2. Consider the following firm's revenue function: R₁ = P10Q₁ + P200₂ where Qs represent the output level of the two different products the firm produces. The firm's cost is assumed to be: a. C = 2Q1 + Q₁ Q₂ +2Q/²/2 2 Present a plot of the marginal cost of the firm with respect to its first product. Interpret
2. Consider the following firm's revenue function: R₁ = P10Q₁ + P200₂ where Qs represent the output level of the two different products the firm produces. The firm's cost is assumed to be: a. C = 2Q1 + Q₁ Q₂ +2Q/²/2 2 Present a plot of the marginal cost of the firm with respect to its first product. Interpret
Chapter1: Making Economics Decisions
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Transcribed Image Text:2. Consider the following firm's revenue function:
R₁ = P10Q₁ + P200₂
where Qs represent the output level of the two different products the firm produces. The firm's
cost is assumed to be:
C = 2Q1 + Q₁ Q₂ +2Q²/2
b.
c.
2
a. Present a plot of the marginal cost of the firm with respect to its first product. Interpret
your result.
Present the profit function of the firm.
Present the first order conditions of the firm's problem. Solve for the optimal levels of
production.
d. Depict the supply functions of both goods.
e. Check the second order condition of the optimization problem.
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Follow-up Question
Please continue to solve from part d to part e. Thank you so much.
![### Question 2: Firm's Revenue and Cost Functions
#### Revenue Function
Consider the following firm's revenue function:
\[ R_1 = P_{10}Q_1 + P_{20}Q_2 \]
where \( Q \)'s represent the output levels of the two different products the firm produces.
#### Cost Function
The firm's cost is assumed to be:
\[ C = 2Q_1^2 + Q_1Q_2 + 2Q_2^2 \]
### Tasks
a. **Present a plot of the marginal cost of the firm with respect to its first product. Interpret your result.**
b. **Present the profit function of the firm.**
c. **Present the first-order conditions of the firm’s problem. Solve for the optimal levels of production.**
d. **Depict the supply functions of both goods.**
e. **Check the second-order condition of the optimization problem.**
---
#### Detailed Explanation for Each Task:
##### a. Plotting the Marginal Cost
The marginal cost with respect to \( Q_1 \) can be derived from the cost function \( C \). The plot should demonstrate how the marginal cost changes as the output level \( Q_1 \) varies.
*Interpretation*: This will show the incremental cost of producing one additional unit of product 1.
##### b. Profit Function
The profit function, \(\pi\), can be derived by subtracting the cost function \( C \) from the revenue function \( R_1 \):
\[ \pi = R_1 - C \]
Substitute the given expressions for \( R_1 \) and \( C \) to find the explicit profit function.
##### c. First-Order Conditions
The first-order conditions involve taking the partial derivatives of the profit function with respect to \( Q_1 \) and \( Q_2 \), and then setting these derivatives equal to zero to solve for the optimal output levels.
\[
\frac{\partial \pi}{\partial Q_1} = 0
\]
\[
\frac{\partial \pi}{\partial Q_2} = 0
\]
##### d. Supply Functions
Determine the supply functions of both goods \( Q_1 \) and \( Q_2 \) based on the solutions from the first-order conditions.
##### e. Second-Order Condition
Check the second-order partial derivatives to ensure that the solution obtained](https://content.bartleby.com/qna-images/question/362a762f-914f-42d1-b0fa-542c2b1764ea/66df4ee8-243b-4516-95c1-c909ef85e927/yyzlj7_processed.png)
Transcribed Image Text:### Question 2: Firm's Revenue and Cost Functions
#### Revenue Function
Consider the following firm's revenue function:
\[ R_1 = P_{10}Q_1 + P_{20}Q_2 \]
where \( Q \)'s represent the output levels of the two different products the firm produces.
#### Cost Function
The firm's cost is assumed to be:
\[ C = 2Q_1^2 + Q_1Q_2 + 2Q_2^2 \]
### Tasks
a. **Present a plot of the marginal cost of the firm with respect to its first product. Interpret your result.**
b. **Present the profit function of the firm.**
c. **Present the first-order conditions of the firm’s problem. Solve for the optimal levels of production.**
d. **Depict the supply functions of both goods.**
e. **Check the second-order condition of the optimization problem.**
---
#### Detailed Explanation for Each Task:
##### a. Plotting the Marginal Cost
The marginal cost with respect to \( Q_1 \) can be derived from the cost function \( C \). The plot should demonstrate how the marginal cost changes as the output level \( Q_1 \) varies.
*Interpretation*: This will show the incremental cost of producing one additional unit of product 1.
##### b. Profit Function
The profit function, \(\pi\), can be derived by subtracting the cost function \( C \) from the revenue function \( R_1 \):
\[ \pi = R_1 - C \]
Substitute the given expressions for \( R_1 \) and \( C \) to find the explicit profit function.
##### c. First-Order Conditions
The first-order conditions involve taking the partial derivatives of the profit function with respect to \( Q_1 \) and \( Q_2 \), and then setting these derivatives equal to zero to solve for the optimal output levels.
\[
\frac{\partial \pi}{\partial Q_1} = 0
\]
\[
\frac{\partial \pi}{\partial Q_2} = 0
\]
##### d. Supply Functions
Determine the supply functions of both goods \( Q_1 \) and \( Q_2 \) based on the solutions from the first-order conditions.
##### e. Second-Order Condition
Check the second-order partial derivatives to ensure that the solution obtained
Solution
by Bartleby ExpertFollow-up Question
Please continue to solve from part d to part e. Thank you so much.
![## Problem 2: Firm's Revenue and Cost Analysis
Consider the following firm's revenue function:
\[ R_1 = P_{10}Q_1 + P_{20}Q_2 \]
where \( Q_i \) represents the output level of the two different products the firm produces. The firm's cost is assumed to be:
\[ C = 2Q_1^2 + Q_1Q_2 + 2Q_2^2 \]
### Tasks:
a. **Present a plot of the marginal cost of the firm with respect to its first product. Interpret your result.**
b. **Present the profit function of the firm.**
c. **Present the first order conditions of the firm’s problem. Solve for the optimal levels of production.**
d. **Depict the supply functions of both goods.**
e. **Check the second order condition of the optimization problem.**
To thoroughly understand the firm's revenue, cost structure, and production decisions, each task should be carefully analyzed and addressed, utilizing the given equations and economic principles.](https://content.bartleby.com/qna-images/question/362a762f-914f-42d1-b0fa-542c2b1764ea/ce423bda-c9a7-4ee0-b93b-0bc875bbd94d/6bt9w8f_processed.png)
Transcribed Image Text:## Problem 2: Firm's Revenue and Cost Analysis
Consider the following firm's revenue function:
\[ R_1 = P_{10}Q_1 + P_{20}Q_2 \]
where \( Q_i \) represents the output level of the two different products the firm produces. The firm's cost is assumed to be:
\[ C = 2Q_1^2 + Q_1Q_2 + 2Q_2^2 \]
### Tasks:
a. **Present a plot of the marginal cost of the firm with respect to its first product. Interpret your result.**
b. **Present the profit function of the firm.**
c. **Present the first order conditions of the firm’s problem. Solve for the optimal levels of production.**
d. **Depict the supply functions of both goods.**
e. **Check the second order condition of the optimization problem.**
To thoroughly understand the firm's revenue, cost structure, and production decisions, each task should be carefully analyzed and addressed, utilizing the given equations and economic principles.
Solution
by Bartleby ExpertKnowledge Booster
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