Question

Industrial Production (a) Economists often use the expression expression "rate of growth" in relative rather than absolute terms. For example, let \(u=f(t)\) be the number of people in the labor force at time \(t\) in a given industry. (We treat this function as though it were differentiable even though it is an integer-valued step function.) Let \(v=g(t)\) be the average production per person in the labor force at time \(t .\) The total production is then \(y=u v\) . If the labor force is growing at the rate of 4\(\%\) per year year \((d u / d t=\) 0.04\(u\) ) and the production per worker is growing at the rate of 5\(\%\) per year \((d v / d t=0.05 v),\) find the rate of growth of the total production, y. (b) Suppose that the labor force in part (a) is decreasing at the rate of 2\(\%\) per year while the production per person is increasing at the rate of 3\(\%\) per year. Is the total production increasing, or is it decreasing, and at what rate?

Short answer

In the first scenario, the total production is growing at a rate of 9% per year. In the second scenario, the total production is increasing at a rate of 1% per year.

Step-by-step solution

Use Product Rule for the first part

Given \(u' = 0.04u\) (rate of growth of labor force) and \(v' = 0.05v\) (rate of growth of labor), we use the product rule to find the rate of change in total production \(y = uv\). This implies: \[y' = u'v + vu'\] Substituting the known values, we get, \(y' = 0.04uv + 0.05uv\)

Simplify calculation

We simplify the equation obtained in Step 1 to calculate the rate of growth in total production. \[y' = (0.04 + 0.05)uv = 0.09uv\] Thus, the total production is growing at a rate of 9% per year.

Use Product Rule for the second part

Now, given that \(u' = -0.02u\) (rate of decrease of labor force) and \(v' = 0.03v\) (rate of growth of labor), we use the product rule to find the rate of change in total production \(y = uv\). This implies: \[y' = u'v + vu'\] Substituting the known values, we get, \(y' = -0.02uv + 0.03uv\)

Simplify calculation

We simplify the equation obtained in Step 3 to calculate the rate of growth in total production. \[y' = (-0.02 + 0.03)uv = 0.01uv\] Thus, the total production is increasing at a rate of 1% per year. The fact that the total change is positive implies that production is increasing.

Key concepts

Product Rule Differentiation

Understanding product rule differentiation is essential for analyzing how two or more functions that multiply together evolve as one of the functions changes. In calculus, the product rule is a formula that determines the derivative of a product of two functions. Specifically, if you have functions u(t) and v(t), the derivative of their product y(t) = u(t)v(t) is found by applying the product rule: \[ y'(t) = u'(t)v(t) + u(t)v'(t) \]

• The function u'(t) represents the derivative of u with respect to t, indicating how u changes over time.
• Similarly, v'(t) represents the derivative of v, reflecting changes in v over time.

When the rate of change for each function is known, the product rule allows us to efficiently determine the rate of change for the entire product. This is particularly useful in economics when we're looking at the interaction of various rates, such as labor force and production per worker.

Rate of Change

The rate of change in a function is a measure of how quickly values of the function increase or decrease over a certain period of time. In the context of the exercise provided, rate of change is used to describe the growth of both the labor force and production per worker in an industrial setting. This rate is expressed as a percentage, calculated by the derivative of the function with respect to time (denoted as du/dt for labor force and dv/dt for production per worker). Specifically, if u = f(t) represents the number of people in the labor force at time t and v = g(t) represents the average production per person, then the rate of growth (or decrease) of these values is found by taking their derivatives: \[ u' = \frac{du}{dt} ,\quad v' = \frac{dv}{dt} \] For example, if the labor force is growing at 4% per year (du/dt = 0.04u), and production per worker is growing at 5% per year (dv/dt = 0.05v), the rates indicate how rapidly these functions are changing. By knowing these rates, we can predict future values and make informed decisions in planning and analysis.

Economics Calculus Applications

Calculus has profound applications in economics, helping to quantify and analyze dynamic changes within economic systems. For instance, economists often employ calculus to deduce the rate of production growth, optimal resource allocation, and trends in economic variables over time. The exercise we examined showcases one such application where differentiation and the product rule are used to calculate the rate of growth in total industrial production.Using these calculus tools, economists can not only predict future growth but also adjust strategies accordingly. For example, understanding the rate at which total production is increasing or decreasing enables a business to make decisions regarding investment, employment, and resource management. If the total production is seen to be increasing, it might suggest a favorable condition for expansion. Conversely, if the production is decreasing, it may signal that measures should be taken to improve efficiency or cut costs.Overall, calculus serves as a powerful tool in the economist’s toolbox, providing a mathematical foundation for making strategic decisions and interpreting the possible future states of economic variables based on their current rates of change.