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.

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} \\]

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 \\]

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.

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.