Question

Suppose a product can be produced using virgin ore at a marginal cost given by \(M C_{1}=0.5 q_{1}\) and with recycled materials at a marginal cost given by \(M C_{2}=5+0.1 q_{2}\). (a) If the inverse demand curve were given by \(p=10-0.5\left(q_{1}+q_{2}\right),\) how many units of the product would be produced with virgin ore and how many units with recycled materials? (b) If the inverse demand curve were \(p=20-0.5\left(q_{1}+q_{2}\right),\) what would your answer be?

Short answer

Answer: In case (a), 7.5 units will be produced using virgin ore and 12.5 units using recycled materials. In case (b), 22.5 units will be produced using virgin ore and 37.5 units using recycled materials.

Step-by-step solution

Find the equilibrium condition

To find the equilibrium condition, we can set the inverse demand curve equal to the marginal cost curve for virgin ore, and equal to the marginal cost curve for recycled materials for both \(q_1\) and \(q_2\): \(10 - 0.5(q_1 + q_2) = 0.5q_1\) for virgin ore; \(10 - 0.5(q_1 + q_2) = 5 + 0.1q_2\) for recycled materials.

Solve the system of equations

Now, we need to solve the two equations with two unknowns \(q_1\) and \(q_2\). Start by solving the virgin ore equation for \(q_1\): \(q_1 = \frac{10 - 0.5q_2}{0.5}\) Next, substitute the expression for \(q_1\) into the recycled materials equation: \(10 - 0.5\left(\frac{10 - 0.5q_2}{0.5} + q_2\right) = 5 + 0.1q_2\) Now, simplify and solve for \(q_2\): \(10 - 10 + 0.5^2q_2 - 0.5q_2 = 5 + 0.1q_2\) \(q_2 = \frac{5}{0.4}\) \(q_2 = 12.5\) Finally, substitute the value of \(q_2\) into the expression for \(q_1\) and solve for \(q_1\): \(q_1 = \frac{10 - 0.5(12.5)}{0.5}\) \(q_1 = 7.5\) #Solution for case(a):# In case (a), 7.5 units of the product will be produced using virgin ore and 12.5 units using recycled materials. #Case (b):# For case (b), we are given the following information: - Inverse demand curve: \(P = 20 - 0.5(q_1+q_2)\) The marginal cost curves are the same as in case (a).

Set a new equilibrium condition

The new equilibrium condition is determined by setting the inverse demand curve for case (b) equal to the marginal cost curves: \(20 - 0.5(q_1 + q_2) = 0.5q_1\) for virgin ore; \(20 - 0.5(q_1 + q_2) = 5 + 0.1q_2\) for recycled materials.

Solve the new system of equations

Now, solve the two equations with two unknowns for case(b) in the same way as done for case(a). Start by solving the virgin ore equation for \(q_1\): \(q_1 = \frac{20 - 0.5q_2}{0.5}\) Next, substitute the expression for \(q_1\) into the recycled materials equation: \(20 - 0.5\left(\frac{20 - 0.5q_2}{0.5} + q_2\right) = 5 + 0.1q_2\) Now, simplify and solve for \(q_2\): \(20 - 20 + 0.5^2q_2 - 0.5q_2 = 5 + 0.1q_2\) \(q_2 = \frac{15}{0.4}\) \(q_2 = 37.5\) Finally, substitute the value of \(q_2\) into the expression for \(q_1\) and solve for \(q_1\): \(q_1 = \frac{20 - 0.5(37.5)}{0.5}\) \(q_1 = 22.5\) #Solution for case(b):# In case (b), 22.5 units of the product will be produced using virgin ore and 37.5 units using recycled materials.

Key concepts

Marginal Cost

Understanding the concept of marginal cost is critical in environmental economics and relates to the additional cost incurred by producing one more unit of a good or service. This notion is at the core of production analysis and helps us explore how businesses make decisions related to production levels.

In the provided exercise, we have two marginal cost functions: one for producing with virgin ore, represented by the equation \(MC_{1} = 0.5 q_{1}\), and one for producing with recycled materials, represented by \(MC_{2} = 5 + 0.1 q_{2}\). Marginal cost is crucial because it influences how much of a product will be produced and at what points production methods may change.

For example, as production of a product using virgin materials increases, the marginal cost increases linearly, indicating each additional unit becomes progressively more expensive to produce. Conversely, the production using recycled materials starts with a higher initial cost but increases at a lower rate per unit, meaning it could be more cost-effective at larger scales. This plays into how equilibrium is established, as firms will produce where the price equals their marginal cost—the point where they can maximize profits.

Inverse Demand Curve

The inverse demand curve is a way of expressing demand in terms of price as a function of quantity, effectively flipping the conventional demand curve on its head. Where a regular demand curve shows quantity demanded at various prices, the inverse demand curve, as seen in our exercise, expresses price as a function of quantity demanded.

When the inverse demand curve is given by the equation \(p = 10 - 0.5(q_{1} + q_{2})\), it indicates that if no products are produced, the highest price consumers are willing to pay for the first unit is $10. As more products are introduced into the market (increasing \(q_{1} + q_{2}\)), the price consumers are willing to pay decreases. This type of demand curve is crucial for setting prices and for determining the equilibrium in a market, alongside the producers’ marginal costs.

To achieve maximum efficiency, producers must align their production decisions with consumers' willingness to pay, reflected by the inverse demand curve. It represents the maximum price that can be charged before consumers start to drop off or reduce the quantity they are willing to purchase.

Equilibrium Condition

The equilibrium condition in environmental economics describes the point where supply meets demand, and both producers and consumers are in balance. At this point, the quantity supplied equals the quantity demanded, and no excess supply or shortages are present in the market.

In the context of the provided exercise, finding the equilibrium condition involves setting the inverse demand curve equal to the marginal cost of production for both methods (virgin ore and recycled materials). This establishes the price and quantity at which the market clears, meaning all units produced can be sold without any leftover surplus or unmet demand.

The exercise asks us to determine how many units will be produced with each method under two different inverse demand curve scenarios. Solving for the equilibrium in each case shows how changes in the market conditions, reflected by different demand curves, influence production decisions. Producers adjust quantities to ensure that the price matches their marginal costs, allowing them to remain competitive while maximizing profits. This concept emphasizes the balance achieved in markets through the interaction of various economic forces.