Question

A firm producing hockey sticks has a production function given by \\[ q=2 \sqrt{k l} \\] In the short run, the firm's amount of capital equipment is fixed at \(k=100 .\) The rental rate for \(k\) 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 \(S C, S A C,\) 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 \(S A C\) and the \(S M C\) curves for the firm. Indicate the points found in part (b). d. Where does the \(S M C\) curve intersect the \(S A C\) curve? Explain why the \(S M C\) curve will always intersect the \(S A C\) curve at its lowest point. Suppose now that capital used for producing hockey sticks is fixed at \(\bar{k}\) in the short run. c. Calculate the firm's total costs as a function of \(q, w, v,\) and \(\bar{k}\) f. Given \(q, w,\) and \(v,\) how should the capital stock be chosen to minimize total cost? g. Use your results from part (f) to calculate the long-run total cost of hockey stick production. h. For \(w=\mathrm{s} 4, v=\$ 1,\) graph the long-run total cost curve for hockey stick production. Show that this is an envelope for the short-run curves computed in part (c) by examining values of \(\bar{k}\) of \(100,200,\) and 400

Short answer

1. Short-run total cost curve and short-run average cost curve: Given k = 100, w = 4, and v = 1, the Short-run Total Cost (STC) and Short-run Average Cost (SAC) functions are: STC = 4 * (((q / 2)^2) / 100) + 1 * 100 = (q^2 / 50) + 100 SAC = STC / q = (q^2 / 50q) + 100 / q 2. Short-run marginal cost function: SMC = d(STC) / dq = (2q / 50) = q / 25 3. Calculate cost values for various production levels: For q = 25, 50, 100, and 200: | q | STC | SAC | SMC | |------|-----------------------|------------------------|--------| | 25 | (25^2 / 50) + 100 | (25^2 / 50 * 25) + 4 | 1 | | 50 | (50^2 / 50) + 100 | (50^2 / 50 * 50) + 2 | 2 | | 100 | (100^2 / 50) + 100 | (100^2 / 50 * 100) + 1 | 4 | | 200 | (200^2 / 50) + 100 | (200^2 / 50 * 200) + 0.5 | 8 | 4. Graphing SAC and SMC curves and points: [Plot the SAC and SMC curves as functions of q, and mark points from the table above on the respective curves] 5. Intersection of SMC curve and SAC curve: Set SMC = SAC and solve for q: q / 25 = (q^2 / 50q) + 100 / q => q = 50 Hence, SMC intersects SAC at q = 50. The SMC curve will always intersect the SAC curve at its lowest point because the SMC represents the additional cost of producing one more unit, and the SAC represents the average cost of producing all units. When these two values are equal, the firm has reached its most efficient point of production. 6. Calculating the firm's total costs as a function of q, w, v, and k-bar: Replacing k with k-bar in the STC equation: STC = 4 * (((q / 2)^2) / k-bar) + 1 * k-bar 7. Choosing the capital stock to minimize total cost: Differentiating STC with respect to k-bar and setting the result equal to zero: d(STC) / dk-bar = - (4 * (q^2 / 8)) / k-bar^2 + 1 = 0 Solving for k-bar: k-bar = sqrt((4 * (q^2 / 8))) 8. Calculating long-run total cost of hockey stick production: After substituting the optimal k-bar found in step 7 into the total cost function, we get: LRTC = 4 * (((q / 2)^2) / sqrt((4 * (q^2 / 8)))) + 1 * sqrt((4 * (q^2 / 8))) 9. Graphing long-run total cost curve for hockey stick production: For w = 4 and v = 1, plot the LRTC function as a function of q. On the same graph, plot the short-run total cost curves for k-bar of 100, 200, and 400. The LRTC curve should show as an envelope for the short-run curves.

Step-by-step solution

1. Short-run total cost curve and short-run average cost curve

To find the Short-run Total Cost (STC) we need to calculate the total cost of capital (which is fixed) and the total cost of labor. The cost of capital is given by \(v\cdot k\) and the cost of labor is given by \(w\cdot l\). Given the production function: \[q = 2\sqrt{kl}\] Since capital is fixed at \(k=100\) and we have wage rate \(w = 4\), the total cost will be the summation of labor cost and capital cost: STC = Labor cost + Capital cost \[STC = w\cdot l + v\cdot k\] Now, we need to find l (labor) from the production function using q = 2√(kl) and then put this value in STC to isolate it on one side of the equation. From the production function, we have: \[l = \frac{(\frac{q}{2})^2}{k}\] Now putting this value of l in STC, we get: \[STC = w\cdot \frac{(\frac{q}{2})^2}{k} + v\cdot k\] To find the short-run average cost (SAC) we will divide the short-run total cost by the quantity of output: \[SAC = \frac{STC}{q}\] Substitute STC from above and obtain the SAC.

2. Short-run marginal cost function

To find the Short-run Marginal Cost (SMC) we need to take the first derivative of the STC: \[SMC = \frac{d(STC)}{dq}\] Substitute the STC obtained above and differentiate it with respect to q.

3. Calculate cost values for various production levels

We will now calculate the SC, SAC, and SMC at q = 25, 50, 100, and 200. Plug the given value of q into the respective functions (STC, SAC, and SMC) found in steps 1 and 2.

4. Graphing SAC and SMC curves and points

Using the results obtained in step 3, plot the SAC and SMC curves for the firm. Indicate the points found in part (b) corresponding to production levels of 25, 50, 100, and 200 hockey sticks.

5. Intersection of SMC curve and SAC curve

Identify the point at which the SMC curve intersects the SAC curve. To do this, equate the SMC and SAC functions derived in step 1 and step 2, and solve for q. Then explain why the SMC curve will always intersect the SAC curve at its lowest point.

6. Calculating the firm's total costs as a function of q, w, v and k-bar

We are given a new scenario where the capital used is fixed at k-bar. To find the firm's total cost as a function of q, w, v, and k-bar, simply replace k with k-bar in the STC equation derived in step 1.

7. Choosing the capital stock to minimize total cost

Given q, w, and v, derive the optimal value of k-bar that minimizes total cost. To do this, differentiate the total cost function (with k-bar) with respect to k-bar and set the result equal to zero. Solve for k-bar.

8. Calculating long-run total cost of hockey stick production

Use the results obtained in step 7 to calculate the long-run total cost (LRTC) of hockey stick production. LRTC is the total cost function obtained in step 6 after substituting the optimal k-bar found in step 7.

9. Graphing long-run total cost curve for hockey stick production

For w = 4 and v = 1, graph the long-run total cost curve for hockey stick production using the LRTC function found in step 8. On the same graph, show that this curve is an envelope for the short-run curves computed in part (c) by examining the values of k-bar of 100, 200, and 400.

Key concepts

Short-run Total Cost

The Short-run Total Cost (STC) comprises two main parts: the fixed cost of capital and the variable cost of labor. Capital, in this context, refers to assets like machinery and equipment, while labor is the human effort needed for production. In the short-run, one of these inputs is fixed, typically capital, while labor can vary. The production function, given as \( q = 2 \sqrt{k l} \), helps in determining how much labor is needed to produce a given quantity.

• Fixed Cost: This is the cost related to capital. Since capital is fixed at \( k = 100 \) and its rental rate is \( v = 1 \), the fixed cost is constant at \( 100 \times 1 = 100 \).
• Variable Cost: This involves the cost of labor, which varies with production levels. The wage rate is \( w = 4 \), and using the production function, we can find the labor requirement for different levels of output.

The STC function is expressed as the sum of these costs: \[ STC = w \cdot l + v \cdot k \]Inserting the expression for labor from the production function allows us to further analyze costs depending on output \( q \). This understanding of STC is crucial as it sets the stage for further analysis of other cost measures.

Marginal Cost

The Marginal Cost (MC) is a reflection of how the total cost changes with a one-unit increase in production. Calculating the Short-run Marginal Cost (SMC) involves differentiating the STC with respect to \( q \), the quantity of output. Essentially, it shows how much extra cost is involved in producing one more hockey stick.To compute:\[ SMC = \frac{d(STC)}{dq} \]This derivative provides insight into the cost dynamics as production scales. The SMC helps firms to decide whether producing additional units is economically viable, aiding in maximizing profitability and operational efficiency.When plotted, the SMC curve typically intersects the SAC curve at its lowest point. This intersection represents the most efficient scale of production, where costs per unit are minimized.

Long-run Total Cost

Unlike the short-run, in the long-run, a firm can adjust all its inputs, both capital and labor. This flexibility allows firms to find the most cost-effective combination of inputs, minimizing the cost of production for different output levels.The Long-run Total Cost (LRTC) is defined for varying levels of capital and labor, optimizing cost with given prices of inputs \( w \) and \( v \). To find this, steps involve determining the optimal solution for input combinations by minimizing the cost function derived from the production constraints.The envelope curve, or the LRTC, is shaped by the most cost-efficient paths through different short-run cost configurations. Here, capital is varied, and the firm chooses the level of \( k \) that results in the lowest cost for each production target, enabling decisions about expansion or reduction in facilities.

Capital and Labor

Capital and Labor are the backbone inputs in production processes. The interplay between them determines not only output but also the cost structure of a firm.

• Capital: Representing long-term investments, it includes machinery, equipment, or any assets required for production. In the short-run, capital is often considered fixed, which means adjustments cannot be made easily or quickly.
• Labor: It is variable in the short-run, allowing firms to adjust the workforce depending on production needs. The costs associated with labor are influenced by wage rates and the amount of labor required as defined by the production function.

Understanding how these inputs interact, particularly through the use of a production function like \( q = 2 \sqrt{k l} \), is essential for efficient production planning. The ultimate goal for any firm is to find the ideal balance between capital and labor that minimizes costs while meeting output requirements, especially important when making both short-run and long-run strategic decisions.