Consumer Theory

Question 1

What does an indifference curve contain?

Answer 1

It contains equally preferred bundles.
Equal preference = same utility level.
Therefore, all bundles in an indifference curve have the same utility level.

Question 2

What is the difference between goods, bads and neutrals?

Answer 2

A good is a commodity unit which increases utility (gives a more preferred bundle).
A bad is a commodity unit which decreases utility (gives a less preferred bundle).
A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

Question 3

What do perfectly complementary indifference curves look like?

Answer 3

L-shaped.

Question 4

What do Cobb-Douglas indifference curves look like?

Answer 4

All curves are hyperbolic, asymptoting to, but never touching the axis.

Question 5

What is the marginal utility of commodity “I”?

Answer 5

The rate-of-change of total utility as the quantity of commodity “i” consumed changes:
Mui = ∂U/∂xi

Question 6

What is the general equation for an indifference curve?

Answer 6

U(x1, x2) = k
Totally differentiating this identity gives:
(∂U/∂x1)dxi + (∂U/∂x2)dx2 = 0

Question 7

What is the Marginal rate of substitution?

Answer 7

dx2/dx1 = -(∂U/∂x1)/(∂U/∂x2) = - x2/x1

Question 8

What is a consumer’s ordinary demand?

Answer 8

This is the most preferred affordable bundle at the given prices and budget.
Ordinary demands are denoted by:
x1*(p1, p2, m) and x2*(p1, p2, m)
OR
q1*(p1, p2, Y) and q2*(p1, p2, Y)

Question 9

What happens when xi* > 0 (or q1i*)?

Answer 9

e.g. x1* > 0 or x2* > 0 etc., then the demanded bundle is interior.

Question 10

What happens when xi* = 0 (or q1i*)?

Answer 10

If x1* = 0 or x2* = 0 etc., then the ordinary demand (x1, x2) is at a corner solutions to the problem of maximising utility subject to a budget constraint.
e.g. perfect substitutes case

Question 11

When is the budget exhausted?

Answer 11

If buying (x1, x2) costs $m.

Question 12

What 2 conditions does (x1, x2) satisfy?

Answer 12

1. The budget is exhausted: p1x1* + p2x2* = m
2. The slope of the budget constraint (p1/p2) and the slope of the indifference curve containing (x1, x2) are equal at (x1, x2). AKA they are tangent.

Question 13

How do you derive the optimal demands/solutions?

Answer 13

Through the substitution method or the lagrangian method. However, for the exam you will have to know the langrangian method.

Question 14

What is the optimality condition for an interior solution (q1 > 0, q2 > 0)?

Answer 14

MRS = -U1/U2 = - p1/p2 = MRT

At an interior solution, the slope of the IC curve equals the slope of the budget line (tangency condititon).

Question 15

What is the lagrangain method?

Answer 15

It is used to solve equality constraint optimisation (either max or min) problems. It can be generalized to handle problems with more choice variables and more constraints.

Question 16

How do you use the lagrangian method to derive optimal demands?

Answer 16

The first step is to set up the Lagrangian function (or L) which is the sum of the original objective 
function O(c1, c2), where c1 and c2 are the choice variables, and the left-hand side of the constraint Y – f(c1, c2) = 0 multiplied by the constant λ. The lagrangian function has to be maximized (for a max 
problem) w.r.t. c1, c2 and λ.

1. Set up the lagrange function
2. Compute the FOCs
3. Find the standard tangency condition and solve for either q1 or q2
4. Sub q1 or q2 into budget constraint (Eq 3 from FOC) to find optimal demands

Question 17

What is the lagrange equation?

Answer 17

max c1, c2, λ = O(c1, c2) + λ[Y – f(c1, c2)]

Question 18

How is the critical value of L found?

Answer 18

Through the first-order conditions:
∂L/∂c1 = 0 <=> …
∂L/dc2 = 0 <=> …
∂L /∂λ = 0 <=> …

Question 19

How do you find the standard tangency condition using the lagrangian method?

Answer 19

By equating the answers of the first 2 partial derivatives from the FOCs.
MRS = - U1/U2 = -p1/p2 = MRT

Question 20

When can the lagrangian method not be used?

Answer 20

When dealing with perfect substitutes and perfect complements.

With perfect substitutes (U = q1 + q2)
- If p1 < p2 then q1* = Y/p1 and q2* = 0
- If p1 > p2 then q1* = 0 and q2* = Y/p2
- If p1 = p2 then q1* and q2* are undetermined
With perfect complements (U = min(q1, q2)
- Q1 = q2 = q so Y = q(p1 + p2) and q* = Y/(p1 + p2)

Question 21

What is an engel curve?

Answer 21

A plot of quantity demanded against income.

Question 22

How do you find engel curve equations?

Answer 22

Rearrange the optimal demand equations to isolate Y.

Question 23

How do you find the engel curve equation for cobb-douglas preferences?

Answer 23

Rearrange the optimal demand equations to isolate y. e.g.
x1* = ay/(a + b)p1 and x2* = by/(a + b)p2
rearranged:
y = (p1(a + b)/a)x1* => engel curve for good 1
y = (p2(a + b)/b)x2* => engel curve for good 2

Question 24

What do cobb-douglas preferenced engel curves look like?

Answer 24

Engel curve for good 1:
Vertical axis = y
Horizontal axis = x1
Straight 45-degree line = engel curve

Engel curve for good 2:
Vertical axis = y
Horizontal axis = x2
Straight 45-degree line = engel curve

Question 25

How do you find the engel curve equation for perfectly complementary preferences?

Answer 25

U(x1, x2) = min(x1, x2)
The ordinary demand equations are x1* = x2* = y/(p1 + p2)

Rearranged to isolate y, these are:
Y = (p1 + p2)x1* => engel curve for good 1
Y = (p1 + p2)x2* => engel curve for good 2

Question 26

What do perfectly complementary preferenced engel curves look like?

Answer 26

Engel curve for good 1:
Vertical axis = y
Horizontal axis = x1
Straight 45-degree line = engel curve

Engel curve for good 2:
Vertical axis = y
Horizontal axis = x2
Straight 45-degree line = engel curve

Question 27

How do you find the engel curve equation for perfectly substitutable preferences?

Answer 27

U(x1, x2) = x1 + x2
The ordinary demand equations are:
x1*(p1, p2, y) = {0, if p1 > p2
		{y/p1, if p1 < p2
x2*(p1, p2, y) = {0, if p1 < p2
		{y/p2, if p1 > p2

Suppose p1 < p2, then x1* = y/p1 and x2* = 0
Rearranged for y:
Y = p1x2 and x2* = 0

Suppose p1 > p2, then x1* =0 and x2 = y/p2
Rearranged for y:
x1 = 0 and Y = p2x1

Question 28

What do perfectly substitutable preferenced engel curves look like?

Answer 28

2 scenarios, if x1 = 0 and if x2 = 0

If x1 = 0 and Y = p2x1
Engel curve for good 1
Vertical axis = y
Horizontal axis = x1
Vertical line on top of y-axis = engel curve 

Engel curve for good 2
Vertical axis = y
Horizontal axis = x2
Straight 45-degree line = engel curve

If Y = p1x2 and x2 = 0:
Engel curve for good 1
Vertical axis = y
Horizontal axis = x1
Straight 45-degree line = engel curve 

Engel curve for good 2
Vertical axis = y
Horizontal axis = x2
Vertical line on top of y-axis = engel curve

Question 29

Are all engel curves straight lines?

Answer 29

No, engel curves are only are straight lights if he consumer’s preferences are homothetic.

Question 30

When are a consumer’s preferences homothetic?

Answer 30

If: (x1, x2) < (y1, y2)  (kx1, kx2) < (ky1, ky2)
For every k >0

Therefore: a consumer’s preferences are only homothetic if the consumers MRS is the same anywhere on a straight line drawn from the origin.

Question 31

What is an example of consumer preferences that are not homothetic?

Answer 31

Quasilinear preferences are not homothetic:
U(x1, x2) = f(x1) + x2
e.g. (U(x1, x2) = √x1 + x2

Question 32

What do quasi-linear preferenced engel curves look like?

Answer 32

Note that quasi-linear indifference curves intersect both axes and each curve is a vertically shifted copy of the others.

Engel curve for good 2
Vertical axis = y
Horizontal axis = x1
Engel curve = vertical line, changes to 45-degree line about half way up axis

Engel curve for good 1
Vertical axis = y
Horizontal axis = x2
Engel curve = straight 45-degree line, changes to vertical line at point x-bar*

Question 33

What is the difference between engel curves for normal and inferior goods?

Answer 33

Normal good = good for which quantity demanded rises with income. Therefore, a normal good’s engel curve is positively sloped (note: the previous engel curve examples used normal goods).

Inferior good = good for which quantity demanded falls as income increases. Therefore, an inferior good’s engel curve is negatively sloped.

Question 34

What is an ordinary good?

Answer 34

A good is called ordinary if the quantity demanded of it always increases as its own price decreases.

Question 35

What is a giffen good?

Answer 35

The quantity demanded of a good rises as its own price increases.