Global economic development trends

Solution. Let - be the cost of the equipment at time t. Then - is the rate of change in value due to wear. According to the condition of the problem, we obtain the following equation:
(1)
where k – is the coefficient of proportionality. Sign “-” indicates a decrease in the cost of equipment over time.
The initial cost specifies the initial condition for the resulting equation . Thus, we get the Cauchy problem: find a particular. Solution of the differential equation at cet’s separate the variables:

Integrating the last equation, we get
, where


Fend C by substituting the condition . We have . Then a particular solution of equation (1) is equal to
. Integrating the last equation, we get
where .
Find C by substituting the condition . We have . Then a particular solution of equation (1I is equal to
(2)
Thus, knowing the wear factor k and the initial cost of the equipment , using formula (2), you can find the actual cost of the equipment after any time t.
2) Playing with nature
This game is played by two players with opposite interests. Therefore, the actions of each player are aimed at increasing the gain (reducing the loss). However, in some tasks leading to gaming, there lack of information about the conditions in which the action is carried out (weather, consumer demand, etc.). These conditions do not depend on the conscious of another player, but on objective reality. Such games are called games with nature. A person in games with nature tries to act prudently, the second player (nature) acts randomly. The game conditions are given by the matrix

Let player A have strategies , and nature states . The simplest situation is when the probability of each state of nature . Then, if all possible are taken into account, then

If player A choose a pure strategy , then the expected payoff is

Application of game theory for solving economic problems. Changes in the situation in the consumer market due to the seasonality of products affect the development of strategies for companies. In conditions of market uncertainty, it is rather difficult to choose the optimal strategy, but by applying mathematical methods and taking into account a certain direction, you can get the maximum profit and in such conditions.
The strategy of behavior in the market under conditions of uncertainty reduces the factor of change, which makes it more likely to predict profit.
Consumer demand and sales of many types of goods depends on the season. It is noticed that for some goods the peak of demand falls on the summer, for some – for the spring – autumn period, for some – for the winter. In this regard, it is particular interest for companies to develop an strategy in transition periods.
If the information about the states of nature is incomplete, then we can use the principle of insufficient reason of Laplace, according to which we can assume that all states of nature are equally probable. Then the optimal strategy is one that achieves

_{i.e. strategy for which the arithmetic mean of the elements of the corresponding row is maximum.
There are a number of criteria that are used when choosing the optimal strategy.}

1.