The Theory of Consumer Equilibrium
The consumer problem is defined thus: assuming a given income (M)
Assuming the consumer has full knowledge of all available goods and services,
and their prices
Assuming that the consumer has a rational set of preferences described by the
utility function (U(X, Y, ...) defined ordinally (not “cardinally”)
And that the consumer can rank all bundles of commodities (eg X, Y, Z) with
the property of “transitivity” such that
if X is preferred to Y (X > Y) and if Y > Z then it must mean that X > Z
or if X is indifferent to Y (X= Y)
and Y = Z then it must follow that X = Z
Then it is possible to have a consistent indifference curve “map” of convex
indifference curves with negative slopes
Result: utility curves will have these properties:
Every combination of commodities MUST have an indifference curve through it
Indifference curves cannot intersect (text-books)
If a commodity has the property that consumers always wants more of it, then
we can call them MIB (More is Better) commodities
Indifference curves will be negatively sloped (convex, not concave) (text-books)
Indifference curves will be associated with higher utility in the “North East”
direction.
N
Convex curves
Y
W
X
E
S
There may be odd utility curve maps
The text books give you all kinds of variations from the normal indifference
curves, for some of which it is impossible establish unique equilibrium points.
For example
or
Y
Y
B
B
A
A
O`
X
O
X
What kind of commodities are X and Y in these above two examples? Have a
read through your text-books to see the justification for these shapes of curves
And why they do not satisfy the indifference curve requirements (eg can A and
B be on the same indifference curves?
We won’t bother with such unusual sets of preferences in lectures.
How does neoclassical consumer theory define the problem?
Given a particular level of money income (M)
Given two MIB (More Is Better) commodities X and Y with prices Px and Py
Given a utility function U = U(X,Y)
How does the consumer maximise her utility, subject to the budget constraint?
Y
O
X
From EC102: you should know the geometric solution
Just add the budget line to the indifference map:
If the consumer buys only Y then point B represents quantity M/Py of Y
If the consumer buys only X, then point C represents quantity M/Px of X
Join the two points A and B up and you get the budget line. And the point E.
Y
B
.
.
OB=M/Py
E
O
.
C
OC= M/Px
X
Why is E the optimum point?
Where the budget line is tangential to one of the indifference curves
At E, the slope of the budget line = slope of the indifference curve
And any movement away from E, in any direction, will immediately represent
either lower utility, or be unfeasible from the point of view of the budget
constraint (have a go at establishing that)
Y
B (M/Py)
E*
X
O
C (M/Px)
Geometric derivation of consumer equilibrium: check N&S
In going from A to B, the total level of utility remains the same (same indifference curve).
If MUy = Change in utility given a unit change in y
Than change in utility going from A to O = (dy* MUy)
Similarly going from O to B represents a change in utility (dx*MUx)
A
Y
B (M/Py)
dy
O
*
dx
B
*
E
X
O
C (M/Px)
The two changes add up to 0
(dy* MUy) + (dx*MUx) = 0
(dy* MUy) = - (dx*MUx)
Hence dy/dx = - MUx/MUy
i.e MRSxy = dy/dx
And as dx becomes smaller and smaller, dy/dx becomes the slope of the indifference
curve.
A
Y
B (M/Py)
dy
dx
*
B
*
E
X
O
C (M/Px)
Maximum utility: slope of budget line= slope of indifference curve
Rearrange Y Py + X Px = M to get the standard linear equation form.(Y=mX+c)
Y Py = M - X Px
hence Y = -(Px/Py) X + (M/Py)
Y = -(Px/Py) X + (M/Py)
Hence the slope of the budget line is negative= -(Px/Py) and at E = slope of the
indifference curve.
= -MUx/MUy
= MRSxy= -(Px/Py)
Y
B
OB=M/Py
.
E
O
.
.
C
OC= M/Px
X
For geometric derivation of consumer equilibrium
Must read through the text-books and explain, using geometry and clear structured logic,
Explain why points D, F, and G cannot represent the consumer’s equilibrium condition
where she maximises her utility, subject to the budget constraint.
Y
B (M/Py)
*
D
G
F E
*
*
X
O
C (M/Px)
The calculus approach to Consumer Equilibrium
Simpler to derive the consumer equilibrium condition using elementary calculus
We shall just derive the “first order conditions”.
For the second order conditions, those interested in doing advanced microeconomics, go and read text like Henderson and Quandt (in the library).
What the second order conditions mean in “common sense” (pretty much the
same as what we are going to illustrate with one variable- why the equilibrium
point will be a maximum utility point, and not a minimum.
[For cost minimisation problems, the second order conditions will have to
establish that the cost will be a minimum and not a maximum.]
Why consumer equilibrium condition is an “optimisation” problem?
Look at the budget line, BC. Suppose X = dalo, Y = rice.
You know the consumer doesn’t want to eat only dalo: so B is probably not an optimal
mount, given the money he has to spend. He would be better off if he buys less dalo and
a bit of rice.
And you know that he does not want to buy only rice, i.e. C is not optimum either.
He would like to perhaps buy less rice and some dalo.
Somewhere in between B and C, he will probably maximise his utility.
Y
B (M/Py)
E*
O
C (M/Px)
X
First, derive the conditions for a “maximum” for one variable
Let us assume there is a function U = U(x) whose curve is given thus
A
U
F
U = U (x)
dU
θ
D
G
dX
x
How do you find the maximum point A? In calculus, the first derivative of U
with respect to the variable X (dU/dX) gives you the slope of the curve.
In the RHS diagram, slope of line DF = tan θ = opposite/adjacent =
FG/DG = dU/dX and as dX becomes extremely, small, then dU/dX becomes the
slope of the curve DF.
(remember tan θ = opposite side/adjacent side in a right angled triangle)
So U(x) is a maximum when the slope of the curve = 0
U
A
U = U (x)
`
θ
x
When the function is at a maximum, the slope of the tangent will = 0 (ie. the
tangent to the curve is horizontal). i.e.
dU/dx = 0
Note in calculus dU/dX is different from dU/dX:
The latter (curly d’s) means taking the derivative of U with respect to X, holding
all other variables constant. i.e a partial derivative.
How know that the point M is maximum and not minimum?
You take the second derivative with respect to x: this gives you the rate at which the
slope is changing: how is it changing from before M, through M and after M?
Y
M
=0
= +ve
= -ve
= +ve
= -ve
=0
X
the slope d(dU/dx goes from being +ve to 0 to -ve. i.e. slope is decreasing
i.e d(dU/dx) = some negative value
dx
ie d2U/dx2 = - ve
Conversely for a minimum for the function, d2Y/dx2 would be + ve
The “Lagrangian Multiplier” Method- the basics
The Utility function (U(X,Y) by itself cannot have a maximum?
Why not?
As long as X or Y or both can increase, then Utility will keep increasing.
The consumer, however, cannot keep increasing his consumption of X and Y
indefinitely, since there is a finite budget, M.
Hence the consumer equilibrium mximum is called a “constrained maximization
problem”.
Maximize U(X,Y) subject to the constraint M = Px*X + Py*Y
The “Lagrangian Multiplier” Method
Write the equation in the form: L is a function of X, Y and λ, in two parts.
L = U (X,Y) + λ (M - PxX - PyY)
(1)
Where U (X,Y) is the utility function to be maximised, subject to the constraint in
brackets, written to = 0.
If M is the income to be spent, X and Y are the amounts of the two commodities, and Px
and Py are their given prices
then
M = PxX + PyY
Hence M - PxX - PyY = 0
This is the underlined bit in (1) - the “budget constraint” written in a form to equal 0.
While the meaning of λ will be understood in due course.
Conditions such that we maximise utility U(X,Y),
subject to the budget constraint
i.e. we are looking for a “constrained maximum”.
L = U (X,Y) + λ(M - PxX - PyY)
(1)
Differentiate with respect to the three variables X, Y and λ.
dL/dX = dU/dX - λ Px = 0
(2)
dL/dY = dU/dY - λ Py = 0
(3)
and M - PxX - PyY = 0
(4)
Taking equations 2, 3 and 4
dU/dX = λ Px
hence dU/dX
Px
=λ
(5)
hence dU/dY
Py
=λ
(6)
and
dU/dY = λ Py
and from equation (4)
M = Px X+ Py Y (hence the budget constraint is satisfied)
From (5) and (6)
dU/dX = dU/dY
Px
Py
(7)
What does (7) represent?
dU/dX = dU/dY
(7)
Px
Py
dU/dX is the change in utility due to the change in consumption of X
and if divided by the price of X then gives you the change in utility due to a
change in consumption of X from the extra dollar spent on X.
While the RHS bit of the equation (dU/dY)/Py is similarly, the change in utility
due to a change in consumption of Y per dollar spent on Y.
Logically, in equilibrium, they MUST be the same.
Otherwise the consumer would rearrange his consumption towards whichever
item gives her the bigger utility per dollar spent.
i.e keep moving money from X to Y or Y to X until the equality holds
Alternatively
You can of course rearrange equation (6) and get the other version which you
are more familiar with from your geometrical exposition.
dU/dX = Px
dU/dY
Py
(8)
LHS is Marginal Rate of Substitution of X for Y
= Slope of the indifference curve
= RHS = ratio of the goods prices.
= slope of the budget line
i.e utility maximization where the budget line is tangential to the indifference
curve
The second order condition
Essentially becomes an equation which ensures that the utility curve is convex and not
concave:
d 2 Y/dX 2 < 0 ie slope has to decrease.
Suppose that the utility curves were concave, then simply satisfying the first order
condition would not imply that E (the tangential point) was a utility maximisation point,
subject to the budget constraint
Is this the maximum utility point? What seems to be the utility maximising point for this
consumer? What is happening to the second derivative here?
B
E
C