Question

Suppose the production function for widgets is given by \\[ q=k l-0.8 k^{2}-0.2 l^{2} \\] where \(q\) represents the annual quantity of widgets produced, \(k\) represents annual capital input, and \(l\) represents anntal labor input. a. Suppose \(k=10 ;\) graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that \(k=10\), graph the \(M P_{l}\) curve. At what level of labor input does \(M P_{l}=0\) ? c. Suppose capital inputs were increased to \(k=20 .\) How would your answers to parts (a) and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

Short answer

a. Maximum AP at \( l=20 \), producing 40 widgets. b. \( MPl=0 \) at \( l=25 \). c. With \( k=20 \), max AP at \( l=40 \); \( MPl=0 \) at \( l=50 \). d. Decreasing returns to scale since doubling inputs results in less than double output.

Step-by-step solution

Finding Total and Average Productivity

Given the production function \( q = 10l - 0.8(10)^2 - 0.2l^2 = 10l - 80 - 0.2l^2 \), simplify this as: \( q = 10l - 0.2l^2 - 80 \). The total productivity, \( TP = q = 10l - 0.2l^2 - 80 \). The average productivity, \( AP = \frac{q}{l} = 10 - 0.2l - \frac{80}{l} \). To find the maximum of \( AP \), set the derivative \( \frac{d(AP)}{dl} = 0 \).

Deriving the Average Productivity Formula

Differentiate \( AP = 10 - 0.2l - \frac{80}{l} \) with respect to \( l \): \( \frac{d(AP)}{dl} = -0.2 + \frac{80}{l^2} \). Set the derivative equal to zero: \( -0.2 + \frac{80}{l^2} = 0 \). Solve for \( l \) to find the level of labor input maximizing average productivity.

Solving for Maximum Average Productivity

\( \frac{80}{l^2} = 0.2 \) yields \( 80 = 0.2l^2 \) or \( l^2 = 400 \). Hence, \( l = 20 \). Substitute \( l = 20 \) into the total production function: \( q = 10(20) - 0.2(20)^2 - 80 = 200 - 80 - 80 = 40 \) widgets.

Calculating Marginal Productivity of Labor

Given \( q = 10l - 0.2l^2 - 80 \), the marginal productivity of labor (MPl) is the derivative of \( q \) with respect to \( l \): \( MPl = \frac{dq}{dl} = 10 - 0.4l \). Set \( MPl = 0 \) to find when it reaches this value.

Solving for Zero Marginal Productivity

Set \( 10 - 0.4l = 0 \), yielding \( 0.4l = 10 \) or \( l = 25 \). Thus, \( MPl = 0 \) when Labor input \( l = 25 \).

Changing Capital to 20

With \( k = 20 \), the new production function is \( q = 20l - 0.8(20)^2 - 0.2l^2 = 20l - 320 - 0.2l^2 \). Total productivity becomes \( q = 20l - 0.2l^2 - 320 \), and average productivity is \( AP = \frac{q}{l} = 20 - 0.2l - \frac{320}{l} \). Solve similarly for maximum AP and zero MP based on these equations.

Solving with Increased Capital input

With \( k = 20 \), solve \( \frac{d(AP)}{dl} = 0 \): \( -0.2 + \frac{320}{l^2} = 0 \) implies \( l^2 = 1600 \) or \( l = 40 \). Total output at that labor input is \( q = 20(40) - 0.2(40)^2 - 320 = 480 \). For zero MP, solve \( 20 - 0.4l = 0 \), yielding \( l = 50 \).

Analyzing Returns to Scale

To examine returns to scale, compare outputs with a proportional increase in both \( k \) and \( l \). If they double both and check if production doubles, they're constant returns. If production more than doubles, it's increasing returns, otherwise decreasing.

Key concepts

Total Productivity

In economics, total productivity refers to the total output produced by a firm or production process using given inputs. Here, we consider a production function, which outlines the relationship between inputs and output. For widgets, the production function is given by\[ q = kl - 0.8k^{2} - 0.2l^{2} \]where \(k\) is the capital input and \(l\) is the labor input. Total productivity involves calculating the total number of widgets produced for different inputs. By substituting specific values of \(k\) and \(l\) into this function, we can find the total widgets produced. This is crucial for firms to understand how changes in their input affect their output.
For instance, when capital \(k\) is set to 10, the equation simplifies to \[ q = 10l - 0.2l^{2} - 80 \] which we can solve to analyze total production based on varying labor inputs. Total productivity helps understanding how different scales of production impact overall efficiency.

Average Productivity

Average productivity is a measure of the efficiency of labor or capital, essentially calculating the output per unit of input. For labor, it is defined as the total output divided by the labor input, represented by the formula:\[ AP = \frac{q}{l} \]In this context, substituting the simplified production function for widgets,\[ AP = \frac{10l - 0.2l^2 - 80}{l} = 10 - 0.2l - \frac{80}{l} \]This equation demonstrates how average productivity changes as labor changes. To identify the level of labor input required for maximum average productivity, we differentiate the AP function with respect to \(l\) and set the derivative to zero:\[ \frac{d(AP)}{dl} = -0.2 + \frac{80}{l^2} = 0 \]Solving gives us the point at which AP is maximized, providing insight into the optimal labor use for maximizing productivity and determining at what point additional labor becomes less effective.

Marginal Productivity

Marginal productivity measures the additional output generated when an additional unit of input is added. Mathematically, it's described as the derivative of the output with respect to the input, allowing us to see the effect on output from small changes in input.Given the production function,\[ q = 10l - 0.2l^2 - 80 \]the marginal productivity of labor \(MPl\) is found by taking the derivative:\[ MPl = \frac{dq}{dl} = 10 - 0.4l \]To determine where marginal productivity is zero, set the equation to zero and solve for \(l\):\[ 10 - 0.4l = 0 \]This shows when labor becomes so abundant that adding more labor doesn't increase total output, known as a point of diminishing returns. Understanding this helps firms optimize their labor usage, ensuring they don't waste resources on unproductive labor.

Returns to Scale

Returns to scale analyze how changes in input affect production output when both labor and capital inputs are scaled simultaneously. This concept is fundamental in determining how efficiently a firm can increase production and how costs relate to output. In this exercise, we compare the outputs when inputs are doubled to see if the output doubles: 1. **Constant Returns to Scale:** If output doubles as inputs double, production is efficient and scalable. 2. **Increasing Returns to Scale:** If output more than doubles, increasing scale is beneficial. 3. **Decreasing Returns to Scale:** If output less than doubles, it indicates inefficiencies in scaling. By analyzing the widget production function under different input scenarios, we identify how well the production adapts to changes in input size. This knowledge aids in strategic planning and investment decisions, ensuring firms understand their production capabilities and limitations.