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: Show that the rate of technical change can also be measured using the profit function. Answer: The rate of technical change can be measured using the profit function by applying the formula: \\[ \frac{\partial \ln q}{\partial t} = \frac{\Pi(P, v, w, t)}{Pq} \cdot \frac{\partial \ln \Pi}{\partial t}. \\] This alternative approach is useful when data on actual input levels are not available, and we only need to know the share of profits in total revenue and the proportionate change in profits over time, while holding all prices constant.

Step-by-step solution

Introducing the Profit Function

First, let's define the profit function for this firm. The profit function, denoted \(\Pi(P, v, w, t)\), is given by the firm's total revenue (\(Pq\), where \(P\) is the price of the produced good) minus its total costs (\(vk+wl\), where \(v\) and \(w\) are the input costs of capital \(k\) and labor \(l\) respectively): \\[ \Pi(P, v, w, t) = Pq - (vk+wl). \\]

Differentiating the profit function with respect to time

Now, let's differentiate the profit function with respect to time (\(t\)) while holding the input prices (\(P\), \(v\), and \(w\)) constant: \\[ \frac{\partial \Pi(P, v, w, t)}{\partial t} = P\frac{\partial q}{\partial t} - v\frac{\partial k}{\partial t} - w\frac{\partial l}{\partial t}. \\]

Connecting the technical change to the profit function

As we need to find a relationship between the technical change rate and profit function, let's now multiply both sides of the equation derived in Step 2 by \(f\) and rearrange for \(\frac{\partial \ln q}{\partial t}\): \\[ \frac{\partial \ln q}{\partial t} = \frac{f_{t}}{f} = \frac{(P\frac{\partial q}{\partial t} - v\frac{\partial k}{\partial t} - w\frac{\partial l}{\partial t})f}{Pq} = \frac{\Pi(P, v, w, t)}{Pq}\cdot\frac{\partial \ln \Pi}{\partial t}. \\] We have shown that the rate of technical change can also be measured using the profit function as \\[ \frac{\partial \ln q}{\partial t} = \frac{\Pi(P, v, w, t)}{Pq} \cdot \frac{\partial \ln \Pi}{\partial t}. \\] This alternative approach can be useful when data on actual input levels (i.e., \(k\) and \(l\)) are not available, as we only need to know the share of profits in total revenue and the proportionate change in profits over time, while holding all prices constant.

Key concepts

Production Function

Understanding a firm’s production function is crucial when analyzing its efficiency and capabilities. A production function, represented as \( q = f(k, l, t) \), describes the relationship between the quantity of output produced (\( q \)) and the input factors such as capital (\( k \)), labor (\( l \)), and often time (\( t \)) signifying technological progress or other changes over time.

The inclusion of time in this function allows us to measure how production efficiency changes due to technological advancements, enhancing our understanding of a firm's dynamic production capabilities. This can be quantitatively analyzed by determining the proportional rate of technical change, which indicates how output responds to technological progress over time.

Profit Function

The profit function represents a firm’s financial health by calculating the difference between total revenue and total costs. Expressed as \( \Pi(P, v, w, t) = Pq - (vk+wl) \), where \( P \) is the price of the output, \( q \) is the quantity of goods produced, \( v \) and \( w \) are the input prices of capital and labor, respectively, and \( k \) and \( l \) are the quantities of capital and labor used.

By analyzing the profit function, especially how it changes over time with respect to \( t \), we can gain insights into how technical change impacts profits, keeping input prices constant. This approach is especially advantageous when direct measurement using production inputs is not possible.

Proportional Rate of Technical Change

The proportional rate of technical change is a measure of the intensity and pace at which technology impacts the production process. Mathematically, it is represented as \( \frac{\partial \ln q}{\partial t} = \frac{f_{t}}{f} \). This essentially captures the percentage change in output due to a one-unit change in time, holding all other factors constant.

Moreover, the exercise demonstrates that this rate of technical change can be equivalently expressed through the profit function as \( \frac{\partial \ln q}{\partial t} = \frac{\Pi(P, v, w, t)}{Pq} \cdot \frac{\partial \ln \Pi}{\partial t} \). This method leverages the idea that profits incorporate all the effects of technical change, making it a proxy for assessing the rate at which a firm’s technology is improving when direct measurement from the production function is not feasible.