Question

Suppose there are a fixed number of 1,000 identical firms in the perfectly competitive concrete pipe industry. Each firm produces the same fraction of total market output, and each firm's production function for pipe is given by \\[ q=\sqrt{K L} \\] Suppose also that the market demand for concrete pipe is given by \\[ Q=400,000-100,000 P \\] where \(Q\) is total concrete pipe. a. If \(w=v=\$ 1,\) in what ratio will the typical firm use \(K\) and \(L\) ? What will be the long-run average and marginal cost of pipe? b. In long-run equilibrium what will be the market equilibrium price and quantity for concrete pipe? How much will each firm produce? How much labor will be hired by each firm and in the market as a whole? c. Suppose the market wage, \(w\), rose to \(\$ 2\) while \(v\) remained constant at \(\$ 1 .\) How will this change the capital-labor ratio for the typical firm, and how will it affect its marginal costs? d. Under the conditions of part (c), what will the long-run market equilibrium be? How much labor will now be hired by the concrete pipe industry? e. How much of the change in total labor demand from part (b) to part (d) represented the substitution effect resulting from the change in wage and how much represented the output effect?

Short answer

Answer: Due to the substitution effect, the firm will change its production by using less labor and more capital, as the optimal ratio changes from L=K to L=2K. However, a numerical value for the substitution effect cannot be determined with the given information. The output effect results in a change in total labor demand from 200,000 units to 0, as there is no demand for concrete pipes in the new scenario.

Step-by-step solution

Find the optimal ratio of K and L using the production function

The production function is given by: \\[ q = \sqrt{K L} \\] Since we are in a competitive market, the firms will minimize their cost of production. To find the optimal ratio of K and L, we need to find the marginal product of K (MPK) and marginal product of L (MPL) from the production function and set the ratio of marginal products equal to the ratio of their costs. Differentiating the production function with respect to K and L respectively, we get MPK and MPL: \\[ MPK = \frac{\partial q}{\partial K} = \frac{1}{2} \cdot \frac{L}{\sqrt{K L}} = \frac{L}{2 \sqrt{K L}} \\] \\[ MPL = \frac{\partial q}{\partial L} = \frac{1}{2} \cdot \frac{K}{\sqrt{K L}} = \frac{K}{2 \sqrt{K L}} \\] Now, equating the ratio of marginal products to the ratio of their costs (w = 1 and v = 1): \\[ \frac{MPK}{MPL} = \frac{w}{v} \Rightarrow \frac{\frac{L}{2 \sqrt{K L}}}{\frac{K}{2 \sqrt{K L}}} = \frac{1}{1} \\]

Simplify the ratio and solve for K and L

Simplifying the ratio, we get: \\[ \frac{L}{K} = 1 \\] So, \\[ L = K \\]

Find long-run average and marginal cost

We are given that \(w = v = 1\). In the long run, average cost (AC) and marginal cost (MC) are equal in a competitive market and we can calculate AC as: \\[ AC = \frac{wL + vK}{q} \\] Substituting \(L=K\) and the production function \(q = \sqrt{KL}\), we get: \\[ AC = \frac{1 \cdot K + 1 \cdot K}{\sqrt{K \cdot K}} = \frac{2K}{\sqrt{K^2}} = 2 \\] Thus, the long-run average cost and marginal cost are both \(2\). #b. Long-run market equilibrium price and quantity, firm production and labor hired#

Find the long-run supply curve

In the long-run, price is equal to the marginal cost in perfectly competitive markets. Therefore, the long-run supply curve in this case is a horizontal line at the price P = MC = \(2\).

Find the market equilibrium price and quantity

To find the market equilibrium price and quantity, we need to equate the market demand function and market supply quantity. The market demand function is given by: \\[ Q=400,000-100,000 P \\] The market supply, given that there are 1,000 identical firms in the industry, is: \\[ Q = 1,000 \cdot q \\] Since we have found that P = 2, we can substitute this into the demand equation and find the value for Q: \\[ Q = 400,000 - 100,000 \cdot 2 = 400,000 - 200,000 = 200,000 \\] Thus, the long-run market equilibrium price (P) is 2, and the market equilibrium quantity (Q) is 200,000.

Calculate firm production and labor hired

Given that there are 1,000 identical firms in the industry, each firm will produce the same amount of output. Therefore, individual firm production (q) is: \\[ q = \frac{Q}{1,000} = \frac{200,000}{1,000} = 200 \\] Now, we have found that \(L = K\) and can use the production function to find the values of K and L for each firm: \\[ q = \sqrt{K L} \Rightarrow 200 = \sqrt{K \cdot K} \Rightarrow K = L = 200 \\] So, each firm will hire 200 units of labor, and in the market as a whole, a total of 200,000 units of labor will be hired. #c. Change in capital-labor ratio and marginal costs when w rises to 2#

Find the new optimal ratio of K and L

Now, we are given that \(w = 2\) and \(v = 1\). Repeat the process we used in part a to equate the ratio of marginal products to the ratio of their costs: \\[ \frac{MPK}{MPL} = \frac{w}{v} \Rightarrow \frac{\frac{L}{2 \sqrt{K L}}}{\frac{K}{2 \sqrt{K L}}} = \frac{2}{1} \\]

Simplify the ratio and solve for K and L

Simplifying the ratio, we obtain the new optimal K and L: \\[ \frac{L}{K} = 2 \\] So, \\[ L = 2K \\]

Calculate the increase in marginal costs

Since price is equal to marginal cost in perfectly competitive markets, an increase in the wage rate (w) will lead to an increase in marginal costs. Since \(w = 2\), the MC will also double. In this case, MC will increase from \(2\) to \(4\). #d. Long-run market equilibrium, and labor hired by the industry#

Find the new long-run supply curve

In the long-run, price is equal to the marginal cost in perfectly competitive markets. The marginal cost is now \(4\). Therefore, the new long-run supply curve is a horizontal line at the price P = MC = \(4\).

Find the new market equilibrium price and quantity

Since we have found the new long-run supply curve (P = 4), we can substitute this into the demand equation and find the value for Q: \\[ Q = 400,000 - 100,000 \cdot 4 = 400,000 - 400,000 = 0 \\] Thus, the long-run market equilibrium price (P) is 4, and the market equilibrium quantity (Q) is 0.

Calculate labor hired by the industry

Since there is no demand for concrete pipes in the new scenario, the industry will not hire any labor. #e. Substitution and output effects#

Substitution effect

The substitution effect refers to the change in the firm's production due to the increase in the price of labor relative to capital. In this case, the substitution effect can be calculated as the difference between labor hired in part (b) (L = 200) and labor hired when \(w = 2\) (L = 2K). Since we are only given the relationship between \(L\) and \(K\), a numerical value for the substitution effect can not be determined.

Output effect

The output effect refers to the change in the firm's production due to the change in output as a result of the price increase. In this case, the output effect is represented by the change in total labor demand from part (b) to part (d) (200,000 to 0). The entire change in total labor demand can be attributed to the output effect as there is no demand for concrete pipes in the new scenario.

Key concepts

Production Function

Understanding the production function is essential for grasping how firms make decisions about their use of inputs to produce outputs. In the context of our exercise, the production function was given by the equation \( q = \[ q=\sqrt{K L} \] \), where \( q \) is the quantity of output (concrete pipes), \( K \) represents capital, and \( L \) represents labor. It's a simple yet powerful tool for determining the relationship between input quantities and the maximum output a firm can produce.

A perfectly competitive firm, such as those in this concrete pipe industry, will strive to use their inputs in the most efficient way possible to maximize output. This is due to the fact that they are price takers, which means they have no control over the market price and can only control their production costs. By analyzing the marginal product of capital (\( MPK \)) and labor (\( MPL \)), firms can adjust their input mix to ensure that the marginal cost per unit of output is minimized—leading to optimal input usage ratios.

Market Demand

Market demand reflects the total amount of a good or service that consumers are willing to purchase at various price levels. In this scenario, the market demand function was given by \( Q=400,000-100,000 P \), where \( Q \) represents the total demand for concrete pipes, and \( P \) stands for the price of the pipe.

For students analyzing market demand within a perfectly competitive market, it is key to understand that the demand curve slopes downward, denoting an inverse relationship between price and quantity demanded. As the price falls, the quantity demanded increases, and vice versa. This is a fundamental principle of economics that holds true across most markets. The demand curve, along with the supply curve, determines the market price and quantity in long-run equilibrium, which is a crucial concept for firms looking to strategize their production and pricing.

Long-run Equilibrium

In long-run equilibrium, firms in a perfectly competitive market will produce such that price equals marginal cost (\( P = MC \)), and there will be no economic profit, as all economic profits are competed away. In other words, long-run equilibrium occurs when firms are making just enough profit to keep them in the market, but not enough to attract new entrants.

For our concrete pipe example, the long-run equilibrium price and quantity were determined by equating the market demand with the market supply, considering that 1,000 identical firms produce the same amount of output. Understanding this concept helps students to identify what happens to individual firms and the market when conditions, such as input costs, change. It shows the dynamic nature of markets and how external changes can lead to a new equilibrium.

Capital-Labor Ratio

The capital-labor ratio is an expression of the amount of capital used in production compared to the amount of labor. This ratio is significant because it reflects the firm's production techniques—whether they are more capital-intensive or labor-intensive.

In the textbook exercise, firms initially used an equal combination of labor and capital (\( L = K \)). However, changes in the wage rate altered the cost optimization, leading to a reevaluation of this ratio. An increase in the wage rate changes the capital-labor ratio because the firm will look to substitute the more expensive input—labor—with capital if possible. A diligent student will recognize that a changing capital-labor ratio affects not only production costs but also decisions about technology adoption and the long-term planning of a company.

Marginal Costs

Marginal cost is the cost of producing one additional unit of a good or service. It is a cornerstone concept in economics because it directly influences pricing and output decisions. In perfectly competitive markets, the price of goods is determined by marginal costs in the long run.

For didactic purposes, it's beneficial to illustrate that when a firm’s marginal costs change—such as when the wage rate increases, doubling the marginal cost from \( 2 \) to \( 4 \)—it responds by adjusting its output to where price still equals marginal cost. This effects overall market supply and can shift firms into a new long-run equilibrium. By understanding how marginal costs affect firm and market behavior, students develop a nuanced view of the interconnectedness of economic actors within a market structure.