Question

The output of an economic system \(Q,\) subject to two inputs, such as labor \(L\) and capital \(K\) is often modeled by the Cobb-Douglas production function \(Q=c L^{a} K^{b} .\) When \(a+b=1,\) the case is called constant returns to scale. Suppose \(Q=1280, a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40.\) a. Find the rate of change of capital with respect to labor, \(d K / d L.\) b. Evaluate the derivative in part (a) with \(L=8\) and \(K=64.\)

Short answer

Answer: The rate of change of capital with respect to labor when \(L=8\) and \(K=64\) is \(-1\).

Step-by-step solution

Isolate K from the production function.

Divide both sides by \(cL^a\) to isolate K: $$K = \frac{Q}{cL^a}$$

Differentiate K with respect to L.

Differentiate both sides with respect to L using the chain rule: $$\frac{dK}{dL} = \frac{-aQ}{cL^{a+1}}$$

Substitute the given values.

Plug in the given values for \(Q, a, c,\) and \(L=8\): $$\frac{dK}{dL} = \frac{-\dfrac{1}{3}(1280)}{40(8^{\dfrac{4}{3}})}$$

Simplify the expression.

Simplify the expression as much as possible: $$\frac{dK}{dL} = \frac{-1280}{40(32)}$$ $$\frac{dK}{dL} = \frac{-32}{32}$$ So, $$\frac{dK}{dL} = -1$$ Now, we'll find the rate of change when \(L=8\) and \(K=64\)

Calculate \(dK/dL\) when \(L=8\) and \(K=64\).

Plug in the values for \(L\) and \(K\) into the expression we derived in step 4: $$\frac{dK}{dL} = -1$$ So, the rate of change of capital with respect to labor when \(L=8\) and \(K=64\) is \(-1\).

Key concepts

Understanding the Rate of Change

The 'rate of change' is a fundamental concept in calculus and economics that describes how a quantity, such as capital in production, changes in response to changes in another quantity, like labor. In the context of the Cobb-Douglas production function, the rate of change of capital with respect to labor, denoted as \(\frac{dK}{dL}\), represents how much capital usage would need to adjust to compensate for a change in the labor input while maintaining a constant output level.

In practical terms, if we know the rate of change, we can predict how altering one input, let's say by adding one more unit of labor, will affect the other input, capital in this case. A negative rate of change, as seen in the given exercise solution where \(\frac{dK}{dL} = -1\), suggests that an increase in labor by one unit would require a decrease of one unit of capital to keep output constant. This direct relationship helps businesses optimize input combinations for cost efficiency and maximum productivity.

Partial Derivatives in Action

Partial derivatives are used to measure how a function changes as one of the variables changes, while all other variables are held constant. In the context of the Cobb-Douglas production function, we examine how a change in labor, \(L\), impacts the output, \(Q\), while keeping capital, \(K\), constant, and vice versa. This involves taking the derivative of the function with respect to one variable at a time.

When we find the partial derivative of \(K\) with respect to \(L\) as in the given exercise, we are essentially investigating how changes in the labor input would influence the amount of capital required. This information can be crucial for managers and economists trying to understand the dynamic between labor and capital inputs, and how they can be altered to affect production outcomes in an economy or business.

The Chain Rule to Differentiate Complex Functions

The chain rule is a derivative rule that allows us to compute the derivative of composite functions. Whenever one quantity depends on another, which in turn depends on some third quantity, we can use the chain rule to find the rate at which the first quantity changes with respect to the third one.

In our particular problem, the rate of change of capital with respect to labor, \(\frac{dK}{dL}\), is found using the chain rule. The output \(Q\) is a function of both capital, \(K\), and labor, \(L\), and if we rearrange the production function to express \(K\) as a function of \(L\), any change in \(L\) will create a 'chain' of changes that also affects \(K\). By applying the chain rule as demonstrated in the steps provided in the solution, we can precisely calculate how a small change in labor input will modify the required capital input, a vital analysis for economic planning and management.