The town of Podunk has decided to provide security services to its residents by hiring workers \((L)\) and guard dogs \((D) .\) Security services \((S)\) are produced according to the production function \\[ s=V_{L D} \\] and residents of the town wish to consume 10 units of such services per period. a. Suppose that \(L\) and \(D\) both rent for \(\$ 1\) per period. How much of each input should the town hire to produce the desired services at minimal cost? What will that cost be? b. Suppose now that Podunk is the only hirer of people who work with guard dogs and that the supply curve for such workers is given by \\[ L=10 z v \\] where \(w\) is the per-period wage of guard dog handlers. If dogs continue to rent for \$1 per period, how much of each input should the town hire to produce the desired services at minimal cost? What will those costs be? What will the wage rate of dog handlers be?

1. Write the production function and constraints: We have the production function \(s = V_{LD}\), and the constraint is that the residents want to consume 10 units of security services per period, which means \(s = 10\). 2. Substitute the constraint into the production function: Replacing \(s = 10\), we get \(10 = V_{LD}\), or \(V = \frac{10}{LD}\). 3. Set up the cost function: As both labor and guard dogs cost \(1 per period, the total cost function is given by \)C(L,D) = L + D$. 4. Minimize the cost function: We need to minimize the cost function \(C(L,D)\) subject to constraint \(V = \frac{10}{LD}\). To do that, we can use the method of Lagrange multipliers. Let's set up the Lagrangian function: \(\mathcal{L}(L, D, \lambda) = L + D + \lambda(\frac{10}{LD} - V)\). 5. Differentiate the Lagrangian function with respect to L, D, and λ: \\[\frac{\partial \mathcal{L}}{\partial L} = 1 - \frac{10\lambda}{D^2} = 0\\] \\[\frac{\partial \mathcal{L}}{\partial D} = 1 - \frac{10\lambda}{L^2} = 0\\] \\[\frac{\partial \mathcal{L}}{\partial \lambda} = \frac{10}{LD} - V = 0\\] 6. Solve the system of equations: From the first and second equation, we get: \\[\frac{10\lambda}{D^2} = \frac{10\lambda}{L^2}\\] And since \(0\ne \lambda\), we can cancel out the \(10\lambda\) term to get: \\[D^2 = L^2\\] Therefore, \(D = L\). Substituting this back into the third equation, we get: \\[\frac{10}{L^2} = V\\] Then \(L^2 = 5\). Thus, \(L = \sqrt{5}\) and \(D = \sqrt{5}\). 7. Calculate the minimum cost: The minimal cost is given by \(C(\sqrt{5}, \sqrt{5}) = \sqrt{5} + \sqrt{5} = 2\sqrt{5}\). So, to produce the desired services at minimal cost, the town should hire \(\sqrt{5}\) workers and \(\sqrt{5}\) guard dogs, and the cost will be \(2\sqrt{5}\).

1. Write the labor supply function: The labor supply function is given as \(L = 10w\) where w is the wage per period for guard dog handlers. 2. Write the cost function with the new labor supply constraint: Now the cost function becomes \(C(w, D) = 10w + D\). 3. Substitute the labor supply function into the constraint on the production function: Replacing \(L = 10w\) in the constraint \(V = \frac{10}{LD}\), we obtain \(V = \frac{1}{wD}\). 4. Minimize the cost function using the Lagrange multipliers method: We set up the Lagrangian function: \(\mathcal{L}(w, D, \lambda) = 10w + D + \lambda(\frac{1}{wD} - V)\). 5. Differentiate the Lagrangian function with respect to w, D, and λ: \\[\frac{\partial \mathcal{L}}{\partial w} = 10 - \frac{\lambda}{w^2D} = 0\\] \\[\frac{\partial \mathcal{L}}{\partial D} = 1 - \frac{\lambda}{wD^2} = 0\\] \\[\frac{\partial \mathcal{L}}{\partial \lambda} = \frac{1}{wD} - V = 0\\] 6. Solve the system of equations: From the first and second equation, we get: \\[\frac{\lambda}{w^2D} = 10\frac{\lambda}{wD^2}\\] And since \(0\ne \lambda\), we can simplify and get: \\[wD = 10D\\] Then, \(w = 10\). Substituting this back into the third equation, we get: \\[V = \frac{1}{wD} = \frac{1}{10D}\\] 7. Determine the inputs to produce the desired services at minimal cost: Since \(w = 10\), we can use the labor supply function to find \(L = 10w = 100\). Then, we use the constraint to find \(D\). We have \(V = \frac{1}{10D}\), so \(D = \frac{1}{10V}\). Plug in the value of V from part a, \(V = \frac{10}{(\sqrt{5})^2} = 2\), we get \(D = \frac{1}{20}\). 8. Calculate the total costs and per-period wage of dog handlers: The total cost is \(C(w, D) = 10w + D = 10\cdot10 + \frac{1}{20} = 100 + \frac{1}{20}\). The per-period wage of dog handlers is \(w = 10\). In order to produce the desired services at minimal cost when labor supply is determined by the given curve, the town should hire 100 workers and \(\frac{1}{20}\) guard dogs. The wage rate of dog handlers will be \(10, and the total cost will be \)100 + \frac{1}{20}$.