Economics 401: Test 1
Thursday 27th January 2011
Answer all questions. Each is worth 10 marks.
1. What are the two main approaches to consumer behaviour in economics? What
elements do they have in common, and how do they differ? What restrictions do they place
on a two-good system of ordinary demand equations?
x1 (p1 , p2 , w)
x2 (p1 , p2 , w)
ANSWER
The two approaches are the choice-based approach and the preference-based approach.
The first is founded on the weak axiom of revealed preference; the second on preference
relations. Both approaches assume standard budget constraints in Rn+ and therefore are
consistent with the implications of imposing such budget constraints. The CB approach assumes the consumer’s choices are consistent with the weak axiom. The PB approach assumes
the consumer’s choices obey the many assumptions imposed on preference relations (reflexivity, completeness, transitivity, continuity, monotonicity or local non-satiation, convexity,
etc.).
The restrictions on the demand system include the following.
Walras’s Law
p1 x1 (p1 , p2 , w) + p2 x2 (p1 , p2 , w) = w
Homogeneity
for all α > 0,
xj (αp1 , αp2 , αw) = xj (p1 , p2 , w) ,
j = 1, 2
Denoting the Slutsky substitution matrix by S, these identities imply
S11 S12
p1 p2
= 0 0
S21 S22
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and
S11 S12
S21 S22
p1
p2
=
0
0
2. Let X be a choice set and (B, C (·)) be a choice structure on X.
(a) Using mathematics, define the weak axiom of revealed preference (WARP).
(b) Do each of the following satisfy WARP? Justify each answer.
(i) X = {a, b, c},B = {B1 , B2 , B3 } , B1 = {a, b} , B2 = {b, c} , B3 = {c, a} , C (B1 ) =
{a} , C (B2 ) = {b} , C (B3 ) = {c};
(ii) X = {a, b, c},B = {B1 , B2 } , B1 = {a, b} , B2 = {a, b, c} , C (B1 ) = {a} , C (B2 ) =
{a, b};
(iii) X = {a, b, c, d},B = {B1 , B2 } , B1 = {a, b, c} , B2 = {a, b, d} , C (B1 ) = {a, c} , C (B2 ) =
{a, d};
ANSWER
(a) WARP can be defined as:
(*) Assume B1 ∈ B, B2 ∈ B, {a, b} ⊂ B1 , {a, b} ⊂ B2 then
a ∈ C(B1 ) and b ∈ C(B2 ) ⇒ a ∈ C(B2 ).
Given this definition it can be proved that the following definition (**) is equivalent to
(*).
(**) Assume B1 ∈ B, B2 ∈ B, {a, b} ⊂ B1 , {a, b} ⊂ B2 then
a ∈ C(B1 ) and b ∈ C(B2 ) ⇒ {a, b} ⊂ C(B1 ) and {a, b} ⊂ C(B2 ).
(b) Now consider (i) and (iii). Here WARP is satisfied trivially because the assumptions
of WARP are not met. In (i) we don’t have two budget sets with two elements in common.
In (iii) we do have two budget sets with two elements in common but then we get stuck on
the next assumption of WARP because one of the two elements in common, b, is not a best
element in either budget set. For (ii), we can see that WARP does not hold because the (**)
definition of WARP is contradicted.
3. Answer both parts.
(a) What does it mean to say that a utility function, u(·), represents a preference relation
on some choice set X? Prove that if u(·) represents preference relation %, this preference
relation must be complete and transitive.
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(b) Now suppose X = R2+ and (x11 , x12 ) (x21 , x22 ) when x11 > x21 , or x11 = x21 and x12 > x22 .
Is this preference relation complete, transitive and continuous? Defend your answers.
ANSWER
(a) A utility function u : X → R represents a preference relation % on the choice set X
if
∀x, y ∈ X, x % y ⇔ u(x) ≥ u(y).
Completeness: We need to show that ∀x, y ∈ X, either x % y or y % x or both. Now
note that u(x) and u(y) are real numbers so either u(x) ≥ u(y) in which case x % y or
u(y) ≥ u(x) in which case y % x; if u(x) = u(y) we know both are true.
Transitivity: Suppose x, y, z ∈ X, and x % y and y % z. We need to show x % z. Since
u represents %, u(x) ≥ u(y) and u(y) ≥ u(z). Since these are real numbers u(x) ≥ u(z) ⇒
x % z.
(b) This is an example of lexicographic preferences where good 1 is the dominant good.
Lexicographic preferences are complete and transitive but not continuous.
Completeness: Consider any two distinct points in R2+ — (x11 , x12 ) (point a) and (x21 , x22 )
(point b). If x11 > x21 , a b. If x21 > x11 , b a. If x11 = x21 , since a and b are distinct, it must
be that either x12 > x22 in which case a b or x22 > x12 in which case b a.
Transitivity: Let a, b, c ∈ R2+ where a = (x11 , x12 ), b = (x21 , x22 ) and c = (x31 , x32 ). Suppose
a b and b c. We want to prove a c. Now notice that a b and b c ⇒ x11 ≥ x31 .
There are two cases (i) and (ii). In case (i), x11 = x31 and then we know that x11 = x21 = x31 in
which case it must be that x12 > x22 > x32 , and thus x12 > x32 which means a c. In case (ii)
x11 > x31 and then immediately we know a c.
Continuity: Suppose
{xn } ⊂ R2+ and
lim
x =x
n→∞ n
{yn } ⊂ R2+ and
lim
y =y
n→∞ n
and
and xn yn , ∀n. Then if were continuous we would be able to deduce that x y. Here is
one counterexample. Let xn = (1/n, 0) and yn = (0, 1). Then x = (0, 0), y = (0, 1), xn yn
∀n but y x. So lexicographic preferences are complete and transitive but they violate
continuity.
4. Consider a price-taking consumer in a two-good world. Let w denote money wealth,
(p1 , p2 ) prices, and xj (p1 , p2 , w) , j = 1, 2 ordinary demand functions. Fill in the following
table as precisely as you can for someone whose preferences are represented by:
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u (x1 , x2 ) = min (2x1 + x2 , x1 + 2x2 ) .
Hint: Draw the u (x1 , x2 ) = 12 indifference line first.
x1 (p1 , p2 , w) x2 (p1 , p2 , w)
Range
ANSWER
u (x1 , x2 ) = min (2x1 + x2 , x1 + 2x2 ) means
2x1 + x2 if 2x1 + x2 ≤ x1 + 2x2 or x1 ≤ x2
u (x1 , x2 ) =
x1 + 2x2 if x1 > x2
Thus the equations for the u (x1 , x2 ) = 12 indifference curve could be written as
12 = 2x1 + x2 if x1 < x2
12 = 2x1 + x2 if x1 = x2 or x1 = x2 = 4
12 = x1 + 2x2 if x1 > x2
So the u = 12 indiference curve starts on the good 1 axis at (12, 0); moves along a straight
line with a |slope| = 1/2 to (4, 4); then along another line segment with a |slope| = 2 from
(4, 4) to the good 2 axis at (0, 12).
With this image in mind we can see the demand equations are:
x1 (p1 , p2 , w)
x2 (p1 , p2 , w)
w/p1
0
p1 x1 (p1 , p2 , w) + p2 x2 (p1 , p2 , w) = w
w/ (p1 + p2 )
w/ (p1 + p2 )
p1 x1 (p1 , p2 , w) + p2 x2 (p1 , p2 , w) = w
0
w/p2
Range
p1 /p2 < 1/2
p1 /p2 = 1/2
1/2 < p1 /p2 < 2
p1 /p2 = 2
2 < p1 /p2
5. This question applies what we have done in class this term to think about a carbon
tax. Suppose a typical Canadian household currently buys 200 litres of gasoline per month
at $1 per litre and spends $1800 per month buying goods (and services) other than gasoline.
Suppose the government is considering a carbon tax that will raise the price of gasoline
from $1 to $2 per litre. If this household’s bahaviour is consistent with WARP, and the
government compensates this household for having to pay the tax by giving it a tax rebate
of $200 per month, state as precisely as you can how much gasoline the household will
choose to purchase each month with the carbon tax and cash rebate in effect. How would
your answer change if you assumed the household’s choices were consistent with maximizing
a Cobb-Douglas utility function in the two goods — “gasoline” and “dollars spent on goods
other than gasoline”?
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ANSWER
Before the carbon tax the household is spending $1800 on things other than gas and $200
on gas, so its monthly budget is $2000. After the carbon tax and the rebate its monthly
income is 2000 + 200 = $2200. So it has just enough money to buy the original bundle of
goods — 200 litres of gas at $2.00 per litre plus $1800 of other stuff. This fits the definition
of a compensated price change and if WARP holds it must be that gas purchased ≤ 200
litres.
If we assume the household’s utility function is Cobb Douglas then
u (gas,$) = gasa $1−a , for some 0 < a < 1 and
the demand for gas = a times wealth/price of gas
Thus the initial data identify a = 1/10 through
2000
1
With the carbon tax and tax rebate in place we have
gas = 200 = a
1 2200
2200
=
= 110.
2
10 2
This shows that if the household’s choices were consistent with maximizing a CD utility
function it would cut its gas purchases from 200 litres per month to 110 litres per month
with the carbon-tax-rebate plan.
gas = a
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