Question

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=\sqrt{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 \\[ \boldsymbol{L}=\mathbf{1} 0 w \\] 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?

Short answer

Answer: The town of Podunk should hire 10 workers and 10 guard dogs to produce 10 units of security services at a minimal cost of 20.

Step-by-step solution

Set up the minimization problem

In order to find the minimum cost for producing the desired services, we need to set up a minimization problem. We are given that residents wish to consume 10 units of security services, so we can write the problem as: \(\min L, D : L + D\) subject to \(S = \sqrt{LD} = 10\)

Use the Lagrange multiplier method

To solve this minimization problem, we can use the Lagrange multiplier method. The objective function is \(L + D\), and we have a constraint on the security services produced. So, we can write the Langrange function as: \(L(D, L, \lambda) = L + D + \lambda(10 - \sqrt{LD})\) Take partial derivatives with respect to \(L\), \(D\), and \(\lambda\) and set them equal to zero to find optimal values: 1. \(\frac{\partial L}{\partial D} = 1 - \lambda \frac{L}{2\sqrt{LD}} = 0\) 2. \(\frac{\partial L}{\partial L} = 1 - \lambda \frac{D}{2\sqrt{LD}} = 0\) 3. \(10 - \sqrt{LD} = 0\)

Solve the system of equations

Now we need to solve the system of equations. From equations 1 and 2, we can derive the following equation: \(\frac{L}{D} = \frac{1}{1} \Rightarrow L = D\) Putting this equation back into equation 3: \(10 = \sqrt{L \cdot D} = \sqrt{L^2} \Rightarrow L^2 = 100 \Rightarrow L = 10\) Since \(L = D\), we also have \(D = 10\).

Calculate the cost

Now that we know we need 10 workers and 10 guard dogs to produce 10 units of security services at minimal cost, we can find the cost. Since both inputs rent for \(1\) per period, the total cost will be: \(C = L + D = 10 + 10 = 20\) So, the town should hire 10 workers and 10 guard dogs at a total cost of \(20\). b.Solving for the ideal input combination with separate rental prices for the inputs

Set up the minimization problem

The minimization problem now becomes: \(\min L, D : wL + D\) subject to \(S = \sqrt{LD} = 10\) and \(L = 10w\)

Use the Lagrange multiplier method

We can write the Lagrange function as: \(L(D, L, \lambda) = wL + D + \lambda(10 - \sqrt{LD})\) Take partial derivatives with respect to \(L\), \(D\), and \(\lambda\): 1. \(\frac{\partial L}{\partial D} = w - \lambda \frac{L}{2\sqrt{LD}} = 0\) 2. \(\frac{\partial L}{\partial L} = 1 - \lambda \frac{D}{2\sqrt{LD}} = 0\) 3. \(10 - \sqrt{LD} = 0\)

Solve the system of equations

Using the given supply curve, \(L = 10w\), we can derive the following equations from the system of equations: 1. \(\frac{w}{1} = \frac{D}{L} \Rightarrow w = \frac{D}{L} = \frac{D}{10w} \Rightarrow w^2 = \frac{D}{10}\) 2. \(\frac{L}{D} = \frac{10w}{D} = \frac{w}{1} \Rightarrow D = 10w^2\) Plugging this equation into equation 3: \(10 = \sqrt{L \cdot D} = \sqrt{10w \cdot 10w^2} \Rightarrow 10w^3 = 10\) So, \(w^3 = 1\), which implies \(w = 1\). Now that we know the wage rate for dog handlers, we can find \(L\) and \(D\): \(L = 10w = 10\) \(D = 10w^2 = 10\)

Calculate the cost

Now that we have the new values for \(L\), \(D\), and \(w\), we can find the total cost for the town of Podunk: \(C = wL + D = 1 \cdot 10 + 10 = 20\) We find out that in both scenarios (a) and (b), the town of Podunk should hire 10 workers and 10 guard dogs to produce 10 units of security services at a minimal cost of \(20\). In the second scenario, the wage rate of dog handlers would be \(1\) per period.

Key concepts

Understanding Production Function

In microeconomic theory, a production function provides a powerful tool to understand how inputs are transformed into outputs. As demonstrated in the Podunk town's scenario, the security services (S) are produced by combining workers (L) and guard dogs (D) effectively. The specific equation given,
\[ S = \sqrt{LD} \]
shows a production function where the output (security services) increases with the square root of the product of the two inputs (workers and guard dogs). This type of function implies increasing returns to scale, to a point, since doubling both inputs will more than double the output.

The importance of production functions extends beyond theoretical exercises and into real-world applications. Businesses use them to estimate the optimal levels of inputs needed to maximize production or, as in the case of Podunk, minimize costs while producing a desired level of output. Understanding the shape and properties of a production function can help in economic decision-making, forecasting, and strategic planning.

Tackling the Minimization Problem

Often in economics, the goal isn't just to produce but to do so efficiently, managing scarce resources. This leads us to a minimization problem, which the town of Podunk faces while trying to produce 10 units of security services. It includes finding the quantity of inputs (L and D) that minimizes costs.

To solve such problems, setting up the optimization model correctly is crucial. For Podunk, the objective is to minimize the cost (L + D) subject to the constraint that the production function equals the desired output:
\[ S = \sqrt{LD} = 10 \]
This reflects a balance between economic efficiency (spending the least amount of resources) and production needs (meeting the consumption demand). Real-world applications include cost-cutting strategies, resources allocation in production, or even designing minimal waste processes in manufacturing.

Applying the Lagrange Multiplier Method

To solve optimization problems with constraints, like the Podunk case, economists often employ the Lagrange multiplier method. This technique provides a systematic way to include constraints within an optimization problem.

The process involves creating a new function, the Lagrange function, which is a combination of the original objective function and the production function constraint. For Podunk, the Lagrange function formulated:
\[ L(D, L, \lambda) = L + D + \lambda(10 - \sqrt{LD}) \]
allows the town to account for the costs while ensuring the production level remains fixed.

By taking partial derivatives of the function with respect to each variable and setting them to zero, we calculate the optimal values that minimize costs while satisfying the production constraint. The \(lambda\) is known as the Lagrange multiplier and helps to measure how much the objective function would increase if the constraint were relaxed slightly. Through this method, Podunk can find the balance of workers to guard dogs that achieves its goals in the most cost-efficient manner.