Question

A number of stores offer film developing as a service to their customers. Suppose that each store offering this service has a cost function \(C(q)=50+0.5 q+0.08 \eta^{2}\) and a marginal cost \(M C=0.5+0.16 \eta\) a. If the going rate for developing a roll of film is \(\$ 8.50\), is the industry in long-run equilibrium? If not, find the price associated with long- run equilibrium. b. Suppose now that a new technology is developed which will reduce the cost of film developing by 25 percent. Assuming that the industry is in long run equilibrium, how much would any one store be willing to pay to purchase this new technology?

Short answer

a) No, the industry is not in long-run equilibrium at the price of $8.50 for developing a roll of film. The price that would put the industry in equilibrium will be derived from the root(s) of the quadratic equation obtained from the cost function and marginal cost. b) The maximum amount any one store would be willing to pay to purchase the new technology will be the difference between the original and new total costs at the quantity computed from the equilibrium condition after the cost is reduced by 25 percent.

Step-by-step solution

Determine if the industry is in long-run equilibrium

In long-run equilibrium, price equals marginal cost. Given that the rate for developing a roll of film is $8.50 (P), and the marginal cost in the problem is \(MC=0.5+0.16q\), set \(P = MC\) to find the quantity \(q\). So, \(8.5 = 0.5 + 0.16q\). Solving for \(q\) will give us the quantity at which the industry is at equilibrium.

Calculate the price that leads to long-run equilibrium

If the computed quantity in Step 1 is negative, it means the industry is not in equilibrium at the given price. To find the price that would put the industry in equilibrium, set the marginal cost equal to the cost function equation. This gives us a quadratic equation to solve for \(q\). The roots of the quadratic equation will then be used to find equivalent prices.

Compute the new cost and maximum amount a store can pay for new technology

If the cost of film developing is reduced by 25 percent, a new cost function is obtained by multiplying each term of the original cost function by 0.75. Assuming that the industry is in long-run equilibrium, the price will not change. Replace the cost function with the new cost function in the equilibrium condition \(MC=P\), and solve for \(q\). The difference between the original total cost and the new total cost for the computed quantity would give the maximum amount a store can pay for this new technology.

Key concepts

Cost Function in Microeconomics

In microeconomics, the cost function represents the relationship between the cost of producing goods and the quantity of goods produced. Specifically, it shows how production costs change with variations in output levels. The cost function has a fixed and a variable component. The fixed cost remains the same regardless of how much is produced, while the variable cost increases with the production quantity.

For example, if a store has a cost function represented by the equation \(C(q)=50+0.5q+0.08q^2\), the \(50\) represents fixed costs, such as rent and salaries that do not change with the amount of film developed. The terms \(0.5q\) and \(0.08q^2\) are variable costs, which include expenses like chemicals and materials that rise with increased film developing.

Understanding a cost function is crucial for businesses to determine the most efficient production levels and to forecast expenses, which in turn affects the pricing strategy and overall competitiveness in the market.

Marginal Cost

Marginal cost is a cornerstone concept in microeconomics that refers to the cost of producing one additional unit of a good. It is an important tool for businesses when making production decisions. Marginal cost is calculated by taking the derivative of the total cost function with respect to quantity, which indicates how costs change as production increases by one unit.

In our example, the marginal cost is given by the equation \(MC=0.5+0.16q\), meaning that for each additional roll of film developed, the store incurs an additional cost starting from \(0.5\) and increasing with the quantity produced. Companies are interested in marginal costs because they play a critical role in determining the number of goods to produce. In the long-run equilibrium, firms will tend to produce at a quantity where the price of the good equals its marginal cost.

Equilibrium Price

Equilibrium price is the point where the supply of goods matches the demand for those goods. In this state, the market is considered to be balanced, and there is no tendency for the price to change. For the film developing stores, the equilibrium price is where the cost of producing an additional roll of film equals what customers are willing to pay.

To determine if an industry is in long-run equilibrium, one must compare the current price to the marginal cost of production. If these are not equal, then the industry is not in equilibrium and adjustments will occur. In our example, if the going rate for developing a film is \($8.50\), but the marginal cost is different, the stores will adjust their output until the price equals the marginal cost so that the industry reaches the long-run equilibrium.

Technological Change in Economics

Technological change can dramatically affect economics by altering production processes, resulting in changes to costs and efficiencies. In our example problem, the introduction of a new technology reduces the cost of film developing by 25%. Such a change can enhance a firm's competitive position, allowing it to potentially charge lower prices or reap higher profits.

From the perspective of an individual store, the value of the new technology depends on how much it reduces costs and increases profit margins. If we assume that the industry is in long-run equilibrium, the maximum amount a store would be willing to pay for the new technology equates to the savings generated by the lower costs that the technology would provide, without changing the equilibrium price, keeping demand constant. Understanding the financial implications of technological advancements is essential for any business to remain competitive and progressive in the modern economic environment.