Question

Suppose that a firm's fixed proportion production function is given by \\[ q=\min (5 k, 10 l) \\] a. Calculate the firm's long-run total, average, and marginal cost functions. b. Suppose that \(k\) is fixed at 10 in the short run. Calculate the firm's short-run total, average, and marginal cost functions. c. Suppose \(v=1\) and \(w=3 .\) Calculate this firm's long-run and short-run average and marginal cost curves.

Short answer

Using the given values for \(v = 1\) and \(w = 3\), we have found the following long-run and short-run cost functions for the firm: 1. Long-run average cost (LAC) function: LAC(q) = - \(\frac{1}{5} + \frac{3}{10}\), if k ≤ l/2 - \(\frac{3}{5}\), if k > l/2 2. Long-run marginal cost (LMC) function: LMC(q) = - \(\frac{1}{2}\), if k ≤ l/2 - \(\frac{3}{5}\), if k > l/2 3. Short-run average cost (SAC) function: SAC(q) = \(\frac{10}{q} + \frac{3}{10}\) 4. Short-run marginal cost (SMC) function: SMC(q) = \(\frac{3}{10}\) These cost functions are crucial for determining the firm's optimal production levels and can be used to analyze pricing strategies, resource allocation, and business expansion in both the short and long run.

Step-by-step solution

Since the production function is given as \(q = \min(5k, 10l)\), we can express it in terms of one input. We can write this as: \\[ q = 5k \text{ if } 5k \leq 10l \\ q = 10l \text{ if } 5k > 10l \\] #Step 2: Derive the long-run total cost (LTC) function#

To find the LTC function, we need to express the cost in terms of both capital (\(k\)) and labor (\(l\)). In the long run, the firm can vary both \(k\) and \(l\). We are given that \(v=1\) and \(w=3\), so total cost is \(C = vk + wl\). Thus, the LTC function can then be formed as follows depending on the production function: \\[ C(q) = k + 3l = \begin{cases} \frac{q}{5} + 3(\frac{q}{10}) & \text{if } k \leq \frac{l}{2} \\ \frac{q}{5}(\frac{3}{2}) + \frac{q}{10} & \text{if } k > \frac{l}{2} \end{cases} \\] #Step 3: Calculate the long-run average cost (LAC) function#

LAC is defined as the long-run total cost divided by the quantity produced. Thus, using the LTC function above, we can write the LAC function as: \\[ LAC(q) = \frac{C(q)}{q} = \begin{cases} \frac{1}{5} + \frac{3}{10} & \text{if } k \leq \frac{l}{2} \\ \frac{3}{5} & \text{if } k > \frac{l}{2} \end{cases} \\] #Step 4: Calculate the long-run marginal cost (LMC) function#

LMC is the derivative of the LTC function with respect to the output \(q\). Since the LTC function is a piecewise function, we will take the partial derivatives with respect to \(q\) in both cases: \\[ LMC(q) = \frac{dC(q)}{dq} = \begin{cases} \frac{1}{2} & \text{if } k \leq \frac{l}{2} \\ \frac{3}{5} & \text{if } k > \frac{l}{2} \end{cases} \\] #Step 5: Find the short-run total, average, and marginal cost functions with fixed capital at 10#

In part (b), capital is fixed at \(k=10\). Therefore, the production function can be expressed as \(q = 5\times10 = 50\) if \(50\leq10l\), or \(q = 10l\) if \(50>10l\). To find the total cost function, we substitute \(k=10\) into the cost function, obtaining \(C(q) = 10 + 3l\). Since the cost in the short run only contains one variable, we solve this cost function for \(l\) in terms of the output: \\[ C(q) = 10 + 3(\frac{q}{10}) => C(q) = 10 + \frac{3q}{10} \\] Thus, the short-run total cost (STC) function is obtained: \\[ STC(q) = 10 + \frac{3q}{10} \\] Next, we find the average cost by dividing the STC function by the output, \(q\), obtaining the short-run average cost (SAC) function: \\[ SAC(q) = \frac{STC(q)}{q} = \frac{10}{q} + \frac{3}{10} \\] Lastly, we calculate the short-run marginal cost (SMC) function, which is the derivative of the STC function with respect to the output: \\[ SMC(q) = \frac{dSTC(q)}{dq} = \frac{3}{10} \\] #Step 6: Solve long-run and short-run cost functions with given values of \(v\) and \(w\)#

With \(v=1\) and \(w=3\), we can now calculate the long-run and short-run average and marginal cost functions given in the previous steps: Long-run average cost: \\[ LAC(q) = \begin{cases} \frac{1}{5} + \frac{3}{10} \\ \frac{3}{5} \end{cases} \\] Long-run marginal cost: \\[ LMC(q) = \begin{cases} \frac{1}{2} \\ \frac{3}{5} \end{cases} \\] Short-run average cost: \\[ SAC(q) = \frac{10}{q} + \frac{3}{10} \\] Short-run marginal cost: \\[ SMC(q) = \frac{3}{10} \\]

Key concepts

Long-Run Total Cost

In microeconomics, the long-run total cost (LTC) is a vital concept which reflects the total cost incurred by a firm when all inputs are variable. Unlike the short run, where some factors like capital might be fixed, the long run allows a business to adjust all its inputs to find the most efficient way to produce goods or services.

The LTC is derived by summing the product of the quantity of each input with its respective cost. In our exercise, we look at a firm that operates under a fixed proportion production function, represented by \( q = \text{min}(5k, 10l) \). To calculate the LTC function, two scenarios are considered based on the relationship between the inputs capital (k) and labor (l). For teaching purposes, it's essential to remember that the LTC curve typically has a characteristic 'U' shape, indicating economies and diseconomies of scale at different output levels.

Average Cost Functions

The average cost functions serve as an indicator of efficiency by showing the cost per unit of output. The long-run average cost (LAC) function is obtained by dividing the LTC by the total output (q). The concept is crucial for understanding at which scale a firm operates most efficiently.

In the fixed proportion production context, the LAC depends on whether capital is less than or equal to half of labor, or greater. Optimization of production costs in microeconomics often revolves around finding the minimum point of the LAC, which corresponds to the 'optimum' size of the firm for cost-efficient production. When explaining this concept to students, using diagrams to illustrate how LAC varies with the output can be particularly helpful.

Marginal Cost Functions

Another cornerstone concept in microeconomics is the marginal cost functions. The long-run marginal cost (LMC) function represents the additional cost of producing one extra unit of output. It is derived by taking the derivative of the LTC function with respect to output, which shows the change in cost from an infinitesimally small change in the quantity produced.

This measure is pivotal in production decision-making, as it indicates the moment when producing one more unit becomes more expensive than the previous. Therefore, LMC helps determine the most profitable output level. For students, understanding the relationship between the marginal cost and average cost functions is essential, as it underpins the supply behavior of a firm.

Short-Run Total Cost

The short-run total cost (STC) function is a reflection of the firm's total cost when at least one factor of production is fixed. In our textbook example, capital (k) is fixed in the short run, bringing in a different perspective on cost calculation compared to the long run. The STC is important for businesses to understand how costs behave when they cannot fully adjust all inputs.

It's crucial for students to grasp how the STC differs from the LTC owing to the presence of fixed costs, which do not change with the level of output. These costs become sunk in the short run but can be adjusted in the long run. Specifically explaining how fixed costs influence the average and marginal costs in the short run can provide a more comprehensive understanding of a firm's immediate cost structure.

Fixed Proportion Production Function

Underlying these discussions is the fixed proportion production function<\/b>. This function signifies that output is limited by a fixed ratio of the inputs, which in our exercise are capital (k) and labor (l). The fixed proportion relationship is given by \( q = \text{min}(5k, 10l) \), indicating that the quantity of output is constrained by whichever input is the limiting factor.

Students should learn that this type of production function indicates a rigid production technology where inputs are strong complements to each other, as opposed to being easily substitutable. Consequently, understanding this function aids in analyzing the relationship between input quantities and costs, and it underscores the importance of balanced growth of input usage for cost management.