Microeconomics Cost and Production Analysis Study Guide

Q1. Derive the cost equations for a firm with production function $Q = LK^2$, given labor price $w = $2$, capital price $r = $5$, and fixed capital $K = 10$ in the short run.

Background

Topic: Cost Structure and Production Functions

This question tests your understanding of how a firm's production function determines its cost structure, especially in the short run when one input (capital) is fixed.

Key Terms and Formulas

• Production function: $Q = LK^2$

• Short-run: Capital ($K$) is fixed at $K = 10$

• Cost equation: $C = wL + rK$

• Fixed cost (FC): Cost associated with fixed input ($K$)

• Variable cost (VC): Cost associated with variable input ($L$)

Step-by-Step Guidance

1. Start by substituting the fixed value of $K$ into the production function: $Q = L \times 10^2$.

2. Simplify the production function to express $L$ in terms of $Q$: $Q = 100L \implies L = \frac{Q}{100}$.

3. Write the total cost equation using $C = wL + rK$, substituting $w = 2$, $r = 5$, and $K = 10$.

4. Express $C$ in terms of $Q$ by substituting $L = \frac{Q}{100}$ into the cost equation.

5. Set up formulas for fixed cost ($FC$) and variable cost ($VC$) using the cost equation.

Try solving on your own before revealing the answer!

Final Answer:

Total cost equation: $C = 2L + 5 \times 10 = 2 \left(\frac{Q}{100}\right) + 50 = \frac{Q}{50} + 50$

Fixed cost: $FC = 5 \times 10 = 50$

Variable cost: $VC = 2L = 2 \left(\frac{Q}{100}\right) = \frac{Q}{50}$

These formulas allow you to calculate total, fixed, and variable costs for any output $Q$ in the short run.

Q2. Using the cost equations from Q1, fill out the table for $Q = 0$ to $Q = 5$ with columns for $C$, $F$, $VC$, $MC$, $AFC$, $AVC$, and $AC$.

Background

Topic: Cost Table Construction

This question tests your ability to apply cost formulas to calculate various cost measures for different output levels.

Key Terms and Formulas

• Total cost ($C$): $C = \frac{Q}{50} + 50$

• Fixed cost ($F$): $F = 50$

• Variable cost ($VC$): $VC = \frac{Q}{50}$

• Marginal cost ($MC$): $MC = \frac{dC}{dQ}$

• Average fixed cost ($AFC$): $AFC = \frac{F}{Q}$

• Average variable cost ($AVC$): $AVC = \frac{VC}{Q}$

• Average cost ($AC$): $AC = \frac{C}{Q}$

Step-by-Step Guidance

1. For each $Q$ from $0$to $5$, use the formulas above to calculate $C$, $F$, and $VC$.

2. Calculate $MC$ by finding the change in $C$ as $Q$ increases by 1 unit.

3. Compute $AFC$, $AVC$, and $AC$ for each $Q$ (except $Q = 0$, where division by zero is undefined).

4. Fill in the table with these values, showing your calculations for each step.

Try solving on your own before revealing the answer!

Final Answer:

Sample table for $Q = 0$ to $Q = 5$:

Q	C	F	VC	MC	AFC	AVC	AC
0	50	50	0	-	-	-	-
1	50.02	50	0.02	0.02	50	0.02	50.02
2	50.04	50	0.04	0.02	25	0.02	25.02
3	50.06	50	0.06	0.02	16.67	0.02	16.69
4	50.08	50	0.08	0.02	12.5	0.02	12.52
5	50.10	50	0.10	0.02	10	0.02	10.02

Each value is calculated using the formulas from Q1. Marginal cost is constant because the variable cost increases linearly with $Q$.

Q2. (a) What is the slope of an Isocost line if $w = $10$ and $r = $5$?

Background

Topic: Isocost Lines

This question tests your understanding of how input prices determine the slope of an isocost line, which represents combinations of labor and capital that cost the same.

Key Terms and Formulas

• Isocost line equation: $C = wL + rK$

• Slope of isocost line: $-\frac{w}{r}$

Step-by-Step Guidance

1. Write the isocost equation: $C = wL + rK$.

2. Rearrange to solve for $K$ in terms of $L$: $K = \frac{C}{r} - \frac{w}{r}L$.

3. Identify the slope as $-\frac{w}{r}$.

4. Substitute $w = 10$ and $r = 5$ into the slope formula.

Try solving on your own before revealing the answer!

Final Answer:

Slope of the isocost line: $-\frac{10}{5} = -2$

This means for every additional unit of labor, capital must decrease by 2 units to keep total cost constant.

Q2. (b) Draw two Isocost lines for $C = $100$ and $C = $150$ using $C = wL + rK$.

Background

Topic: Isocost Line Visualization

This question tests your ability to graphically represent isocost lines for different total costs.

Key Terms and Formulas

• Isocost line equation: $K = \frac{C}{r} - \frac{w}{r}L$

Step-by-Step Guidance

1. For $C = 100$, write the equation: $K = \frac{100}{5} - \frac{10}{5}L = 20 - 2L$.

2. For $C = 150$, write the equation: $K = \frac{150}{5} - \frac{10}{5}L = 30 - 2L$.

3. Plot both lines on a graph with $L$ on the x-axis and $K$ on the y-axis.

4. Label each line according to its total cost.

Try solving on your own before revealing the answer!

Final Answer:

Isocost lines:

• $C = 100$: $K = 20 - 2L$

• $C = 150$: $K = 30 - 2L$

Each line shows the combinations of $L$ and $K$ that cost $100$and $150$, respectively.

Q3. (a) Draw a diagram showing the least-cost combination of $L$ and $K$ for producing output $Q$ using an Isoquant and an Isocost curve.

Background

Topic: Isoquants and Isocosts

This question tests your ability to use graphical analysis to find the least-cost input combination for a given output.

Key Terms and Formulas

• Isoquant: Curve showing all input combinations that produce a given output $Q$

• Isocost: Line showing all input combinations with the same total cost

• Least-cost combination: Point where the isoquant is tangent to the isocost line

Step-by-Step Guidance

1. Draw an isoquant for output $Q$ on a graph with $L$ and $K$ axes.

2. Draw an isocost line for a given total cost.

3. Identify the point of tangency between the isoquant and isocost line.

4. This point represents the least-cost combination of $L$ and $K$ for producing $Q$.

Try solving on your own before revealing the answer!

Final Answer:

The least-cost combination is found at the tangency point between the isoquant for $Q$ and the isocost line. At this point, the slopes of the isoquant and isocost are equal: $\frac{MP_L}{MP_K} = \frac{w}{r}$.

Q3. (b) If the price of capital ($r$) increases, what happens to the least-cost combination of $L$ and $K$ for producing $Q$?

Background

Topic: Input Price Changes and Cost Minimization

This question tests your intuition about how changes in input prices affect the firm's choice of input combinations.

Key Terms and Formulas

• Isocost slope: $-\frac{w}{r}$

• Least-cost combination: Point of tangency between isoquant and isocost

Step-by-Step Guidance

1. Consider how an increase in $r$ affects the slope of the isocost line.

2. The isocost line becomes steeper, meaning capital becomes relatively more expensive.

3. The firm will substitute away from capital toward labor, changing the least-cost combination.

4. Expect the new combination to use more labor and less capital for the same output $Q$.

Try solving on your own before revealing the answer!

Final Answer:

When $r$ increases, the isocost line becomes steeper. The least-cost combination shifts to use more labor and less capital, as capital is now relatively more expensive.

Q3. (c) Draw a second isocost line to show the new least-cost combination after the price of capital increases.

Background

Topic: Graphical Analysis of Input Price Changes

This question tests your ability to visually represent the effect of input price changes on the firm's cost-minimizing choices.

Key Terms and Formulas

• Isocost line: $K = \frac{C}{r} - \frac{w}{r}L$

• Isoquant: Curve for output $Q$

Step-by-Step Guidance

1. Draw the original isocost line and isoquant for $Q$.

2. Draw a new isocost line with a steeper slope, reflecting the higher price of capital.

3. Identify the new tangency point, which shows the new least-cost combination.

4. Label the new isocost line and the new input combination.

Try solving on your own before revealing the answer!

Final Answer:

The new isocost line is steeper, and the new least-cost combination is at a point with more labor and less capital. This visually demonstrates the substitution effect due to the increase in $r$.

Q4. Using the diagram below, fill out a table for $Q = 0$ to $Q = 4$ with columns for total cost ($C$), marginal cost ($MC$), and average cost ($AC$), given $w = $5$ and $r = $10$.

Background

Topic: Long-Run Cost Calculation

This question tests your ability to use graphical information to calculate cost measures for different output levels in the long run.

Key Terms and Formulas

• Total cost ($C$): $C = wL + rK$

• Marginal cost ($MC$): $MC = \frac{\Delta C}{\Delta Q}$

• Average cost ($AC$): $AC = \frac{C}{Q}$

Step-by-Step Guidance

1. For each $Q$ (from 0 to 4), use the diagram to find the corresponding values of $L$ and $K$.

2. Calculate total cost for each $Q$ using $C = wL + rK$.

3. Compute marginal cost as the change in total cost when $Q$ increases by 1.

4. Calculate average cost for each $Q$ (except $Q = 0$).

5. Fill in the table with these values, showing your calculations for each step.

Try solving on your own before revealing the answer!

Final Answer:

Sample table (using diagram values):

Q	C	MC	AC
0	0	-	-
1	100	100	100
2	140	40	70
3	170	30	56.67
4	200	30	50

Values are calculated using $w = 5$, $r = 10$, and the $L$ and $K$ values from the diagram for each $Q$.

Q5. End of chapter question #3.6 from Chapter 7 of Perloff.

Background

Topic: Application of Cost and Production Concepts

This question is likely testing your ability to apply cost and production concepts to a specific scenario or problem from the textbook.

Key Terms and Formulas

• Refer to the textbook for the exact question and relevant formulas.

Step-by-Step Guidance

1. Read the textbook question carefully and identify what is being asked.

2. Determine which cost or production formulas are relevant to the problem.

3. Set up the equations and solve step-by-step, stopping before the final calculation.

Try solving on your own before revealing the answer!

Final Answer:

Refer to Chapter 7, question 3.6 in Perloff for the full solution. Apply the relevant cost and production formulas as discussed above.