The marginalizes – that is what will be called the supporters of the law of diminishing marginal utility – will use such a “law “to explain the value of goods , based on the idea that individuals seek the highest possible satisfaction, that is, they have a hedonistic behavior, and they are rational, that is, they act in pursuit of that objective. The problem of the consumer, which is supposed to be rational and hedonistic, is to select the basket of goods that maximizes its utility, given its limited resources, it is subject to one. Such selection depends, therefore, on the form of its utility function -of its tastes- and also on the price of goods in this way, the budget constraint.
More precisely, the selection is made in such a way that the relation between the marginal utility and the price of each good is equal for all the goods of the chosen basket.
In effect, if this were not the case, the consumer could increase his utility by modifying the composition of the basket. If, for example, the relation between marginal utility and price were greater for good A than for good B, it is logical that the consumer had an interest in selling B and buying A with the result of the operation; the basket considered would not correspond to a maximum profit. Such reasoning is valid whatever the assets A and B considered.
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The condition of “optimality “that has just been established-equality of the ratios between marginal profits and prices-can be stated as follows: the basket that maximizes utility under the condition of a budget constraint is such that the marginal utility of each good is proportional to the price of the product. Same, being equal the coefficient of proportionality for all. This coefficient depends on income because if it increases, it budget constraints less “adjusted” so that the consumption of goods increases and marginal profits decrease; Now, as fixed prices are assumed, the relationship between marginal profits and prices, that is, our coefficient of proportionality, decreases. The macroeconomists
They call this relationship the marginal utility of income.
One of the important consequences of the principle of maximization is that it provides a powerful justification for the use of mathematical techniques. Indeed, insofar as the utility depends on the quantities consumed, it can be represented as a function of those quantities, which in turn can be represented as a vector whose elements are the numbers that represent the quantities of each one. Of the goods, for example, the basket represented by 3 kilos of carrots, 5 liters of milk and a pair of shoes, is represented by the vector.
Insofar as the marginal utility has implicit the idea of the variation of the utility, the appropriate mathematical concept to represent it is the derivative. As in general in the function of utility involved several goods, carrots, milk, shoes, it admits several derivatives called partial, one for each good.
Now, as in the search for extreme, maximum and minimum points. Of a function the derivative calculation is made to intervene in general, the interest of the mathematical formalization is clear. It can also be indicated that the custom of identifying “marginalize” and “search for extremes by the calculation of derivatives”.