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MANAGERIAL AND DECISION ECONOMICS Manage. Decis. Econ. 21: 305–312 (2000) Demonstrating Cournot and Collusive Equilibria Using Computer Spreadsheets Charles E. Hegji* Department of Economics, Auburn Uniersity Montgomery, Montgomery, AL, USA DOI: 10.1002/mde.984 INTRODUCTION The work of Smith and Smith (1988), Smith et al. (1990) and others clearly demonstrates that com- puter spreadsheets can be used to model standard microeconomic concepts such as marginal rev- enue, marginal cost and value of marginal product. More recently, Cahill and Kosicki (2000) demonstrated spreadsheet methodologies for teaching more advanced microeconomic topics, such as mapping indifference curves from utility functions and the Hicksian decomposition for a change in the price of a product. The present paper builds on these approaches by demonstrat- ing how spreadsheet models can be used to en- hance students’ understanding of Cournot equilibrium and the outcome of collusive agree- ments between firms. This choice has been made because the Cournot model is often difficult to teach to undergraduate students, particularly because of its heavy reliance on mathematics. Also, the link between the Cournot and collusive outcomes in a duopoly setup is sometimes not as clearly presented as could be. The spreadsheets presented in this paper are an attempt to overcome these potential diffi- culties and provide an alternative vehicle for dis- cussing these topics. Before continuing I would like to mention that although the spreadsheets discussed in this paper have been developed using Microsoft Excel, similar types of spreadsheets can be constructed using Lotus1-2-3 and Corel Quatro Pro. AN EXAMPLE OF COURNOT EQUILIBRIUM USING REACTION FUNCTIONS We begin by developing a Cournot spreadsheet using the traditional reaction function approach. The example developed here is based on a nu- meric example presented by Mansfield (1999). The spreadsheet is a modification of one developed in Hegji’s Spreadsheet Exercises to Accompany Man - agerial Economics: Theory, Applications, and Cases (Hegji, 1999). Assume that two hypothetical companies, the Carpenter Company and Hanover Corporation, are the only two producers of a particular type of machine bearing. The market demand for this machine bearing is given by: P =320 Q =320 (Q H +Q C ), (1) where P is price per crate of bearings and Q is hundreds of crates demanded per week. Accord- ing to the above equation, market price depends on total market output. Market output is the sum of the output of the Carpenter Company and Hanover Corporation. For simplicity, Carpenter and Hanover are as- sumed to have identical linear cost curves. These curves are given by TC C =20Q C , (2) TC H =20Q H . Each firm’s total costs are a function of its out- put, with a constant marginal cost of twenty. The Cournot model assumes that each firm takes the market price and its rival’s output as given. The firm then maximizes profits. Under these assumptions the profits of the two compa- nies are given by * Correspondence to: Department of Economics, Auburn Uni- versity Montgomery, Montgomery, AL 36117, USA. Copyright © 2000 John Wiley & Sons, Ltd.

Demonstrating Cournot and Collusive equilibria using computer spreadsheets

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Page 1: Demonstrating Cournot and Collusive equilibria using computer spreadsheets

MANAGERIAL AND DECISION ECONOMICS

Manage. Decis. Econ. 21: 305–312 (2000)

Demonstrating Cournot and CollusiveEquilibria Using Computer Spreadsheets

Charles E. Hegji*

Department of Economics, Auburn Uni�ersity Montgomery, Montgomery, AL, USA

DOI: 10.1002/mde.984

INTRODUCTION

The work of Smith and Smith (1988), Smith et al.(1990) and others clearly demonstrates that com-puter spreadsheets can be used to model standardmicroeconomic concepts such as marginal rev-enue, marginal cost and value of marginalproduct. More recently, Cahill and Kosicki (2000)demonstrated spreadsheet methodologies forteaching more advanced microeconomic topics,such as mapping indifference curves from utilityfunctions and the Hicksian decomposition for achange in the price of a product. The presentpaper builds on these approaches by demonstrat-ing how spreadsheet models can be used to en-hance students’ understanding of Cournotequilibrium and the outcome of collusive agree-ments between firms.

This choice has been made because the Cournotmodel is often difficult to teach to undergraduatestudents, particularly because of its heavy relianceon mathematics. Also, the link between theCournot and collusive outcomes in a duopolysetup is sometimes not as clearly presented ascould be. The spreadsheets presented in this paperare an attempt to overcome these potential diffi-culties and provide an alternative vehicle for dis-cussing these topics. Before continuing I wouldlike to mention that although the spreadsheetsdiscussed in this paper have been developed usingMicrosoft Excel, similar types of spreadsheets canbe constructed using Lotus1-2-3 and CorelQuatro Pro.

AN EXAMPLE OF COURNOTEQUILIBRIUM USING REACTION

FUNCTIONS

We begin by developing a Cournot spreadsheetusing the traditional reaction function approach.The example developed here is based on a nu-meric example presented by Mansfield (1999). Thespreadsheet is a modification of one developed inHegji’s Spreadsheet Exercises to Accompany Man-agerial Economics: Theory, Applications, andCases (Hegji, 1999).

Assume that two hypothetical companies, theCarpenter Company and Hanover Corporation,are the only two producers of a particular type ofmachine bearing. The market demand for thismachine bearing is given by:

P=320−Q=320− (QH+QC), (1)

where P is price per crate of bearings and Q ishundreds of crates demanded per week. Accord-ing to the above equation, market price dependson total market output. Market output is the sumof the output of the Carpenter Company andHanover Corporation.

For simplicity, Carpenter and Hanover are as-sumed to have identical linear cost curves. Thesecurves are given by

TCC=20QC, (2)

TCH=20QH.

Each firm’s total costs are a function of its out-put, with a constant marginal cost of twenty.

The Cournot model assumes that each firmtakes the market price and its rival’s output asgiven. The firm then maximizes profits. Underthese assumptions the profits of the two compa-nies are given by

* Correspondence to: Department of Economics, Auburn Uni-versity Montgomery, Montgomery, AL 36117, USA.

Copyright © 2000 John Wiley & Sons, Ltd.

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306 C.E. HEGJI

�C= (320−QC−QH)QC−20QC

=300QC−QC2 −QCQH, (3)

�H= (320−QC−QH)QH−20QH

=300QH−QH2 −QCQH,

where �C is profits of the Carpenter Companyand �H are profits of the Hanover Corporation.

To maximize profits for either company, obtainthe partial derivative of profits with respect tooutput holding the rival’s output constant. Thisresults in the conditions

��C

�QC

=300−2QC−QH=0,

��H

�QH

=300−2QH−QC=0. (4)

Solving the first equation for QC and the secondequation for QH obtains

QC=150−0.5QH, (5)

QH=150−0.5QC.

Equation (5) defines the Carpenter Company’sand the Hanover Corporation’s reaction func-tions. They express each company’s profit-maxi-mizing output given its rival’s output. Theequations show that the optimal output for eachcompany decreases as its rival’s output increases.A Cournot equilibrium occurs when each com-petitor’s assumption about its rival’s behavior andits rival’s actual behavior are consistent. This ideais fairly simple to convey with a computerspreadsheet.

The spreadsheet begins by entering the Carpen-ter Company’s conjecture about the HanoverCorporation’s output in column A. Data is en-tered beginning in row 4. Column B contains theCarpenter Company’s reaction to its guess aboutHanover’s output, using Carpenter’s reactionfunction. The formula b4=150−0.5×a4, etc. isentered down the rows of the column. SinceCournot equilibrium requires that the two compa-ny’s conjectures about each other’s output areconsistent, the spreadsheet uses the CarpenterCompany’s reactions as potential conjectures forthe Hanover Corporation. The Hanover Corpora-tion’s reactions are in turn computed in column Cusing its reaction function. These entries are cal-culated using c4=150−0.5×b4, and so forth.

The completed spreadsheet appears in Figure 1.

Cournot equilibrium occurs in Row 14, with eachfirm producing 100 units.

Graphing the Cournot equilibrium to resemblethe diagrams in most intermediate microeconom-ics texts requires some additional work. This isbecause the Hanover Corporation’s conjecturesare based on the Carpenter Company’s reactions,listed from largest to smallest.

Begin by inserting a column between columns Band C. This will be used to duplicate and reorderthe entries in column B. Copy the entries incolumn B. Use the Special Paste/Values commandto paste the numeric values calculated in columnB into column C. If this is not done the spread-sheet will recalculate the Carpenter Company’sreactions when column B is copied to column C.In column D, the Hanover Company’s reactionsare recalculated based on the entries in C. There-fore, d4=150−0.5×c4, and so forth. Finally,highlight the entries in column C, and Sort fromsmallest to largest. The resulting spreadsheet ap-pears in Figure 2.

The reaction functions are graphed usingExcel’s Chart Wizard after highlighting columnsA, B and D. Choose X–Y Scatter Diagram forthe chart type. This plots the elements in columnB and A as a function of the entries in Column A.Choose any of the several chart sub types avail-able (I like either subtype two or four) and followthe directions. The final product should looksomething like the diagram in Figure 3.

Figure 3 clearly shows the Cournot equilibriumas occurring at the intersection of the two reactionfunctions. Each firm produces 100 units of out-put. The graph can be used as a vehicle fordiscussing Cournot equilibrium. A straightfor-ward exercise would be to have Students experi-ment with alternative reaction functions to seehow different reaction functions result in differentCournot equilibrium.

COURNOT EQUILIBRIUM WITHOUTREACTION FUNCTIONS

As stated earlier, one difficulty in teaching theCournot model is its somewhat heavy reliance onmathematics. Use of reaction functions presup-poses the student understands the meaning of apartial derivative and can evaluate simple partialderivatives. The spreadsheet developed in thepresent section is an attempt to overcome this

Copyright © 2000 John Wiley & Sons, Ltd. Manage. Decis. Econ. 21: 305–312 (2000)

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307COURNOT AND COLLUSIVE EQUILIBRIA

Figure 1. Computing Cournot reaction functions.

Figure 2. Preparing the reaction function for graphing.

problem, by demonstrating Cournot equilibriumwithout the use of calculus.

The spreadsheet in Figure 4 is constructed intwo parts. Computations are done in the upperpart of the spreadsheet with results appearing inthe table at the bottom of the spreadsheet. Com-putations for the Carpenter Company appear onthe left-hand side of the spreadsheet, while com-

putations for the Hanover Corporation appear onthe right. Carpenter’s conjecture about Hanoverappears in cell a2 and Hanover’s conjecture aboutCarpenter appears in cell f2.

The Cournot equilibrium is determined througha series of iterations. Begin by entering a Carpen-ter conjecture of 0 for the Hanover Corporation’soutput. This output and Carpenter’s assumed

Copyright © 2000 John Wiley & Sons, Ltd. Manage. Decis. Econ. 21: 305–312 (2000)

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308 C.E. HEGJI

Figure 3. Graph of reaction functions.

Figure 4. Spreadsheet for determining Cournot equilibrium.

output are substituted into Equation (1) to deter-mine the resulting market price confronting theCarpenter Company. Therefore, cell entry b4 iscomputed using b4=320−$a$2−a4, and soforth. The resulting price along with quantity a4are used to compute Carpenter’s total revenue in

column c, while Carpenter’s total cost are com-puted in column d using c4=20×a4. . . Profitsfor Carpenter are computed in column e as thedifference between total revenues and total costs.As the table at the bottom of the spreadsheetsuggests, if the Carpenter Company assumes that

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309COURNOT AND COLLUSIVE EQUILIBRIA

the Hanover Corporation produces no output,Carpenter Company’s profits will be maximized ifit produces 150 units of output.

This output is used as the Hanover Corpora-tion’s first conjecture about the Carpenter Com-pany’s production. The value entered in cell f2and used along with the other values in column fto compute the resulting market price facingHanover. Therefore, g4=320−$f$2− f4, etc.The remaining entries for the Hanover Corpora-tion are computed in a fashion analogous to thosefor the Carpenter Company. Given these entriesand the conjecture that the Carpenter Company isproducing 150 units results in a profit-maximizingoutput of 80 units for the Hanover Corporation.This appears in the table at the bottom of thespreadsheet in Figure 4.

The table shows that this cannot be a Cournotequilibrium. Carpenter Company’s conjectureconcerning Hanover’s output (QC=0) and theHanover Corporation’s actual profit maximizingoutput (QH=80) differ. The process is thereforerepeated starting with the conjecture QH=10 for

the Carpenter Company. The Cournot equi-librium occurs when the Carpenter Company’sconjecture about the Hanover Corporation’s out-put and the Hanover Corporation’s profit-maxi-mizing output are the same. Similarly forHanover’s conjecture and the Carpenter Compa-ny’s profit-maximizing output. Equilibrium isshown in Figure 5 with both firms producing 100units of output.

The spreadsheet developed in the present sec-tion can be used to help students to understandCournot equilibrium in several ways. First, eachstudent could be asked to determine the Cournotequilibrium on his or her own. A second, andpossibly more intriguing exercise, would be forthe students to determine the Cournot equilibriumworking in teams of two, with the students takingon the roles of the two firms and reaching anequilibrium by bargaining. A third series of possi-bilities could be used in conjunction with either ofthe above two. This would be to experiment withalternative demand functions and cost curves fac-ing the firms and to examine the impact this

Figure 5. Cournot equilibrium.

Copyright © 2000 John Wiley & Sons, Ltd. Manage. Decis. Econ. 21: 305–312 (2000)

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Table 1. Cournot Equilibrium Values for Various Demand and Cost Functions

Carpenter totalMarket demand Hanover total Carpenter Cournot Hanover Cournotcostcost output output

TCC=20QCP=320−Q TCH=20QH 100 100TCC=20QCP=640−Q TCH=20QH 206.67 206.67TCC=20QC TCH=20QHP=320−0.5Q 200 200

P=320−Q TCC=20QC TCH=40QH 106.67 86.67TCC=4QC

2 TCH=4QH2 28.79 28.79P=320−Q

assumption would have on the equilibrium outputand market shares. Results of such experimentsappear in Table 1.

A COLLUSIVE EQUILIBRIUM

Treatment of oligopoly in many microeconomictextbooks includes a comparison of the Cournotequilibrium with a situation in which the firmscollude and set price and quantity as a monopo-list. We now develop a spreadsheet to model thecollusive outcome.

The spreadsheet begins by assuming that thecartel has determined the allocation of outputbetween the two firms. This allocation, ex-pressed as output of the Hanover Corporationrelative to the Carpenter Company, is entered incell b2. Carpenter’s output is entered downcolumn A, starting in row 4. Hanover’s output isentered down column b, using the market share incell b2. Therefore, b4=$b$2×a4, and so forth.Market price is determined in column c using thedemand curve in Equation (1) and the joint out-put of the two firms. For instance, c4=320−a4−b4.

Figure 6. Collusive outcome.

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311COURNOT AND COLLUSIVE EQUILIBRIA

The market price along with the CarpenterCompany’s output determine its total revenue,d4=b4×c4, etc. Carpenter’s total costs are cal-culated using e4=20×a4, and Carpenter’s totalprofits are computed as the difference betweentotal revenue and total costs, that is, f4=d4−e4,etc. Total revenue, total costs and total profits forthe Hanover Corporation are determined in thesame manner as for the Carpenter Company, withthe output in column b replacing the output incolumn a in all the formulae. Total profits for thetwo firms are computed as the sum of the respec-tive elements in column f and column i. Thecompleted spreadsheet appears in Figure 6.

The spreadsheet shows that for the collusiveequilibrium, total market output is 150 units, witheach firm producing 75 units. The equilibriumresults in a market price of $170 and the totalmarket profit $22500 shared equally between thetwo firms. These outputs, market price and profitscan be compared to those for the Cournot equi-librium as points of discussion for the twomodels.

A particularly interesting discussion point re-lates to the market shares for the two firms. In thepresent example the market shares of the firms areequal. An interesting class exercise would be toask the students to experiment with different mar-ket shares for the firms (different values in cell b2)to see the impact this has on the collusive out-come. The students will find that equilibriumprice, output and total market profits will beindependent of the firms’ market shares. Only theprofits of the individual firms will differ. Thereader is encouraged to try this for him or herself.

The spreadsheet in Figure 6 in conjunction withthat in Figure 7 can be used as a basis for thediscussion of the fragility of the collusive agree-ment. Figure 6 shows that both firms prefer thecollusive agreement to the situation where thefirms compete by bidding price down to averagecosts. In this situation profits of both firms arezero. However, Figure 7 shows that if one of thefirms (in our example the Carpenter Company)could capture the entire market by bidding pricedown to average cost, it would do so. In this case

Figure 7. The advantage of cheating on the collusive agreement.

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312 C.E. HEGJI

its profits would equal $22500, the total profitsgenerated by the collusive agreement. This resultcould be used as a basis for discussions of gametheory and strategy in a duopoly setting.

CONCLUSION

I am firmly convinced that computer spreadsheetsare a valuable tool for teaching topics in microe-conomics. They add a quantitative dimension toinstruction that is a supplement to both graphicand mathematical approaches. My experience hasalso been that students appreciate the use ofcomputer spreadsheets in their classes, since manyof them will be using spreadsheets in their futureplaces of employment.

A potentially useful application of spreadsheetsas an aid to teaching microeconomics is in deriv-ing the Cournot and collusive equilibrium in a

two-firm oligopoly. To my knowledge, the teach-ing literature has not yet addressed this problem.I hope that the present note is a useful first step inthis direction.

REFERENCES

Cahill M, Kosicki G. 2000. Exploring economic modelsusing Excel. Southern Economic Journal 66: 770–792.

Hegji CE. 1999. Spreadsheet Exercises to AccompanyManagerial Economics: Theory, Applications, andCases (4th edn). WW Norton: New York.

Mansfield E. 1999. Managerial Economics: Theory, Ap-plications, and Cases (4th edn). WW Norton: NewYork.

Smith LM, Smith LC. 1988. Teaching microeconomicswith microcomputer spreadsheets. Journal of Eco-nomic Education 19: 363–382.

Smith LM, Smith LC, Ellis S. 1990. Some applicationsof microcomputer software in teaching economictheory: a new math. Unpublished manuscript.

Copyright © 2000 John Wiley & Sons, Ltd. Manage. Decis. Econ. 21: 305–312 (2000)