Question

A firm producing hockey sticks has a production function given by In the short run, the firm's amount of capital equipment is fixed at \(K=100\). The rental rate for \(\mathrm{AT}\) is \(v=\$ 1,\) and the wage rate for \(L\) is \(w-\$ 4\) a. Calculate the firm's short-run total cost curve. Calculate the short-run average cost curve. b. What is the firm's short-run marginal cost function? What are the \(S T C, S A T C,\) and \(S M C\) for the firm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks? c. Graph the \(S A T C\) and the \(S M C\) curves for the firm. Indicate the points found in part (b). d. Where does the SMC curve intersect the SATC curve? Fxplain why the SMC curve will al ways intersect the \(S A T C\) curve at its lowest point.

Short answer

In this exercise, we calculated the short-run total cost (TC), short-run average cost (AC), and short-run marginal cost (MC) curves for a firm producing hockey sticks with a production function of q = 2√(K⋅L), where capital (K) is fixed at 100 in the short run. We found the TC, AC, and MC functions and computed these values for various levels of output quantity (q). We also discussed graphing the SATC and SMC curves and determined the point of intersection, which represents the lowest point of the SATC curve. The reason for this intersection point is that when SMC is less than SATC, average costs decrease, and when SMC is greater than SATC, average costs increase. The point where SMC equals SATC is the lowest point of the SATC curve, corresponding to the minimum average cost of production.

Step-by-step solution

A. Calculate Short-Run TC and AC Curves

First, we need to rewrite the production function in terms of L, as K is fixed: \\[ L = \frac{q^2}{4K} \\] To find the total cost (TC) function, multiply the wage rate (w) by the labor input (L) and the rental rate (v) by the capital equipment (K): \\[ TC = wL+vK = 4 \cdot \frac{q^2}{4K}+\$1 \cdot 100 \\] Plug in the given values of K = 100, v = \(1, and w = \)4: \\[ TC = \frac{q^2}{100} + 100 \\] Now, to find the average cost (AC) function, divide TC by the output quantity (q): \\[ AC = \frac{TC}{q} = \frac{q}{100} + \frac{100}{q} \\]

B. Short-Run Marginal Cost Function and Calculations

To find the marginal cost (MC) function, find the first derivative of the total cost (TC) function with respect to the output quantity (q): \\[ MC = \frac{d(TC)}{dq} = \frac{2q}{100} \\] Now, calculate the STC, SATC, and SMC for q = 25, 50, 100, and 200: 1. q = 25: \\[ STC = \frac{25^2}{100} + 100 = 156.25 \\] \\[ SATC = \frac{25}{100} + \frac{100}{25} = 5.25 \\] \\[ SMC = \frac{2(25)}{100} = 0.50 \\] 2. q = 50: \\[ STC = \frac{50^2}{100} + 100 = 250 \\] \\[ SATC = \frac{50}{100} + \frac{100}{50} = 3 \\] \\[ SMC = \frac{2(50)}{100} = 1 \\] 3. q = 100: \\[ STC = \frac{100^2}{100} + 100 = 200 \\] \\[ SATC = \frac{100}{100} + \frac{100}{100} = 2 \\] \\[ SMC = \frac{2(100)}{100} = 2 \\] 4. q = 200: \\[ STC = \frac{200^2}{100} + 100 = 500 \\] \\[ SATC = \frac{200}{100} + \frac{100}{200} = 2.5 \\] \\[ SMC = \frac{2(200)}{100} = 4 \\]

C. Graph SATC and SMC Curves

Plotting the Short-Run Average Total Cost (SATC) and Short-Run Marginal Cost (SMC) curves with the calculated points from part (b). You can use graphing software or graph paper to create the graph. 1. SATC curve: Plot the curve of the function \\[ AC = \frac{q}{100} + \frac{100}{q} \\] 2. SMC curve: Plot the curve of the function \\[ MC = \frac{2q}{100} \\] 3. Points from part (b): Indicate the STC, SATC, and SMC values for q =25, 50, 100, and 200 on the graph.

D. Intersection of SMC and SATC Curves

To find the intersection point of the SATC and SMC curves, equate the AC and MC functions: \\[ \frac{q}{100} + \frac{100}{q} = \frac{2q}{100} \\] Solving for q, we get q = 100. The SMC curve intersects the SATC curve at the minimum point of the SATC curve. This happens because when the SMC is less than SATC, the average cost decreases, and when SMC is greater than SATC, the average cost increases. The point where SMC equals SATC is the point where average cost stops decreasing and starts increasing, hence the lowest point of the SATC curve.

Key concepts

Production Function

A production function is a fundamental concept in economics that describes the relationship between inputs used in production and the quantity of output produced. In our example, the firm producing hockey sticks has a production function that shows how labor and capital are transformed into hockey sticks. Given a fixed amount of capital equipment, denoted as \(K=100\), any change in production primarily involves altering the amount of labor, denoted as \(L\). The formula for the production function is \(L = \frac{q^2}{4K}\), where \(q\) represents the quantity of hockey sticks produced. This formula shows that if more hockey sticks are to be made, more labor is needed, emphasizing the direct relationship between labor input and output level.
Understanding production functions helps firms optimize resource use by determining the necessary combinations of labor and capital for desired output levels. It’s a blueprint for making informed decisions in the short run, where some inputs remain constant, like our fixed capital, \(K\).
Key points to remember:

• Production functions show the output that can be achieved with different input combinations.
• In the short run, some inputs (like capital) remain fixed.
• Labor is often adjusted to meet changing production needs.

Marginal Cost

Marginal cost (MC) is the additional cost incurred from producing an extra unit of output. For the firm producing hockey sticks, the short-run marginal cost is determined by taking the first derivative of the total cost (TC) function with respect to output quantity \(q\). The formula is \(MC = \frac{2q}{100}\). This calculation gives us insight into how costs change with changes in production levels.
For example, when the firm moves production from 25 to 50 hockey sticks, the marginal cost increases, reflecting the added expense of boosting production. It provides a crucial decision-making tool, especially in competitive markets, helping firms determine the optimal level of production to maximize profits.

• MC is calculated from the first derivative of the TC function.
• It represents the cost of producing one more unit.
• It's essential for pricing and production decisions.

Average Cost

Average cost (AC) refers to the total cost per unit of output, calculated by dividing total cost by the quantity of output. In this example, the formula for calculating the average cost is given by \(AC = \frac{q}{100} + \frac{100}{q}\). It shows how the average cost behaves as production scales.
At different output levels, like 25, 50, 100, and 200 hockey sticks, the AC changes, reflecting the firm’s efficiency or inefficiency at each level of production. This information aids the firm in pinpointing output levels that optimize cost-effectiveness. Lower average costs contribute to competitive pricing and profitability.

• AC gives the cost per unit, vital for analyzing cost efficiency.
• Helps identify ideal production levels for minimizing costs.
• Supports strategic pricing policies.

Fixed Capital

Fixed capital represents resources that remain unchanged in the short run, regardless of the level of production. In our scenario, the hockey stick firm's fixed capital is \(K=100\). This constant level of capital highlights the constraints firms face in adjusting to short-term production demands without altering fixed assets.
Fixed capital is a significant factor in cost analysis since it contributes to fixed costs that don’t change with production. Knowing the fixed capital requirements enables better financial planning and resource allocation, crucial for long-term strategic development.

• Fixed capital remains constant in the short run.
• Impacts total and average cost calculations.
• Is crucial for understanding the resource limits in production.