(a) Unconstrained consumer equilibrium
The necessary conditions for maximizing the total utility are:
Sufficient conditions are:
For example, when consumers consume multiple products, the total utility function is:
When tu = u (x, y):
At this time, the necessary conditions for maximizing the total utility are:
;
Sufficient conditions are:
;
(b) Constrained consumer equilibrium
When studying consumption equilibrium, it is usually assumed that:
First, consumers' hobbies are fixed.
Second, the income of consumers is fixed.
Third, the price of goods is fixed.
1. law of marginal utility equality
The optimal configuration is the configuration that makes the marginal utility of each use equal. Because if not, consumers can improve their welfare by reducing the number of one use and expanding the number of another.
Table 3-3 Utility and Savings of Commodity X and Commodity Y
Table 3-4 Utility of Goods per Yuan
4× 5+3× 4+8 = 40 (yuan)
2. Mathematical analysis of the law of marginal utility equality (the condition of consumer equilibrium)
TU=U(X，Y，Z)
max TU=U(X，Y，Z)……………………………………( 1)
s.t. M=PX X+PY Y+PZ Z …………………………(2)
L=U(X，Y，Z)+λ[M—(PXX+PYY+PZZ)] ……(3)
…………………………………(4)
…………………………………(5)
…………………………………(6)
……………………(7)
∵
……………………………(8)
………………(9)
In which,
、 、
、 、
………………………………………( 10)
………………( 1 1)
…………………( 12)
∵ ; ;
∴ …………( 13)
MUX =λPX； MUY =λPY； MUZ =λPZ
………………… ( 14)
………………………( 15)
…………………( 16)
For example, suppose a consumer has 120 yuan, and there are two kinds of goods that can be purchased and consumed. Given that the price of X commodity is 4 yuan, the price of Y commodity is 2 yuan, and the utility function of consumers' consumption of X and Y commodities is, how can consumers achieve the greatest satisfaction?
Solution:
According to the conditions of consumer equilibrium, we must find out the conditions to meet:
∵
∵
Order:
Get:
Simplify it to get: y = 2x, and substitute it into the constraint function to get:
X = 15 (unit), Y = 30 (unit)
So the total utility is:
According to the calculation, consumers will buy 15 units of X goods and 30 units of Y goods, and the maximum utility is one utility unit.