Consumer Theory
Required:
Show that the equilibrium condition and consumer equilibrium under both cardinal and ordinal utility theory are identical.
They both assume that the consumer is rational.
Consumer equilibrium, under cardinal utility theory, is achieved when the sufficient condition is met. That is, the total expenditure is equal to the consumer's income.
If a consumer is assumed to consumes two commodities only X and Y, then:
Utility is a function of Y and X;
U = f(X,Y)…………………………..i
Let the price of the two commodities be Px and Py respectively
Thus total expenditure will be;
E = PxX+ PyY ……………………….ii and since the income of the consumer is assumed to be constant (fixed),
Hence E≤ M,
M≥PxX + PyY…………………………iii but since the consumer is assumed to be rational she spends the entire income on the two commodities;
Thus M = PxX +Py Y………………….iv
Thus from equations ii & iv M =E Since the consumer aims at maximizing the her satisfaction; S = U- E ……………………………v Substituting equations (i) & (ii) into equation (v)
S = XY – PxX + PyY…………………………….vi Mathematically, first order condition, for maxima is achieved when a derivative is equal to zero. The consumer is aiming to maximize S, hence;
[pic],
[pic]
But, [pic]
Therefore,
[pic]
Thus,
[pic]………………………………….vii
It thus follows that the derivative of S with respect to Y will be;
[pic];
[pic];
[pic]…………………………………….viii
Hence 1=1;
[pic][pic]………………………………….ix
[pic]…………………………………….x
The ordinal utility theory uses the indifference curves(IC) to analyze the equilibrium condition of the consumer.
Y
IC
X
The consumer aims at consuming at the highest indifference curve.
The furthest IC from the origin presents the highest satisfaction of two commodities, but is constrained by her fixed income which is represented by the budget line. The consumer
Show that the equilibrium condition and consumer equilibrium under both cardinal and ordinal utility theory are identical.
They both assume that the consumer is rational.
Consumer equilibrium, under cardinal utility theory, is achieved when the sufficient condition is met. That is, the total expenditure is equal to the consumer's income.
If a consumer is assumed to consumes two commodities only X and Y, then:
Utility is a function of Y and X;
U = f(X,Y)…………………………..i
Let the price of the two commodities be Px and Py respectively
Thus total expenditure will be;
E = PxX+ PyY ……………………….ii and since the income of the consumer is assumed to be constant (fixed),
Hence E≤ M,
M≥PxX + PyY…………………………iii but since the consumer is assumed to be rational she spends the entire income on the two commodities;
Thus M = PxX +Py Y………………….iv
Thus from equations ii & iv M =E Since the consumer aims at maximizing the her satisfaction; S = U- E ……………………………v Substituting equations (i) & (ii) into equation (v)
S = XY – PxX + PyY…………………………….vi Mathematically, first order condition, for maxima is achieved when a derivative is equal to zero. The consumer is aiming to maximize S, hence;
[pic],
[pic]
But, [pic]
Therefore,
[pic]
Thus,
[pic]………………………………….vii
It thus follows that the derivative of S with respect to Y will be;
[pic];
[pic];
[pic]…………………………………….viii
Hence 1=1;
[pic][pic]………………………………….ix
[pic]…………………………………….x
The ordinal utility theory uses the indifference curves(IC) to analyze the equilibrium condition of the consumer.
Y
IC
X
The consumer aims at consuming at the highest indifference curve.
The furthest IC from the origin presents the highest satisfaction of two commodities, but is constrained by her fixed income which is represented by the budget line. The consumer