Solution

;

;

.

Accordingly to the initial condition we have

,

then the solution, which is a law of change of the labor productivity, will be

.

Problem 5.2. The total production cost is a function of the value of production. Find the function of the total cost if it is known that marginal cost for all values of is proportional to the average cost. Coefficient of proportionality equals, initial condition: .

Solution. When are total production cost, then - average cost, and - marginal cost. Accordingly to the condition of the problem let us compose equation

.

This is a separable first order differential equation. Separating variables and integrating we will have

;

;

;

.

For finding C we have initial condition

.

Then

- is the function of the total cost.

Problem 5.3. Costs of technological progress is function of time . Find the form of this function, if its growth rate changes according to the law, and it is known that the costs of technological progress at the initial time is .

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