Question

Suppose that a firm's production function exhibits technical improvements over time and that the form of the function is \(q=f(k, l, t) .\) In this case, we can measure the proportional rate of technical change as \\[ \frac{\partial \ln q}{\partial t}=\frac{f_{t}}{f} \\] (compare this with the treatment in Chapter 9 ). Show that this rate of change can also be measured using the profit function as \\[ \frac{\partial \ln q}{\partial t}=\frac{\Pi(P, v, w, t)}{P q} \cdot \frac{\partial \ln \Pi}{\partial t} \\] That is, rather than using the production function directly, technical change can be measured by knowing the share of profits in total revenue and the proportionate change in profits over time (holding all prices constant). This approach to measuring technical change may be preferable when data on actual input levels do not exist.

Short answer

Question: Prove that the proportional rate of technical change can be measured using the profit function given as \(\frac{\partial \ln q}{\partial t}=\frac{\Pi(P, v, w, t)}{P q} \cdot \frac{\partial \ln \Pi}{\partial t}\). Solution: We followed the steps of differentiating the profit function, calculating the proportionate change in profits over time, and finding the connection between the proportional rate of technical change and the profit function. We derived both sides of the given equation and found that they are equal, thus proving the desired result.

Step-by-step solution

The production function with technical improvements over time is given by: \(q = f(k, l, t)\), where \(q\) is the output, \(k\) is the capital input, \(l\) is the labor input, and \(t\) represents the technical progress parameter. The profit function \(\Pi(P, v, w, t)\) is given by: \(\Pi = P q - v k - w l\), where \(P\) is the output price, \(v\) is the price of capital and \(w\) is the price of labor. #Step 2: Define the proportional rate of technical change#

From the question, we know that the proportional rate of technical change can be given as: \(\frac{\partial \ln q}{\partial t} = \frac{f_{t}}{f}\), where \(f_{t}\) represents the partial derivative of \(f\) with respect to \(t\). We need to show that the proportional rate of technical change can also be calculated using the profit function as: \(\frac{\partial \ln q}{\partial t}=\frac{\Pi(P, v, w, t)}{P q} \cdot \frac{\partial \ln \Pi}{\partial t}\). #Step 3: Differentiate the profit function with respect to \(t\)#

We will differentiate the profit function \(\Pi = P q - v k - w l\) with respect to \(t\). We can write \(\Pi = P f(k, l, t) - v k - w l\). Now taking the derivative with respect to \(t\): \(\frac{\partial \Pi}{\partial t} = P \frac{\partial f}{\partial t} - v \frac{\partial k}{\partial t} - w \frac{\partial l}{\partial t}\). Since we are holding all prices constant, we have \(\frac{\partial k}{\partial t} = \frac{\partial l}{\partial t} = 0\). So, \(\frac{\partial \Pi}{\partial t} = P \frac{\partial f}{\partial t}\). #Step 4: Differentiate the profit function with respect to \(f\) to find the shares of profits in total revenue#

Now, we differentiate \(\Pi\) with respect to \(f\), treating \(f\) as a variable and holding all other variables constant: \(\frac{\partial \Pi}{\partial f} = P \cdot \frac{\partial q}{\partial f} = P\). So, the share of profits in total revenue is given by \(\frac{\Pi}{P q}\). #Step 5: Calculate the proportionate change in profits over time#

To find the proportionate change in profits over time, differentiate the natural logarithm of \(\Pi\) with respect to \(t\): \(\frac{\partial \ln \Pi}{\partial t} = \frac{1}{\Pi}\cdot\frac{\partial \Pi}{\partial t} = \frac{P \frac{\partial f}{\partial t}}{\Pi}\). #Step 6: Show the connection between the proportional rate of technical change and the profit function#

Substitute the share of profits in total revenue and the proportionate change in profits over time in the given equation: \(\frac{\partial \ln q}{\partial t}=\frac{\Pi(P, v, w, t)}{P q} \cdot \frac{\partial \ln \Pi}{\partial t} = \frac{\Pi}{P q} \cdot \frac{P \frac{\partial f}{\partial t}}{\Pi}\). Observe that \(\Pi\) cancels in the numerator and denominator, and we get: \(\frac{\partial \ln q}{\partial t} = \frac{\frac{\partial f}{\partial t}}{f}\). Since we know that \(\frac{\partial \ln q}{\partial t} = \frac{f_{t}}{f}\), we can see that both sides of the given equation are equal, thus proving the desired result.