Production and cost focuses on a perfectly competitive

Question

Overview

This course includes two case studies. These exercises are designed to actively involve you in microeconomic reasoning and decision making and to help you apply the concepts covered in the course to complex real-world situations. The case studies provide practice reading and interpreting both quantitative and qualitative analysis. You will then use your analysis to make decisions and predictions. These exercises provide practice communicating reasoning in a professional manner.

Prompt

Case Study One: Production and Cost focuses on a perfectly competitive industry. Each competitive firm in this industry has a Cobb-Douglas production function: q=0.02K^0.5L^0.5. These firms combine capital and labor to produce output. In task 3-2 you will use graphs and equations to analyze competitive firm decisions, the interaction between those decisions, and the competitive market determination of price.

Skills needed to complete this case study:

1. The ability to enter data, enter formulas, and create charts in Excel (Note: use the data provided in the Case Study One Data document.)
2. The ability to use basic algebra

To complete Case Study One, follow the steps below:

1. Use algebra to derive the cost function:

• To solve for K as a function of q and L. Show your work, and verify that you have this solution:K=q²/0.02²L.
• Write the cost function. Cost is equal to the sum of the expenditures to purchase capital plus the expenditures to purchase labor. Each of these expenditures is equal to the price of the input multiplied by the quantity of the input. Use the letter r to denote the price of capital and w to denote the price of labor.

2. Use Excel to create and graph isoquant curves:

• Use column A to store possible values for L. Use the first row to label the column. Put a zero (use the number 0) in the second row. Put the formula =a2+5 in the third row. Copy this formula in rows 4-25.
• Use column B to store the quantity of K that is needed to combine with each possible value of L to produce 5 units. Use the equation from Step 1, with q = 5.
• Use column C to store the quantity of K that is needed to combine with each possible value of L to produce 10 units. Use the equation from Step 1, with q = 10.
• Use column D to store the quantity of K that is needed to combine with each possible value of L to produce 15 units. Use the equation from Step 1, with q = 15.
• Use scatter-plot to graph the isoquants. Print the graph, and use this graph to complete the following table:

Q	quantity of L that must be combined with K=5000 to produce each quantity of output (q)
5
10
15

3. Consider the short run situation in which K is fixed at 5000. Assume r = .05 and w = 40. Open a new Excel worksheet for cost information. Note the difference between your production worksheet, in which the first column stored possible values of L, and this new cost worksheet in which the first column will store possible values of q. The variable represented in the first column will be graphed on the horizontal axis of the scatter-plot. For the isoquant diagram, L is shown on the horizontal axis. The new cost worksheet will be used to graph cost functions, with quantity of output on the horizontal axis

• Use column A to store possible values of q from 0 through 20.
• Use column B to store Total Fixed Cost (TFC). Fixed cost in this example is equal to r*K, with K = 5000.
• Use column C to store Total Variable Cost (TVC). Variable cost in this example is equal to w*L. To compute the quantity of L that must be combined with K=5000 to produce each possible value of q, remember that the production function is:q=0.02K^0.5L^0.5.
• Use algebra to solve for L as a function of q and K. Because K is fixed at 5000 for this short run analysis, the resulting equation will have L on the left-hand-side and the variable q will be combined with several constants on the right-hand-side. Substitute this equation for L to generate the TVC values for column C. Be careful to enclose the entire denominator in parentheses.
• Use column D to store Total Cost (TC) = TFC + TVC.
• Generate AFC in column E by dividing TFC/q
• Generate AVC in column F by dividing TVC/q
• Generate ATC in column G by dividing TC/q
• Generate one estimate of MC in column H by computing the change in TC as output increases. Leave the first row blank. Enter a formula into the third row: =d3-d2. Copy this formula into the remaining columns. This will provide arc elasticity.
• Generate a second estimate of MC in column I. Point elasticity is equal to2(40q)/{0.02²(5000)}
• Use scatter-plot to graph TC, TFC and TVC.
• Use scatter-plot to generate a second graph to show ATC, AFC, AVC and the second estimate of MC.

4. Find equilibrium P & Q in the perfectly competitive market. Demand is represented by the equation: P = 720 - 0.5Q. The perfectly competitive firms are assumed to be identical. The quantity supplied by each individual firm is represented by the firm's MC curve. In order to graph market supply and market demand, however, we need to focus on market quantity, rather than individual firm quantity. The market quantity is equal to Q=nq; where q is the individual firm's quantity.

• Solve for the short-run equilibrium market P & Q using algebra.
• Generate new columns in Excel to represent demand and supply. This will require some strategic thinking. You will want to generate a graph with market quantity on the horizontal axis. That means that you will need to generate a column of numbers to represent possible values of market quantity.
  • Let column J represent market quantity, Q=100q.
  • Store values for demand in column K. Store values of supply in column L. The numbers in column L will be equal to the numbers representing MC in column I. You can simply copy these numbers into column L. Alternately, you could enter the formula for MC, recognizing that the firm-level quantity is equal to the market quantity stored in column J divided by 100.

• Graph demand and supply. Verify that the computed equilibrium P & Q are consistent with the graph.

5. Complete the following table for a firm that is producing the profit-maximizing level of output.

Revenue		$$$
	Optimal firm q
	Short-run equilibrium P
	Revenue = q*P
Cost
	TFC
	TVC (incurred by a firm that is producing the optimal Q)
	TC = TVC + TFC
Profit
	Profit = Revenue - TC

6. Generate a graph to show the optimal quantity that will be produced by each competitive firm, and the resulting profit. This graph will include 4 curves to show:

• The short-run equilibrium price (To generate this horizontal line, use column L to store the values of the equilibrium P. Because this line is horizontal, all of the numbers stored in this column are identical.)
• ATC
• AVC
• MC.

Is your graph is consistent with your profit computation? Explain.

7. Assume that potential entrants will have exactly the same cost function as the existing firms. Will new firms enter the market? Why or why not?

8. You work for a firm that produces an input that is used by these competitive firms. Your marketing vice president has asked you to provide analysis to support the marketing department's strategic planning committee. They understand that the industry is not currently in long-run equilibrium, and they have asked you to help them estimate the output that will be produced and the number of firms that will exist when the industry reaches long-run equilibrium. This will require several steps:

• Draw the pair of graphs that depict long run competitive equilibrium. Note that two things are true in LR equilibrium:
  • P=MC=ATC for individual firm
  • S=D in market

• You will need equations that describe these two facts (fill in the blanks):

o    ATC = MC

{0.5(5000)}/q +40q{0.02²(5000)}= _________________________

• D = S

720-0.5Q = 40Q/n; where Q=nq

• Solve these two equations for q and n.
• To create graphs, complete the following steps:

o    Open a new worksheet

o    Use column A to store possible values of the market quantity. Put zero in the second row. Enter the formula =a2+100 in the third row. Copy this formula into rows 4-20.

o    Enter the formula for market demand into the second column (=720 - 0.5*A2)

o    Enter the formula for the initial market supply into the third column (=40*A2/100); where n=100 is the number of firms in the initial market.

o    Enter the formula for the final market supply into the fourth column (=40*a2/n); where n is the long-run equilibrium number of firms you computed in the previous step.

o    Use scatter-plot to create a graph that shows market demand and both the initial market supply and the long-run equilibrium market supply. Verify that the short-run and long-run equilibrium prices and quantities are consistent with your algebraic solution values.

o    Complete the following table:

	Long Run Equilibrium
Output of each individual firm
Industry output
Number of firms