Consumer Theory Flashcards
What does an indifference curve contain?
It contains equally preferred bundles.
Equal preference = same utility level.
Therefore, all bundles in an indifference curve have the same utility level.
What is the difference between goods, bads and neutrals?
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).
What do perfectly complementary indifference curves look like?
L-shaped.
What do Cobb-Douglas indifference curves look like?
All curves are hyperbolic, asymptoting to, but never touching the axis.
What is the marginal utility of commodity “I”?
The rate-of-change of total utility as the quantity of commodity “i” consumed changes:
Mui = ∂U/∂xi
What is the general equation for an indifference curve?
U(x1, x2) = k
Totally differentiating this identity gives:
(∂U/∂x1)dxi + (∂U/∂x2)dx2 = 0
What is the Marginal rate of substitution?
dx2/dx1 = -(∂U/∂x1)/(∂U/∂x2) = - x2/x1
What is a consumer’s ordinary demand?
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)
What happens when xi* > 0 (or q1i*)?
e.g. x1* > 0 or x2* > 0 etc., then the demanded bundle is interior.
What happens when xi* = 0 (or q1i*)?
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
When is the budget exhausted?
If buying (x1, x2) costs $m.
What 2 conditions does (x1, x2) satisfy?
- The budget is exhausted: p1x1* + p2x2* = m
- 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.
How do you derive the optimal demands/solutions?
Through the substitution method or the lagrangian method. However, for the exam you will have to know the langrangian method.
What is the optimality condition for an interior solution (q1 > 0, q2 > 0)?
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).
What is the lagrangain method?
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.
How do you use the lagrangian method to derive optimal demands?
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.
- Set up the lagrange function
- Compute the FOCs
- Find the standard tangency condition and solve for either q1 or q2
- Sub q1 or q2 into budget constraint (Eq 3 from FOC) to find optimal demands
What is the lagrange equation?
max c1, c2, λ = O(c1, c2) + λ[Y – f(c1, c2)]
How is the critical value of L found?
Through the first-order conditions:
∂L/∂c1 = 0 <=> …
∂L/dc2 = 0 <=> …
∂L /∂λ = 0 <=> …
How do you find the standard tangency condition using the lagrangian method?
By equating the answers of the first 2 partial derivatives from the FOCs.
MRS = - U1/U2 = -p1/p2 = MRT
When can the lagrangian method not be used?
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)
What is an engel curve?
A plot of quantity demanded against income.
How do you find engel curve equations?
Rearrange the optimal demand equations to isolate Y.
How do you find the engel curve equation for cobb-douglas preferences?
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
What do cobb-douglas preferenced engel curves look like?
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
How do you find the engel curve equation for perfectly complementary preferences?
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
What do perfectly complementary preferenced engel curves look like?
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
How do you find the engel curve equation for perfectly substitutable preferences?
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
What do perfectly substitutable preferenced engel curves look like?
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
Are all engel curves straight lines?
No, engel curves are only are straight lights if he consumer’s preferences are homothetic.
When are a consumer’s preferences homothetic?
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.
What is an example of consumer preferences that are not homothetic?
Quasilinear preferences are not homothetic:
U(x1, x2) = f(x1) + x2
e.g. (U(x1, x2) = √x1 + x2
What do quasi-linear preferenced engel curves look like?
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*
What is the difference between engel curves for normal and inferior goods?
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.
What is an ordinary good?
A good is called ordinary if the quantity demanded of it always increases as its own price decreases.
What is a giffen good?
The quantity demanded of a good rises as its own price increases.
