# Output Maximization Subject to a Nonlinear Constraint.

Byline: Jamal Nazrul Islam, Haradhan Kumar Mohajan and Pahlaj
Moolio

ABSTRACT

The main aim of this paper is to derive the mathematical formulation to device an optimal purchasing policy for the service providing agency. An attempt has been made to maximize the output function of an agency subject to a nonlinear budget constraint by assuming that the agency gets price discounts for purchasing larger quantities of other inputs. Such quantity discounts alter the linear budget constraint and result in a nonlinear (convex type) budget constraint. We use the method of Lagrange multipliers and apply the first- order necessary conditions as well as the second-order sufficient conditions for maximization. We also use comparative static analysis and study the behavior of the agency when prices of inputs undergo change, besides providing useful interpretation of the Lagrange multipliers in this specific case. Illustrating an explicit example, we show that the optimization problems play an important role in the real world.

JEL. Classification: C51; C65; C61; D24

Keywords: Maximization, Nonlinear Constraint, Interpretation of Lagrange Multiplier

1. INTRODUCTION

It is quite common to receive a discount on the price of each unit when ordering larger quantities of commodities, or in some markets the prices vary depending on quantities of commodity purchased. Also firms offer a lower per unit price if a consumer is willing to purchase larger quantities of a commodity. Such quantity discounts alter the linear budget constraint, and result in a nonlinear (convex type) budget constraint. This is an extension of the problem considered by Moolio et al (2009) by assuming that a government agency obtains price discounts by purchasing larger quantities of other inputs R . We assume that the government agency is allocated an annual budget B and required to maximize some sort of services to the community. If the agency uses factors K , L, and R in the same sense as used by Moolio et al (2009) to produce and provide services to the community, then its objective is to maximize the output function subject to a nonlinear budget constraint.

Baxley and Moorhouse (1984) suggested this problem. One can read the relevant articles to optimization problems by Islam (1997), Moolio (2002), Moolio and Islam (2008), Moolio et al (2009), and Islam et al (2010). All of the studies cited above are derived under the assumption of linear constraints. Moreover, fundamental relationships between mathematical economics, social choice and welfare theory by introducing utility functions, preference relations and Arrow's Impossibility Theorem are given in detail by Islam et al (2009). A detailed discussion on algebraic production functions and their uses is considered by Humphery (1997). An introduction to the Lagrange multipliers method and its application in the field of power systems economic operation is given by Li (2008). Kalvelagen (2003) discusses the well known optimization problem of utility maximization under a budget constraint, constructing interesting non-trivial variants by assuming some non-linear pricing structures actually observed in daily life.

DeSalvo and Huq (2002) show that under some forms of nonlinear pricing, after a price rise people may buy more of a commodity than would have been bought under linear pricing.

To get intuitive ideas and understanding of the problem at hand, we consider here, explicitly, a simple algebraic function in three variables, and examine the behaviour of the agency, that is, how a change in the costs of input will affect the situation, or if the demand of the services undergoes some changes. We also give suitable interpretation of the Lagrange multiplier in the context of this specific situation, besides using it as a device for transforming a constrained problem into a higher dimensional unconstrained problem.

Organization of remaining paper is as below: section 2 details the building of model, in section 3 we consider an explicit example and find optimal output, section 4 explains the interpretation of Lagrange multipliers, in section 5 second-order sufficient conditions are applied for maximization, section 6 enlightens the comparative static analysis, and concluding remarks are given in section 7.

2. THE MODEL

We consider, for the fixed annual budget, a government agency is charged to produce and provide to the community a quantity Q of the services during a year, with the use of K quantity of capital, L quantity of labour, and R quantity of other inputs into its service oriented production process. The agency uses factors K , L, and R to produce and provide services. Its objective is to maximize the output function:

Q = g (K, L, R) subject to a nonlinear budget constraint: B = rK + wL + r (R) R; where r is the rate of interest or services of capital per unit of capital K, w is the wage rate per unit of labour L, and r is the cost per unit of other inputs R, while g is a suitable production function. The government agency takes these and all other factor prices as given. We assume that second order partial derivatives of the function g with respect to the independent variables (factors) K , L, and R exist.

3. AN EXPLICIT EXAMPLE

In order to get intuitive ideas and an intrinsic understanding of the problem, we consider explicitly a simple algebraic form of the output function in three variables:

Q = g (K, L, R) = KLR (1)

subject to particular nonlinear budget constraint:

B = rK + wL + r (R) R, where r (R) = r0 R [?] r0, with r0 being the discounted price of the inputs R.

Therefore, the budget constraint takes the form:

This is a four dimensional unconstrained problem obtained from (1) and (2) by the use of Lagrange multiplier l, as a device. Assuming that the government agency maximizes it's output, the optimal quantities K , L , R , l of K, L, R, l that necessarily satisfy the first-order conditions, which we obtained by partial differentiation of the Lagrangian function (3) with respect to four variables l, K, L, and R and setting them equal to zero:

equation

This indicates that there will be no effect on the level of labour L, if the interest rate of capital K This also indicates that labour and capital are complementary. increases.

The above analysis relates to the effects of a change in the interest rate of the capital K ; our comparative static results are readily adaptable to the case of a change in the wage rate of labour L . However, the comparative static results are bit different in the case of a change in the cost of other inputs R ; which we analyze here.

Next, we analyze the effect of a change in budget B . Let us suppose that the service-providing agency is provided with an additional budget, and asked to increase its output. Naturally, we can expect that it will increase in its inputs K , L, and R . We examine and verify this mathematically. Again from (26), we get:

7. CONCLUDING REMARKS

We have applied the method of Lagrange multipliers to maximize output function subject to a nonlinear constraint, and derived mathematical formulation to devise optimal purchasing policy for a service providing agency. With the help of an explicit example, we studied the behaviour of the agency applying comparative static analysis; that is, if the price of an input rises, how an agency behaves; as well as it is also demonstrated that if an agency's budget increases how the agency is going to behave. Illustrating an explicit example, we showed that the optimization problems play an important role in real world, as well as in the cases where objective function and constraint have specific meanings; the Lagrange multipliers often have an identifiable significance. This is the fourth paper in the series of our papers published earlier in Indus Journal of Management and Social Sciences.

REFERENCES

Baxley, J.V. and Moorhouse, J.C. 1984. Lagrange Multiplier Problems in Economics. The American Mathematical Monthly, 91(7): 404-412 (Aug. - Sep).

Chiang, A.C. 1984. Fundamental Methods of Mathematical Economics. 3rd ed. Singapore: McGraw-Hill. DeSalvo, J.S., and Huq, M. 2002. Introducing Nonlinear Pricing into Consumer Choice Theory. Journal of Economic Education, 166- 179 (Spring 2002).

Humphery, T.M. 1997. Algebraic Production Functions and Their Uses Before Cobb- Douglas. Federal Reserve Bank of Richmond Economic Quarterly, 83(1) Winter.

Islam, J.N. 1997. Aspects of Mathematical Economics and Social Choice Theory. Proceedings of the Second Chittagong Conference on Mathematical Economics and its Relevance for Development. J.N. Islam (Ed.), Chittagong: University of Chittagong, Bangladesh.

Islam, J.N., Mohajan, H.K., and Moolio, P. 2009. Preference of Social Choice in Mathematical Economics, Indus Journal of Management and Social Sciences, 3(1):17-37 (Spring 2009).

Islam, J.N., Mohajan, H.K., and Moolio, P. 2010. Utility Maximization subject to Multiple Constraints. Indus Journal of Management and Social Sciences, 4(1):15-29 (Spring 2010).

Kalvelagen, E. 2003. Utility Maximization with Nonlinear Budget Constraints Under GAMS.

Li, H. 2008. Lagrange Multipliers and their Applications. Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37921, USA.

Moolio, P. 2002. Theory and Applications of Classical Optimization to Economic Problems. M. Phil. Thesis. Chittagong: University of Chittagong, Bangladesh.

Moolio, P. and Islam, J.N. 2008. Cost Minimization of a Competitive Firm. Indus Journal of Management and Social Sciences, 2(2): 184-160 (Fall 2008).

Moolio, P., Islam, J.N., and Mohajan, H.K. 2009. Output Maximization of an Agency. Indus Journal of Management and Social Sciences, 3(1): 39-51 (Spring 2009).

The material presented by the authors does not necessarily represent the viewpoint of editors and the management of Khadim Ali Shah Bukhari Institute of Technology as well as the author' institute.

1 Emeritus Professor, Research Centre for Mathematical and Physical Sciences, University of Chittagong, Bangladesh. Phone: +880-31-616780.

2 Assistant Professor, Premier University, Chittagong, Bangladesh. Email: haradhan_km@yahoo.com

3 Professor, Pannasastra University of Cambodia, Phnom Penh, Cambodia. Corresponding Author: Pahlaj Moolio. Email: pahlajmoolio@gmail.com

Acknowledgement: Authors would like to thank the editors and anonymous referees for their comments and insight in improving the draft copy of this article. Author furthur would like to declare that this manuscript is original, has not previously been published, not currently on offer to another publisher; and willingly transfers its copy rights to the publisher of this journal.

ABSTRACT

The main aim of this paper is to derive the mathematical formulation to device an optimal purchasing policy for the service providing agency. An attempt has been made to maximize the output function of an agency subject to a nonlinear budget constraint by assuming that the agency gets price discounts for purchasing larger quantities of other inputs. Such quantity discounts alter the linear budget constraint and result in a nonlinear (convex type) budget constraint. We use the method of Lagrange multipliers and apply the first- order necessary conditions as well as the second-order sufficient conditions for maximization. We also use comparative static analysis and study the behavior of the agency when prices of inputs undergo change, besides providing useful interpretation of the Lagrange multipliers in this specific case. Illustrating an explicit example, we show that the optimization problems play an important role in the real world.

JEL. Classification: C51; C65; C61; D24

Keywords: Maximization, Nonlinear Constraint, Interpretation of Lagrange Multiplier

1. INTRODUCTION

It is quite common to receive a discount on the price of each unit when ordering larger quantities of commodities, or in some markets the prices vary depending on quantities of commodity purchased. Also firms offer a lower per unit price if a consumer is willing to purchase larger quantities of a commodity. Such quantity discounts alter the linear budget constraint, and result in a nonlinear (convex type) budget constraint. This is an extension of the problem considered by Moolio et al (2009) by assuming that a government agency obtains price discounts by purchasing larger quantities of other inputs R . We assume that the government agency is allocated an annual budget B and required to maximize some sort of services to the community. If the agency uses factors K , L, and R in the same sense as used by Moolio et al (2009) to produce and provide services to the community, then its objective is to maximize the output function subject to a nonlinear budget constraint.

Baxley and Moorhouse (1984) suggested this problem. One can read the relevant articles to optimization problems by Islam (1997), Moolio (2002), Moolio and Islam (2008), Moolio et al (2009), and Islam et al (2010). All of the studies cited above are derived under the assumption of linear constraints. Moreover, fundamental relationships between mathematical economics, social choice and welfare theory by introducing utility functions, preference relations and Arrow's Impossibility Theorem are given in detail by Islam et al (2009). A detailed discussion on algebraic production functions and their uses is considered by Humphery (1997). An introduction to the Lagrange multipliers method and its application in the field of power systems economic operation is given by Li (2008). Kalvelagen (2003) discusses the well known optimization problem of utility maximization under a budget constraint, constructing interesting non-trivial variants by assuming some non-linear pricing structures actually observed in daily life.

DeSalvo and Huq (2002) show that under some forms of nonlinear pricing, after a price rise people may buy more of a commodity than would have been bought under linear pricing.

To get intuitive ideas and understanding of the problem at hand, we consider here, explicitly, a simple algebraic function in three variables, and examine the behaviour of the agency, that is, how a change in the costs of input will affect the situation, or if the demand of the services undergoes some changes. We also give suitable interpretation of the Lagrange multiplier in the context of this specific situation, besides using it as a device for transforming a constrained problem into a higher dimensional unconstrained problem.

Organization of remaining paper is as below: section 2 details the building of model, in section 3 we consider an explicit example and find optimal output, section 4 explains the interpretation of Lagrange multipliers, in section 5 second-order sufficient conditions are applied for maximization, section 6 enlightens the comparative static analysis, and concluding remarks are given in section 7.

2. THE MODEL

We consider, for the fixed annual budget, a government agency is charged to produce and provide to the community a quantity Q of the services during a year, with the use of K quantity of capital, L quantity of labour, and R quantity of other inputs into its service oriented production process. The agency uses factors K , L, and R to produce and provide services. Its objective is to maximize the output function:

Q = g (K, L, R) subject to a nonlinear budget constraint: B = rK + wL + r (R) R; where r is the rate of interest or services of capital per unit of capital K, w is the wage rate per unit of labour L, and r is the cost per unit of other inputs R, while g is a suitable production function. The government agency takes these and all other factor prices as given. We assume that second order partial derivatives of the function g with respect to the independent variables (factors) K , L, and R exist.

3. AN EXPLICIT EXAMPLE

In order to get intuitive ideas and an intrinsic understanding of the problem, we consider explicitly a simple algebraic form of the output function in three variables:

Q = g (K, L, R) = KLR (1)

subject to particular nonlinear budget constraint:

B = rK + wL + r (R) R, where r (R) = r0 R [?] r0, with r0 being the discounted price of the inputs R.

Therefore, the budget constraint takes the form:

This is a four dimensional unconstrained problem obtained from (1) and (2) by the use of Lagrange multiplier l, as a device. Assuming that the government agency maximizes it's output, the optimal quantities K , L , R , l of K, L, R, l that necessarily satisfy the first-order conditions, which we obtained by partial differentiation of the Lagrangian function (3) with respect to four variables l, K, L, and R and setting them equal to zero:

equation

This indicates that there will be no effect on the level of labour L, if the interest rate of capital K This also indicates that labour and capital are complementary. increases.

The above analysis relates to the effects of a change in the interest rate of the capital K ; our comparative static results are readily adaptable to the case of a change in the wage rate of labour L . However, the comparative static results are bit different in the case of a change in the cost of other inputs R ; which we analyze here.

Next, we analyze the effect of a change in budget B . Let us suppose that the service-providing agency is provided with an additional budget, and asked to increase its output. Naturally, we can expect that it will increase in its inputs K , L, and R . We examine and verify this mathematically. Again from (26), we get:

7. CONCLUDING REMARKS

We have applied the method of Lagrange multipliers to maximize output function subject to a nonlinear constraint, and derived mathematical formulation to devise optimal purchasing policy for a service providing agency. With the help of an explicit example, we studied the behaviour of the agency applying comparative static analysis; that is, if the price of an input rises, how an agency behaves; as well as it is also demonstrated that if an agency's budget increases how the agency is going to behave. Illustrating an explicit example, we showed that the optimization problems play an important role in real world, as well as in the cases where objective function and constraint have specific meanings; the Lagrange multipliers often have an identifiable significance. This is the fourth paper in the series of our papers published earlier in Indus Journal of Management and Social Sciences.

REFERENCES

Baxley, J.V. and Moorhouse, J.C. 1984. Lagrange Multiplier Problems in Economics. The American Mathematical Monthly, 91(7): 404-412 (Aug. - Sep).

Chiang, A.C. 1984. Fundamental Methods of Mathematical Economics. 3rd ed. Singapore: McGraw-Hill. DeSalvo, J.S., and Huq, M. 2002. Introducing Nonlinear Pricing into Consumer Choice Theory. Journal of Economic Education, 166- 179 (Spring 2002).

Humphery, T.M. 1997. Algebraic Production Functions and Their Uses Before Cobb- Douglas. Federal Reserve Bank of Richmond Economic Quarterly, 83(1) Winter.

Islam, J.N. 1997. Aspects of Mathematical Economics and Social Choice Theory. Proceedings of the Second Chittagong Conference on Mathematical Economics and its Relevance for Development. J.N. Islam (Ed.), Chittagong: University of Chittagong, Bangladesh.

Islam, J.N., Mohajan, H.K., and Moolio, P. 2009. Preference of Social Choice in Mathematical Economics, Indus Journal of Management and Social Sciences, 3(1):17-37 (Spring 2009).

Islam, J.N., Mohajan, H.K., and Moolio, P. 2010. Utility Maximization subject to Multiple Constraints. Indus Journal of Management and Social Sciences, 4(1):15-29 (Spring 2010).

Kalvelagen, E. 2003. Utility Maximization with Nonlinear Budget Constraints Under GAMS.

Li, H. 2008. Lagrange Multipliers and their Applications. Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37921, USA.

Moolio, P. 2002. Theory and Applications of Classical Optimization to Economic Problems. M. Phil. Thesis. Chittagong: University of Chittagong, Bangladesh.

Moolio, P. and Islam, J.N. 2008. Cost Minimization of a Competitive Firm. Indus Journal of Management and Social Sciences, 2(2): 184-160 (Fall 2008).

Moolio, P., Islam, J.N., and Mohajan, H.K. 2009. Output Maximization of an Agency. Indus Journal of Management and Social Sciences, 3(1): 39-51 (Spring 2009).

The material presented by the authors does not necessarily represent the viewpoint of editors and the management of Khadim Ali Shah Bukhari Institute of Technology as well as the author' institute.

1 Emeritus Professor, Research Centre for Mathematical and Physical Sciences, University of Chittagong, Bangladesh. Phone: +880-31-616780.

2 Assistant Professor, Premier University, Chittagong, Bangladesh. Email: haradhan_km@yahoo.com

3 Professor, Pannasastra University of Cambodia, Phnom Penh, Cambodia. Corresponding Author: Pahlaj Moolio. Email: pahlajmoolio@gmail.com

Acknowledgement: Authors would like to thank the editors and anonymous referees for their comments and insight in improving the draft copy of this article. Author furthur would like to declare that this manuscript is original, has not previously been published, not currently on offer to another publisher; and willingly transfers its copy rights to the publisher of this journal.

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Date: | Dec 31, 2011 |

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