Question

Suppose that firm \(\mathrm{A}\) has the following short-run production function \(\mathrm{Q}=\mathrm{K}_{\mathrm{c}} \sqrt{\mathrm{L}}\), where \(K\) denotes capital and \(L\) labour. Suppose that the level of capital is fixed at \(\mathrm{k}_{0}=10\) The total cost of firm \(\mathrm{A}\) in the short run is \(\mathrm{STC}=10 \mathrm{wL}\) where \(w\) is the wage paid to each worker. Assume that the wage is \(£ 20\). Using the production function, show how the short-run total cost depends on the quantity produced \(Q\). Plot the short-run total cost on a graph, where you put \(Q\) on the horizontal axis.

Short answer

The short-run total cost, \(\text{STC}\), is given by \(2Q^2\). It is quadratic and increases as \(Q\) increases.

Step-by-step solution

Express Labour as a Function of Output

Start with the given production function \(Q = K_{c} \sqrt{L}\) and substitute the fixed level of capital \(K_0 = 10\), leading to \(Q = 10 \sqrt{L}\). Solve this equation for \(L\):\[\sqrt{L} = \frac{Q}{10}\]\[L = \left(\frac{Q}{10}\right)^2\]

Substitute Labour into Total Cost Equation

We know the short-run total cost is \(\text{STC} = 10 \times w \times L\). With \(w = £20\), substitute \(L\) from Step 1 into the cost equation:\[\text{STC} = 10 \times 20 \times \left(\frac{Q}{10}\right)^2\] Simplify this to obtain:\[\text{STC} = 200 \times \left(\frac{Q^2}{100}\right)\] \[\text{STC} = 2Q^2\]

Plot the Short-Run Total Cost Curve

The equation \(\text{STC} = 2Q^2\) represents a parabola opening upwards, with its vertex at the origin \((0,0)\). On a graph, draw the horizontal axis as quantity \(Q\) and the vertical axis as short-run total cost \(STC\). Plot the curve by computing several points for different values of \(Q\) and connecting them with a smooth curve. For example, for \(Q=10\), \(\text{STC}=2 \times 10^2 = 200\), and for \(Q=20\), \(\text{STC}=2 \times 20^2 = 800\).

Key concepts

Production Function

A production function is a fundamental concept in economics that describes the relationship between inputs and outputs in the production process. In the short-run scenario given, the production function used by firm \(A\) is \(Q = K_{c} \sqrt{L}\). Here, \(Q\) stands for the quantity of output produced, \(K\) symbolizes capital which is fixed at 10 (\(K_0 = 10\)), and \(L\) represents labour.
The production function shows how different inputs, specifically capital and labour in this case, are transformed into outputs. Since capital is fixed in the short run, any increase in output \(Q\) must come from changes in the labour \(L\).

• Mathematically, \(Q = K_{0} \sqrt{L}\) means output is proportionally related to the square root of labour when capital is constant.
• The square root function indicates diminishing returns to labour; doubling the amount of labour input won't double the output.

To understand how much labour is necessary to produce a certain quantity of output, you can rearrange the formula to solve for \(L\), which leads to \(L = \left( \frac{Q}{10} \right)^2\). This step is crucial for determining the cost based on the level of labour needed to achieve that output.

Labour Cost

The concept of labour cost is essential as it forms a significant part of a company's short-run total cost. For firm \(A\), the labour cost can be derived from the short-run total cost function given: \(\text{STC} = 10wL\), where \(w\) is the wage per worker, set at £20.
To calculate the short-run total cost based on production, you first need to express labour \(L\) as a function of output \(Q\) which is done using the production function: \(L = \left( \frac{Q}{10} \right)^2\). Substituting this into the total cost function provides:\[\text{STC} = 10 \times 20 \times \left(\frac{Q}{10}\right)^2\]
Simplifying, you get:\[\text{STC} = 2Q^2\]

• The resulting equation means that the short-run total cost increases with the square of output quantity \(Q\).
• This relationship reflects the rising cost of production as more output requires increasing amounts of labour input, given the fixed capital.

Understanding how labour costs rise with increased production is vital for managing resources efficiently in the short run.

Graphing Economic Functions

Graphing is a potent means of visualizing economic functions, enabling readers to quickly grasp relationships between variables. In the context of firm's short-run cost structure, plotting the equation \(\text{STC} = 2Q^2\) helps us understand how costs change with varying production levels.
To set this up on the graph:

• Place the quantity produced \(Q\) on the horizontal axis.
• Plot the short-run total cost \(STC\) on the vertical axis.

The STC curve, represented by \(2Q^2\), is a parabola that opens upwards. Here's how you can create this graph:

• Calculate several key points. For instance, when \(Q=10\), \(\text{STC} = 2 \times 10^2 = 200\).
• For another point, \(Q=20\) results in a cost of \(\text{STC} = 2 \times 20^2 = 800\).
• Join these points with a smooth curve that starts from the origin (0,0), illustrating the cost of production increases as output rises.

Graphing the function helps visualize how the cost function is quadratic, reflecting the dynamic of labour costs increasing with production. Understanding graphing aids in better comprehension and decision-making regarding operational scales in the short run.