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: • The expression says that: • the marginal utility of this increment is positive • the total utility increases as Xi consumption becomes greater > 0 for all i

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  MRSAX2, X1 = MRSBX2, 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: (i)  (ii)  By dividing (i) and (j), we get:

for all i = 1, 2, 3,…n

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CALCULUS MODELS OF CONSUMER EXCHANGE