Problem 14
Question
An industrial production process costs \(C(q)\) million dollars to produce \(q\) million units; these units then sell for \(R(q)\) million dollars. If \(C(2.1)=5.1, R(2.1)=6.9\) \(M C(2.1)=0.6\), and \(M R(2.1)=0.7\), calculate (a) The profit earned by producing \(2.1\) million units (b) The approximate change in revenue if production increases from \(2.1\) to \(2.14\) million units. (c) The approximate change in revenue if production decreases from \(2.1\) to \(2.05\) million units. (d) The approximate change in profit in parts (b) and (c).
Step-by-Step Solution
Verified Answer
(a) Profit is 1.8 million dollars. (b) Revenue increases by 0.028 million dollars. (c) Revenue decreases by 0.035 million dollars. (d) Profit change is 0.004 and -0.005 million dollars for parts (b) and (c) respectively.
1Step 1: Calculate Profit for 2.1 million units
Profit is calculated as the difference between revenue and cost.\[ P(q) = R(q) - C(q) \]Given: \[ R(2.1) = 6.9, \quad C(2.1) = 5.1 \]So, the profit for producing 2.1 million units is:\[ P(2.1) = R(2.1) - C(2.1) = 6.9 - 5.1 = 1.8 \text{ million dollars} \]
2Step 2: Approximate Change in Revenue for an Increase to 2.14 million units
To find the approximate change in revenue when units increase from 2.1 to 2.14, we use the marginal revenue.Marginal Revenue (MR) is given as the derivative of revenue at a specific point, and it estimates small changes in revenue with respect to changes in quantity.\[ \Delta R = MR(2.1) \times \Delta q \]Given:\[ MR(2.1) = 0.7, \quad \Delta q = 2.14 - 2.1 = 0.04 \]So, the change in revenue is:\[ \Delta R = 0.7 \times 0.04 = 0.028 \text{ million dollars} \]
3Step 3: Approximate Change in Revenue for a Decrease to 2.05 million units
To find the approximate change in revenue when units decrease from 2.1 to 2.05, again, we use the marginal revenue.\[ \Delta R = MR(2.1) \times \Delta q \]Here, the change in quantity is negative:\[ \Delta q = 2.05 - 2.1 = -0.05 \]So, the change in revenue is:\[ \Delta R = 0.7 \times (-0.05) = -0.035 \text{ million dollars} \]
4Step 4: Approximate Change in Profit for Increased Production
We calculate the approximate change in profit by considering both change in revenue and change in cost.The change in profit is given by:\[ \Delta P = \Delta R - \Delta C \]For increased production:\[ MR = 0.7, \quad MC = 0.6, \quad \Delta q = 0.04 \]Change in cost:\[ \Delta C = MC(2.1) \times \Delta q = 0.6 \times 0.04 = 0.024 \text{ million dollars} \]Change in profit:\[ \Delta P = 0.028 - 0.024 = 0.004 \text{ million dollars} \]
5Step 5: Approximate Change in Profit for Decreased Production
Similarly, to find the change in profit for decreased production:\[ \Delta q = -0.05 \]Change in cost:\[ \Delta C = MC(2.1) \times \Delta q = 0.6 \times (-0.05) = -0.03 \text{ million dollars} \]Change in profit:\[ \Delta P = -0.035 - (-0.03) = -0.035 + 0.03 = -0.005 \text{ million dollars} \]
Key Concepts
Marginal RevenueMarginal CostChange in RevenueChange in Profit
Marginal Revenue
Marginal Revenue (MR) represents the additional revenue generated from selling one more unit of a product. It's an essential concept when considering profit maximization. In the context of the given exercise, the marginal revenue at 2.1 million units is provided as 0.7 million dollars. This indicates that for every additional unit produced around this point, the revenue is expected to increase by that amount.
Understanding MR is crucial because it guides decisions on whether production should be increased or decreased. Businesses will often seek to produce up to the point where the marginal revenue equals marginal cost to maximize their profits. Remember, beyond this point, additional production might start to lead to losses since costs begin to outweigh additional revenues. In a nutshell, MR is a guide to optimal production decisions.
Understanding MR is crucial because it guides decisions on whether production should be increased or decreased. Businesses will often seek to produce up to the point where the marginal revenue equals marginal cost to maximize their profits. Remember, beyond this point, additional production might start to lead to losses since costs begin to outweigh additional revenues. In a nutshell, MR is a guide to optimal production decisions.
Marginal Cost
Marginal Cost (MC) is the cost of producing one more unit of output. In this exercise, the marginal cost at 2.1 million units is 0.6 million dollars. This means for each additional unit produced, it costs the company an extra 0.6 million dollars. Understanding MC is vital because it helps firms decide the most cost-effective point up to which they should produce.
A good strategy for profit maximization is to continue increasing production as long as the marginal revenue exceeds the marginal cost. Once these two are equal, any further production doesn’t increase profit and may even decrease it, as costs start to overtake revenue gains.
When MC is less than MR, as with our exercise, it suggests a favorable condition to produce more units, as each extra unit adds more to revenue than it does to costs, increasing overall profits.
A good strategy for profit maximization is to continue increasing production as long as the marginal revenue exceeds the marginal cost. Once these two are equal, any further production doesn’t increase profit and may even decrease it, as costs start to overtake revenue gains.
When MC is less than MR, as with our exercise, it suggests a favorable condition to produce more units, as each extra unit adds more to revenue than it does to costs, increasing overall profits.
Change in Revenue
Change in Revenue is calculated by applying the concept of marginal revenue to the change in production quantity. In the exercise, we find two scenarios to consider: a small increase and a small decrease in production.
For the increase from 2.1 to 2.14 million units, the change in revenue (\(\Delta R\)) is obtained by multiplying the MR at 2.1 (0.7 million dollars) by the change in units (0.04 million units). This results in an additional revenue of 0.028 million dollars.
Conversely, when production decreases from 2.1 to 2.05 million units, the negative change in units (-0.05 million) leads to a decrease in revenue calculated as -0.035 million dollars. Understanding these shifts helps evaluate whether increasing or decreasing production would benefit the company financially. The key is to keep marginal revenue and cost observations in context with overall business goals.
For the increase from 2.1 to 2.14 million units, the change in revenue (\(\Delta R\)) is obtained by multiplying the MR at 2.1 (0.7 million dollars) by the change in units (0.04 million units). This results in an additional revenue of 0.028 million dollars.
Conversely, when production decreases from 2.1 to 2.05 million units, the negative change in units (-0.05 million) leads to a decrease in revenue calculated as -0.035 million dollars. Understanding these shifts helps evaluate whether increasing or decreasing production would benefit the company financially. The key is to keep marginal revenue and cost observations in context with overall business goals.
Change in Profit
Change in Profit examines how changes in revenue and changes in cost impact the overall profitability of production modification. To find this, both the change in revenue (\(\Delta R\)) and the change in cost (\(\Delta C\)) due to production changes are considered.
For the increase scenario (from 2.1 to 2.14 million units), with MR of 0.7 and MC of 0.6, we calculate \(\Delta R - \Delta C\) which equals 0.004 million dollars. This slight profit increase suggests that incrementally scaling production in this region is beneficial.
For decreased production (from 2.1 to 2.05 million units), the calculations yield a profit decrease of -0.005 million dollars. This change shows that reducing production in this case would marginally hurt profits.
Overall, profit changes rely on the precise balance between revenue generations and cost increments. Continual adjustments based on MR and MC are essential for optimizing profit.
For the increase scenario (from 2.1 to 2.14 million units), with MR of 0.7 and MC of 0.6, we calculate \(\Delta R - \Delta C\) which equals 0.004 million dollars. This slight profit increase suggests that incrementally scaling production in this region is beneficial.
For decreased production (from 2.1 to 2.05 million units), the calculations yield a profit decrease of -0.005 million dollars. This change shows that reducing production in this case would marginally hurt profits.
Overall, profit changes rely on the precise balance between revenue generations and cost increments. Continual adjustments based on MR and MC are essential for optimizing profit.
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