Utility function

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Things have mass, volume and temperature. By analogy with the early establishment of consumer choice theory attempts were made to assign each product its utility and measured quantitatively. They failed, however, the concept of utility is convenient. It reflects the satisfaction of a specific set of consumer goods.
The function U (x, y) is called a utility function, if U (x ', y')> U (x ", y") if and only if the set of (x ', y') is preferable to set (x ", y ").
In particular, it follows that the equivalent to each other set of goods utility function takes the same value. In other words, each indifference curve of utility function is constant, and it can be given by the equation of the form U (x, y) = U0. At a higher indifference curve utility function takes the greater value.
The task of selecting the user can be written as

max U (x,y) unnder + = I, x ≥ 0 y ≥ 0.

Marginal utility of goods is called an increase of the utility of consumption of an additional unit of that commodity. For goods X marginal utility is by definition equal

= = .

In the limit when ∆x tends to zero, we obtain

= .

Similarly, the marginal utility of product Y.
Although the utility function U (x, y) is not uniquely defined, for it is a fair hypothesis that diminishing marginal utility: every unit of product increases the overall utility, but by a smaller amount than the previous one. Another name for this hypothesis - the first law of G. Gossen (1810-1858). This means that the marginal utility of goods decreases as its quantity.
Consider any indifference curve, which is given by the equation Let U (x, y) = U₀. The constancy of the utility function means that its total differential dU (x, y) = 0 along a given indifference curve, i.e.

+ = 0,

+ = 0.

Hence, we find that the slope of the indifference curve

= .

On the other hand, as already noted, the slope of the budget line is - p_x/p_y At equilibrium (when the internal solution exists), they both must be equal to each other. Hence, then, equating, we obtain another condition for equilibrium:
= (4.3)

The principle of equal usefulness: the consumer in equilibrium allocates revenue on all goods in such a way as to equate marginal utility per one tenge (U.S. dollar, the ruble) spent on each of the commodities. To equilibrium condition (4.3) can given an economic explanation. Suppose that in equilibrium it does not hold, let, for definiteness,

> . (4.4)

If one reduces the consumption of cakes on the unit, then he would reduce the overall usefulness on MU_x and would save on p_y tenge. On these p_y tenge it can be purchased additional p_y/p_x oranges and thereby increase the overall usefulness to the total change in utility will be
= - + = > 0,

i.e. you can increase the benefits derived by the consumer. This contradicts the fact that the initial state was an equilibrium.
Hence, in equilibrium (4.4) is impossible. Similarly we show that it is impossible to reverse inequality. Therefore, in equilibrium, the equality (4.3) have to be hold.

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