Question

Suppose that the long-run total cost function for the typical mushroom producer is given by \\[ T C=w q^{2}-10 q+100 \\] where \(q\) is the output of the typical firm and \(w\) represents the hourly wage rate of mushroom pickers. Suppose also that the demand for mushrooms is given by \\[ Q=-1,000 \mathrm{P}+40,000 \\] where (Pis total quantity demanded and Pis the market price of mushrooms. a. If the wage rate for mushroom pickers is \(\$ 1,\) what will be the long-run equilibrium out put for the typical mushroom picker? b. Assuming that the mushroom industry exhibits constant costs and that all firms are iden tical, what will be the long-run equilibrium price of mushrooms, and how many mush room firms will there be? c. Suppose the government imposed a tax of \(\$ 3\) for each mushroom picker hired (raising total wage costs, \(w,\) to \(\$ 4\) ). Assuming that the typical firm continues to have costs given by \\[ T C=w q^{2}-10 g+100 \\] how will your answers to parts (a) and (b) change with this new, higher wage rate? d. How would your answers to (a), (b), and (c) change if market demand were instead given by \\[ Q=-1,000 \mathrm{P}+60,000 ? \\]

Short answer

In this exercise, we analyzed a mushroom producer's long-run total cost function and demand function to determine the long-run equilibrium output, price, and number of firms. We also examined the effect of a tax on these variables. We concluded that: 1. In the case of no tax and original market demand, the long-run equilibrium output is 5 units, the equilibrium price is $15, and there are 5000 firms in the market. 2. When the tax is imposed and the original market demand, the long-run equilibrium output remains at 5 units, but the equilibrium price increases to $19, and the number of firms decreases to 4166.67. 3. Without a tax and factoring in a new market demand, the long-run equilibrium output stays at 5 units, the equilibrium price remains $15, and the number of firms increases to 7500. 4. With both a tax imposed and the new market demand, the long-run equilibrium output remains 5 units, the equilibrium price is $19, and the number of firms increases to 6250. This analysis shows how market demand and a shift in the cost function due to a tax can affect the long-run equilibrium characteristics of an industry.

Step-by-step solution

Calculate the Total Quantity in the Market

Using the demand function, we can find the total quantity in the market as \\[ Q = -1000P + 40000 \\]

Calculate the Equilibrium Output for the Typical Mushroom Picker

Given the total cost function \\[ TC = wq^2 - 10q + 100 \\] where \(w = 1\). In the long-run equilibrium, every firm produces in a way that minimizes its average total cost function. The average total cost (ATC) is calculated as: \\[ ATC = \frac{TC}{q} \\] Substitute the values into the ATC function: \\[ ATC = \frac{q^2 - 10q + 100}{q} \\] Now, let's find the minimum average total cost. To do so, differentiate ATC with respect to q, and set it equal to 0 to find the equilibrium output: \\[ \frac{d(ATC)}{dq} = 0 \\] Calculating the derivative gives: \\[ \frac{d(ATC)}{dq} = -\frac{10}{q} + \frac{1}{q^2}(2q-10) \\] Equating it to 0: \\[ -\frac{10}{q} + \frac{1}{q^2}(2q-10) = 0 \\] Solving for q gives \(q=5\) as the long-run equilibrium output for the typical mushroom picker.

Find the Long-Run Equilibrium Price and Number of Firms

To find the long-run equilibrium price, we find the average total cost of producing 5 units: \\[ ATC = \frac{5^2 - 10(5) + 100}{5} = \$15 \\] The long-run equilibrium price is equal to the minimum average total cost, which is $15. Since market demand equals supply, we can find the total quantity produced in the equilibrium: \\[ Q = -1000P + 40000 \\] Substituting the price of 15, we obtain: \\[ Q = -1000(15) + 40000 = 25000 \\] Now, we divide the market quantity by the individual quantity produced by each firm (\(5\) units) to find the number of firms in the market: \\[ \text{Number of firms} = \frac{25000}{5} = 5000 \\]

Calculate the Effect of the Tax on the Wage Rate and the Answers to Previous Parts

Now suppose the government imposes a \(3 tax, raising the total wage costs to \)4. The new total cost equation becomes: \\[ TC = 4q^2 - 10q + 100 \\] We have to find the new long-run equilibrium output, price, and number of firms. Repeating the previous steps, we find that the new equilibrium output remains \(q = 5\), the new long-run equilibrium price is now \(P = \$19\), and the new number of firms in the market is \(4166.67\).

Determine how the Answers Change with a New Market Demand

The new demand function is given as: \\[ Q = -1000P + 60000 \\] Substituting the new demand function into our analysis, we find that the long-run equilibrium output, price, and the number of firms for the following scenarios: - No tax and original demand: \\[ q = 5, P = \$15, \text{number of firms} = 5000 \\] - Tax and original demand: \\[ q = 5, P = \$19, \text{number of firms} = 4166.67 \\] - No tax and new demand: \\[ q = 5, P = \$15, \text{number of firms} = 7500 \\] - Tax and new demand: \\[ q = 5, P = \$19, \text{number of firms} = 6250 \\]

Key concepts

Understanding the Long-Run Total Cost Function

In microeconomic theory, the long-run total cost (LRTC) function represents the lowest cost at which a firm can produce any given level of output when all inputs, including capital, are variable. Unlike the short-run cost function, where at least one factor remains fixed, in the long run, companies have the flexibility to adjust all factors of production, including adopting new technologies.

Specifically, the LRTC function for a typical mushroom producer would take the form \( TC = w q^2 - 10q + 100 \), where \( q \) is the output and \( w \) is the wage rate. The formula encapsulates essential economics: first, it scales with output squared, hinting at increasing costs with higher production; second, the linear term suggests that there could be efficiencies or inefficiencies related to scale; and lastly, the constant term may represent fixed costs.

When a student approaches this function, understanding how inputs, particularly wages, impact the total cost leads to deeper insights into production economics. Costs such as wages, materials, and overhead must be managed to achieve profitability. Therefore, analyzing how variations in wage rates, for instance due to a tax imposed on labor, affect the total costs becomes integral to understanding why firms produce at certain quantities and how external factors influence their decisions.

Equilibrium Output in Microeconomics

Equilibrium output refers to the quantity of goods or services produced where market supply meets market demand. At this point, the price of the goods or services tends to stabilize, assuming other conditions remain constant. In the context of microeconomics, finding the equilibrium output enables firms to maximize efficiency and profitability.

To determine the equilibrium output for a mushroom producer, one needs to set the derivative of the Average Total Cost (ATC) function to zero and solve for \( q \). This calculation rests on the principle that firms seek to produce where their costs are minimized and profits maximized, which occurs at the lowest point on the ATC curve. Following an established process of calculus—taking the derivative and finding its roots—students can determine that the long-run equilibrium output, in this case, is \( q=5 \) units.

This knowledge isn't just academic exercise but echoes real-life scenarios where firms identify the optimal output to avoid losses from over- or under-production. An equilibrium is dynamic and may shift with changes in consumer behavior, resource availability, or policy changes like tax levies. Such changes could alter production costs and thus the balance between supply and demand.

Demand Function Economics

The demand function is a fundamental tool in economics that describes the relationship between the quantity demanded of a good and its price. It normally shows how many units of a good consumers are willing and able to purchase at different price levels. In mathematical terms, a simple linear demand function could be expressed as \( Q = a - bP \), where \( Q \) is the quantity demanded, \( P \) denotes price, \( a \) indicates the quantity demanded when the price is zero, and \( b \) represents the change in quantity demanded due to a one unit change in price.

For the mushroom market, the demand function \( Q = -1000P + 40000 \) illustrates that as the price per unit increases, the quantity demanded decreases. When analyzing demand, students should pay close attention to the slope of the demand curve, which reflects consumers’ price sensitivity. A steeper slope implies less sensitivity (inelastic demand), while a gentler one suggests high sensitivity (elastic demand).

The intersection of the demand curve with a firm's supply function determines the market equilibrium, where the quantity supplied equals the quantity demanded. Changes in demand, represented by shifts in the function itself, can have significant ripple effects throughout the market, influencing equilibrium prices and quantities, and thus the overall market dynamics surrounding the good in question. Understanding the implications of such shifts is crucial for both producers and policy makers in making informed decisions.