Transcript CALCULUS MODELS OF CONSUMER EXCHANGE

CALCULUS MODELS OF
CONSUMER EXCHANGE
Appendix Chapter 2
Microeconomic Policy Analysis
Johnny Patta
Calculus Models on Consumer Exchange
Let U(X1, X2, …, Xn) represent a utility function for a consumer in an
economy with n goods.
Assume that:
– The function is smooth and continuous
– The goods are infinitely divisible
– The more is better
The assumption of the more is better can be expressed as follows:
> 0 for all i
The expression says that:
•the marginal utility of this increment is positive
•the total utility increases as Xi consumption
becomes greater
We defined an indifference set:
• a curve if there are only two goods
• a surface if there are more than two goods
as the locus of all consumption bundles which provide the
consumer with the same level of utility
• If constant level of utility, then as the goods X1, X2, …, Xn
change along the U indifferent surface, it must always be true
that:
= U(X1, X2, …, Xn)
• If we consider only changes in the X’s along an indifferent
surface  the total utility doesn’t change at all
• Suppose the only changes we consider are of X1 and X2:
all the other X’s are being constant. Then we move along
an indifferent curve:
 MRS X2, X1
• Efficiency  MRSA X2, X1 = MRSB X2, X1
• Mathematically, this condition can be shown as follows:
1. Consider: US(MS, TS) and UJ(MJ, TJ)
= the total amount of meat in the economy
= the total amount of tomatoes in the economy
If for any given utility level S of Smith, Jones is getting the
maximum possible utility, we called it efficient
2.Consider also, that in the two person economy,
knowledge of Jones’s consumption of one good allows us
to infer Smith’s consumption
Eg. MS = - MJ
From the equation, we know that the increase in Jones’s
meat consumption causes the following change in
Smith’s:
3.The problem is to choose the level of two variables, meat
and tomato consumption, which maximize the utility
level of Jones:
But the real problem is to maximize subject to the constraints that
total real resources are limited to and
that Smith must get
enough of those resources to yield a utility level of
S:
S
= US(MS, TS)
4. The constraints are as follows:
 MS = – MJ
 TS = – TJ
Then all the constraints can be represented in one
equation:
S
= US[(
– MJ), ( – TJ)]
5. We use Lagrange multipliers to solve the
maximization problem with constraints:
L (MJ, TJ, ) = UJ(MJ, TJ) + { US - US[( – MJ), (
– TJ)]}
The one that maximizes UJ will also maximize L
6.Making use of the chain rule in taking the partial
derivatives:
(i)
(ii)
(iii)
• Remember:
7. Then the equation (i) and (ii) can be simplified
as follows:
(i)
(ii)
8.Subtract the terms with in them form both sides of each
equation and then divide (i) by (ii):
MRSJM, T = MRSSM, T
Using pricing system
• Each consumer will allocate his or her budget in such a way
that the MRS of any two goods Xi and Xj equals to the price
ratio of those two goods Pi/Pj. This can be seen
mathematically as follows, letting I represent the consumer’s
total budget to be allocated
• The consumer wants to choose goods
X1, X2, …, Xn, which maximize utility subject to the following
budget constraint:
I = P1X + P2X2 + … + PnXn
Using Lagrange multipliers:
L = U(X1, X2, …, Xn) + (I - P1X1 - P2X2 - … - PnXn)
Using partial derivation:
By dividing (i) and (j), we get:

(i)

(ii)
for all i = 1, 2, 3,…n
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