglas, a = 0.8 15

17. Discussion Point
After asking one question with particular values for m and p, does it help to have a further question with the same value of p, but a different for m? Why or why not? 16

18. Relationship Between q1, q2 and m
-What happens if we hold p constant and change m?
-All linear relationships, not useful for determining differences between preferences 17

19. Relationship Between q1, q2 and p1
-What happens if we hold m constant and change p?
-Different relationships between different preferences 18

20. Relationship Between q1, q2 and p1
Perfect Substitutes
As p1 rises from 0, q1 falls (while q2 remains at 0) until a particular value (??) when q1 drops to 0 and remains there, while q2 becomes a non-zero constant
Perfect Complements
As p1 rises q1 and q2 both fall in such a way that the ratio q2/q1 remains constant
Cobb-Douglas
As p1 rises q1 falls and q2 remains constant, in such a way that the ratio of expenditures p2q2/p1q1 remains constant 19

21. Discussion Point What if a fourth possibility was added to the list?
-Stone-Geary preferences, with parameters s1, s2 and a.
-Would discovering these preferences take longer than the other types?
-If you had decided the preference was Stone-Geary, how many observations, and of what type, would be needed to find the values of the three parameters? 20

22. Type 4: Stone-Geary U(q1, q2) = (𝑞1 −𝑠1) 𝑎 (𝑞2 −𝑠2) 1−𝑎 -0<a<1
-Extension of Cobb-Douglas. Where individuals have to consume a subsistence level of goods first, and then act according to Cobb-Douglas
U(q1, q2) = (𝑞1 −𝑠1) 𝑎 (𝑞2 −𝑠2) 1−𝑎
-0<a<1 21

23. Example 4 – Stone-Geary
Round m p1 p2 q1 q2 1
100 66 34 2 5
14.8 26 3 6
12.67 24
-At which round can we rule out each preference?
-Can we confirm it’s Stone-Geary, and what are the parameters; a, s1 and s2 22

24. Example 4 – Stone-Geary
Round m p1 p2 q1 q2 1
100 66 34 2 5
14.8 26 3 6
12.67 24
-At which round can we rule out each preference?
-1st Round not Type 1: Perfect Substitutes
-2nd Round not Perfect Complements (not fixed proportions) or Cobb-Douglas (not constant proportion of income)
-Can we confirm it’s Stone-Geary, and what are the parameters; a, s1 and s2
q1 = s1+a(m-p1s1-s2)/p1
q2 = s2+(1-a)(m- p1s1 - s2)
a = 0.8, s1 = 10, s2 = 20 23

25. Discussion Point
Now suppose that we make the exercise more realistic by adding in noise.
Suppose that demands are determined by some preferences, but there is noise in the expression of the demands. So the true demand, as given by the preferences, may be q, but the actually expressed demand may be q±e where e is some noise, some imprecision.
Allow e to take the values -1, 0 or 1 each with equal probability. Let us suppose that the tutor checks that e is indeed random (how?). How does this change the difficulty of inferring the true preferences? 24

26. Example 5
Round m p1 p2 q1 q2 1
100 79 21 2 81 20 3 4 80 19
-Given that we know the distribution of the error is uniform, and can take on {-1, 0, 1}
-We know that the true values are: q1 = 80, q2 = 20
-Therefore we can infer they are Cobb-Douglas with a = 0.8 or Perfect Complements with a = 0.25
-What if the error was more complex?? 25