Question

demand is given by \\[ Q=1,500-50 P \\] a. What is the industry's long-run supply schedule? b. What is the long-run equilibrium price \(\left(P^{*}\right) ?\) The total industry output \(\left(Q^{*}\right) ?\) The output of each firm \(\left(q^{*}\right) ?\) The number of firms? The profits of each firm? c. The short-run total cost function associated with each firm's long-run equilibrium output is given by \\[ C(q)=0.5 q^{2}-10 q+200 \\] Calculate the short-run average and marginal cost function. At what output level does short-run average cost reach a minimum? d. Calculate the short-run supply function for each firm and the industry short-run supply function. e. Suppose now that the market demand function shifts upward to \(Q=2,000-50 P .\) Using this new demand curve, answer part (b) for the very short run when firms cannot change their outputs. f. In the short run, use the industry short-run supply function to recalculate the answers to (b). g. What is the new long-run equilibrium for the industry?

Short answer

Based on the given information, the short-run average cost function is SRAC(q) = 0.5q - 10 + (200/q), and the short-run marginal cost function is SRMC(q) = q - 10. Additionally, the output level at which short-run average cost reaches a minimum is 20 units. Unfortunately, due to insufficient information provided regarding the industry's long-run supply schedule, long-run equilibrium values, short-run supply functions, and the relationship between industry and individual firms, we cannot answer all questions posed. Further information is required to provide a full response to the exercise.

Step-by-step solution

Short-Run Average Cost Function

The short-run average cost function is given by: \(\text{SRAC}(q) = \frac{C(q)}{q} = \frac{0.5q^2 - 10q + 200}{q} = 0.5q - 10 + \frac{200}{q}\) To find the short-run marginal cost (SRMC) function, we need to calculate the first derivative of the total cost function.

Short-Run Marginal Cost Function

The short-run marginal cost function is given by: \(\text{SRMC}(q) = \frac{dC(q)}{dq} = 1q - 10\) To find the output level where the short-run average cost reaches a minimum, we need to set the first derivative of the SRAC function to zero and solve for \(q\).

Output Level Where SRAC Reaches a Minimum

Setting the first derivative of the SRAC function to zero: \(\frac{d(\text{SRAC}(q))}{dq} = \frac{d(0.5q - 10 + \frac{200}{q})}{dq} = 0.5 - \frac{200}{q^2} = 0\) Solving for \(q\): \(q^2 = 400\) \(q = 20\) So, the output level at which short-run average cost reaches a minimum is 20 units. #d. Calculating the Short-Run Supply Function for Each Firm and the Industry# As there's no information provided regarding firm production or the number of firms, we are unable to calculate the short-run supply function for each firm and the industry. #e. Answering Part (b) with the Updated Market Demand Function# The new market demand function is \(Q = 2000 - 50P\). However, due to insufficient information on industry and firm production, we cannot answer this part using the updated market demand function. #f. Recalculating Answers to (b) with Industry Short-Run Supply Function# Since we are unable to calculate the industry short-run supply function, we can't recalculate the answers to part (b) using this information. #g. Finding the New Long-Run Equilibrium for the Industry# To find the new long-run equilibrium for the industry, we would need to find the new long-run equilibrium price, total industry output, and the number of firms. However, due to insufficient information, we cannot find these values and hence, we cannot find the new long-run equilibrium for the industry.

Key concepts

Demand Function

In microeconomic theory, the demand function is a mathematical representation that describes how much of a good or service consumers are willing and able to purchase at different prices, assuming all other factors are held constant (ceteris paribus). This function is crucial for determining how market demands vary with changes in price, helping firms and policy makers make informed decisions.

For example, the demand function provided in the exercise, \(Q=1,500-50P\), illustrates that as the price (P) increases, the quantity demanded (Q) decreases, which aligns with the law of demand. The negative sign before 50 reflects this inverse relationship between demand and price. Understanding and working with this basic type of linear demand function is fundamental in microeconomics and will be encountered often in both theoretical and applied scenarios.

Long-Run Supply Schedule

The long-run supply schedule in microeconomics represents the total amount of goods or services that can be supplied in an industry at different price levels when all inputs can be varied. Unlike the short-run where at least one factor of production is fixed, the long-run assumes that firms can adjust all their inputs and there are no fixed costs.

This schedule is known for its importance in predicting how an industry responds to changing economic conditions over time. In the long run, firms can enter or exit the market, which can affect the overall supply and equilibrium. Industry's long-run supply schedule is often more elastic compared to the short-run supply schedule, given that firms have more time to adjust their production processes and resource allocations.

Short-Run Average Cost

Short-run average cost (SRAC) is a vital concept in microeconomic theory that indicates the average total cost of production per unit of output when at least one factor of production is fixed. It's a blend of variable and fixed costs spread over the output produced. Calculating SRAC informs businesses about the most efficient level of production and when economies of scale are being achieved.

The mathematical representation for the SRAC is derived by dividing the total cost, \(C(q)\), by the quantity of output produced, \(q\). Taking our exercise's cost function, \(C(q)=0.5q^2-10q+200\), and dividing it by output gives the SRAC equation used to find the most cost-effective output level for a firm in the short run. Minimizing SRAC is crucial for firms aiming to optimize their production levels with respect to costs.

Short-Run Marginal Cost

Short-run marginal cost (SRMC) is another core component in microeconomic theory. It reflects the change in total cost that arises when the quantity produced is increased by one additional unit. In essence, it's the cost of producing 'one more' unit of output. SRMC plays a vital role in the profit-maximizing decisions of firms, as it influences pricing and output levels.

The SRMC can be found by taking the first derivative of the total cost function with respect to quantity, \(q\). For the problem at hand, the SRMC is determined by deriving the function, \(C(q)=0.5q^2-10q+200\), resulting in \(SRMC(q)=1q-10\). This calculation helps a firm determine the point at which producing additional units becomes more expensive, commonly referred to as 'decreasing returns to scale'.

Market Equilibrium

Market equilibrium is a key concept in microeconomic theory where the quantity of goods supplied is equal to the quantity demanded, and there is no tendency for change. At this point, the market price is stable, and the desires of both consumers and producers are balanced.

For the market to reach equilibrium, the supplied goods must match the consumer's demand. This balancing act results in the so-called equilibrium price \(P^*\) and quantity \(Q^*\). In the context of the example provided, market equilibrium can be calculated using the intersection point of the industry's supply and demand curves. While long-run equilibrium factors in the entry and exit of firms, short-run equilibrium assumes existing firms can't change their production levels. The analysis of market equilibrium is crucial for economic policy and understanding market dynamics.