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Managerial Economics

Unit Titles

1 Introduction to Managerial Economics

2 The Theory of the Consumer

3 The Theory of the Firm

4 Competitive and Monopolistic Markets

5 Strategic Behaviour and Oligopoly

6 Bargaining and Private Information

7 The Optimal Provision of Incentives

8 Financial Investment, Capital Structure and Corporate Control

Managerial Economics

Unit 1 Introduction to Managerial Economics

Contents Page

What this unit is about 2

1 Introduction to Managerial Economics 3

2 The Main Concepts in Microeconomics 5

3 Optimisation in Economic Analysis 6

4 Properties of Objective Functions and Feasible Sets 7

5 Properties of Solutions 11

6 Constrained Optimisation: the Lagrange Method 12

7 Comparative Statics and the Envelope Theorem 14

8 Conclusions 17

Revision Exercise 17

References 20

2 MANAGERIAL ECONOMICS

UNIVERSITY OF LONDON

What this unit is aboutThis first unit introduces you to the main methods of microeconomics and manage-rial economics. We see how we can formulate problems of optimisation underconstraints, which are central to this course. After explaining the language and themain concepts of microeconomics, we look at the role of optimisation in economicanalysis. We derive some general methods for solving optimisation problems and foranalysing the characteristics of the solutions. This unit lays the foundations for theanalysis in the course.

What you will learn• The scope of microeconomics;

• The nature of managerial economics;

• What is meant by adverse selection;

• The meaning of moral hazard;

• How to define an economic commodity;

• What is a price;

• Who are economic agents;

• The characteristics of a market;

• The meaning and properties of objective function;

• The definition and use of a feasible set;

• When a solution to an optimisation problem exists;

• When is the solution a global optimum;

• When is the solution unique;

• How to use the Lagrange method;

• The nature of Lagrange multipliers;

• What is meant by comparative statics, or sensitivity analysis;

• What is the envelope theorem.

Readings

Gravelle and Rees, Microeconomics, Chapter 1 and Appendices A-G, I and J.

Milgrom and Roberts, Economics, Organization and Management, Chapter 1

UNIT ONE 3

CENTRE FOR FINANCIAL AND MANAGEMENT STUDIES

1 Introduction to Managerial EconomicsWelcome to this course in managerial economics. In this course we will study theway individual economic units – firms, consumers, managers etc. – should go aboutmaking their decisions. In order to address this question, we have to be more preciseabout what the objectives of individual economic units are: what do they seek toachieve, and what are their aims? We also have to be precise about the context inwhich the economic units operate: what are the variables under their control? Andwhat constraints do they face?

Economic agents do not operate in a social vacuum. A crucial aspect of theirbehaviour includes their interactions with other agents. On the one hand, this cancreate new opportunities for individuals. Agents must rely on others in pursuing theirown interests. This may be achieved either by explicit co-operation or by theirunderstanding of the other agents’ selfish pursuit of their own best interests.

On the other hand, the need to interact with other agents can place severe constraintson individual agents’ behaviour. They will not always be able to get their own wayand will be forced to accommodate the other agents’ needs. Sometimes, economicagents may try to behave in a strategic fashion. They will try to anticipate the others’reactions to their own actions, and will therefore endeavour to make decisions whichturn to their best advantage, given that their actions could influence the other agents’choices and behaviour. One way agents can achieve this aim is by entering into someform of contract, which – explicitly or implicitly – takes into account other agents’motivations and seeks to exploit their economic incentives in order to induce them toperform.

In general, thus, we have to examine:

• how individual agents behave;

• what is their motivation;

• what are the constraints they face; and

• how they interact with other agents in the economy.

An important issue in managerial economics arises when agents have imperfectknowledge of the characteristics of the other agents (adverse selection), or when theycannot compel the other agents to behave in a given way (moral hazard). Forinstance, an insurance company may be unable to observe the exact class of risk ofits prospective customers. If it were to offer a standard contract to all its prospectiveclients, high-risk individuals might form a disproportionate number of its customers.This would be a case of adverse selection. A possible way out is for the insurancecompany to offer a range of different contracts to its customers, with differentpremia and penalty structures. If the range of contracts has been optimally designed,high-risk customers will prefer one type of contract, and low-risk customers another.The insurance company will therefore be able to separate high- from low-riskcustomers by offering a range of contracts, and by letting customers choose.

Another example might be an employer who wants to elicit a high level of effortfrom its employees. This would be a case of moral hazard. The employer couldachieve its aim by creating suitable incentives – in the form of performance-relatedpay, appropriate promotion schemes, etc. – which reward higher effort.



The approach to managerial economics which we study in this course relies veryclosely on microeconomic analysis. Indeed, we could define managerial economicsas that branch of microeconomics which helps develop a rational decision making

approach in management. We deal with individual behaviour and motivation,explore how agents may interact with each other, and analyse how best they coulddesign contracts in order to elicit the desired behaviour from other agents. So, forinstance, we look at the optimal production decisions of firms, or at the design oflabour contracts between an employer and its employees.

1.1 The Structure of this Course

The outline of this course is as follows. This unit illustrates the main ideas in micro-economics and managerial economics and reviews the theory of mathematicaloptimisation which we use in this course. Unit 2 deals with the theory of theconsumer. We look at how rational agents can maximise their individual welfare,given the constraints they face. Unit 3 presents the theory of the firm. We analyse thestructure of technology in the short and in the long run, and consider the optimaloutput supply by firms. In the next units we move on from individual consumers andfirms to consider how they interact in markets. In Unit 4 we look at competitivemarkets and monopoly, and in Unit 5 we explore oligopolistic markets. In dealingwith the latter, we make extensive use of game theory, which studies how agents canbehave strategically. Unit 6 considers the important issue of how agents shouldbehave when they have imperfect knowledge of the characteristics of other agents orof how they will behave. Unit 7 examines how agents can devise economic mecha-nisms to elicit information from other agents, and how they can design optimalcontracts which induce the other agents to provide the required incentives. Unit 8brings together the methods of the previous units and applies them to issues infinancial investment, capital structure and corporate control.

You have been given the following textbooks, which constitute the main readings forthis course:

Hugh Gravelle and Ray Rees (2004) Microeconomics, 3rd edition, Prentice-Hall,Harlow;

Paul Milgrom and John Roberts (1992) Economics, Organization and Management,Prentice-Hall, Inc., Englewood Cliffs (New Jersey).

Gravelle and Rees cover all the main topics in microeconomics, including the morerecent and advanced topics in the economics of imperfect information and incen-tives. Milgrom and Roberts deal specifically with the issues of optimal organisation,co-ordination, motivation, and incentives that are crucial for modern managerialeconomics. You will see that Gravelle-Rees and Milgrom-Roberts are very differentin their approach: the first one is more formal in its arguments and more mathemati-cal, whereas the second makes more use of examples and applications. The twotextbooks complement each other quite well, and by studying them both you willexperience a useful range of approaches to microeconomics and managerial eco-nomics

It is useful at this point for you to stop and read the first chapter, ‘Does

Organization Matter?’ in Milgrom and Roberts. This chapter is an introduction to theproblems of business organisation, and explains why the compensation and owner-

UNIT ONE 5


ship structure of firms can be an important determinant of their performance. In thischapter, the authors clearly illustrate the importance of economic decisions inbusiness organisations. They introduce a number of useful concepts:

• co-ordination within a company and with outside suppliers;

• performance-related pay systems;

• the ratchet effect;

• the role of information; and

• incentives,

which will be fundamental to our analysis in the later units of this course.

2 The Main Concepts in MicroeconomicsThis section introduces the main ideas and concepts in microeconomics. The basicnotion is that of a commodity, which constitutes the object of production andexchange in economics. The concept of commodity should be interpreted in a broadsense, including both goods and services. It is important to note that the exactdefinition of commodity must specify its physical characteristics, the location wherethe commodity is available and the date when it is made available. Thus, a commod-ity could be a car, of a particular make and type, in Paris, on a given date. The samecar, on the same date, but in Mexico City, should be regarded as a different com-modity.

Another example of a commodity may be given by the consultancy servicesprovided by a financial analyst, with a given educational and professional back-ground, in Hong Kong, on a given date. The consultancy services provided by thatsame analyst in Hong Kong, but on a different date, should be regarded as a differentcommodity.

The second main concept in microeconomics is price. The price of commoditiesmeasures the terms at which the commodities can be traded with one another. It iscustomary to express prices in terms of monetary units of accounts, such as Singa-pore dollars or South African rands. In microeconomics, the main notion is that ofrelative price between two commodities – let’s call them commodity A and com-modity B. The relative price between A and B is the number of units of B whichhave to be given up in order to purchase one unit of A. For example, if the monetaryprice of commodity A is 200 Singapore dollars, and the monetary price of commod-ity B is 50 Singapore dollars, then the relative price of A in terms of B is 4, since wehave to give up 4 units of B in order to purchase one unit of A.

An important assumption which is often made in microeconomics is that agents donot suffer from money illusion: if all monetary prices were suddenly to double, thereal decisions of agents would be unaffected. This is because the relative pricesbetween commodities would still be the same, reflecting the fact that the terms atwhich commodities are traded with one another have not changed.

The third main concept is that of economic agents. In traditional microeconomics,these are usually classified as consumers and firms. We shall analyse the behaviourof consumers in Unit 2, and the behaviour of firms in Unit 3. Consumers mustallocate their limited resources among the different commodities they can purchase,and firms employ inputs such as capital and labour to produce output. In the



traditional analysis, firms are seen as individual decision makers. In the more recentmicroeconomic theory and in managerial economics, however, it is usually acknowl-edged that firms are complex organisations, and that the individual agents attached toa firm may each have different goals in mind. It is therefore necessary to explorehow firms behave, given the possible conflicts of interest between the members ofthe organisation. We address these and related issues in Units 6, 7 and 8.

The fourth main concept is that of a market. By this we mean the place whereeconomic commodities are traded. It is important to note that a market is notnecessarily a formal market place. Trade occurs whenever agents engage in anexchange of commodities, irrespective of whether this exchange is regulated or not.Also, trade does not require exchange of money: Barter, for instance, is a form oftrade. An important issue in microeconomics is to analyse how markets work, andhow agents behave in markets. In markets with a large number of participants,individuals often have very little power to alter the conditions of exchange. A smallshopkeeper in a large town may have limited control over the prices charged for itsgoods, because higher prices could mean losing most of its customers: they couldjust walk away and buy from the rival shops. By contrast, the only shopkeeper in aremote village could wield some market power, in the sense of enjoying somelatitude in setting prices. Its customers would not be able easily to walk away fromthe shop and purchase the commodities somewhere else, if no rival shop wereavailable. An important component of this course is the analysis of how marketswork, and how agents can behave strategically in a market setting. We exploremarket behaviour in Units 4 and 5.

Please now stop and read pages 1-6 from Gravelle and Rees, Chapter 1, section

A. The textbook introduces the main concepts in microeconomic analysis (com-modities, price, economic agents and markets), and illustrates them by means ofexamples. We shall constantly be referring to these concepts in this course, so it isessential that you are thoroughly familiar with them. In addition, I should like you topay special attention to the discussion of markets and of economic agents.

3 Optimisation in Economic AnalysisIn microeconomics, the assumption is often made that agents behave in a rationalfashion. This means that, when making their decisions, they consider all the possiblealternative courses of action, rank them according to their preferences, and finallychoose the action which they prefer best. Thus, a consumer seeking to maximise herutility given her total income will consider all the possible uses of her income, willrank these uses according to the utility she derives from each of them, and finallywill choose the use which yields the highest utility (Unit 2). Similarly, a firm mightseek to maximise its profit given technology and input prices and given a demandcurve for its output or output price, and will decide on the levels of labour andcapital it employs (Units 3 and 4).

Formally, the process of choice can be modelled as an optimisation problem facedby the economic agent. There is a well-defined objective function that the agentseeks to maximise by optimal choice of the decision variables. The context in whichchoices take place is modelled as a set of constraints on individual behaviour. Thus,the objective function of consumers is their utility function, which they seek to

UNIT ONE 7


maximise; their choice variables are the quantities of the various commodities whichare consumed, and their choice must satisfy the budget constraint which requires thattheir expenditure cannot exceed their total income. The objective function of the firmis the value of its profits; its choice variables are the quantity of inputs employed(capital and labour), the output supplied and the price of its output (unless the firm isoperating under perfect competition), and its constraints are the level of technologyand the demand for its output by consumers.

The general method of microeconomics is therefore to model individual choiceas a problem of optimisation under constraints. This approach is very general, and itis easily extended to the more recent topics in microeconomics and managerialeconomics, such as the economics of imperfect information. Consider, for example,the case of an employer who wants to elicit a higher level of effort from its employ-ees. Its objective function are its profits, its choice variables the remuneration systemoffered to its employees, and its constraints the response of its employees (who canbe thought of as rational and optimising agents in their turn, seeking to maximisetheir own welfare given the remuneration scheme offered). The problem can befairly complicated, but the basic structure is quite straightforward, and alwaysinvolves optimisation under constraints.

Note that optimisation may involve either a maximisation or a minimisation problem.Examples of the latter case are the minimisation of costs of a firm, or the minimisa-tion of the risk faced by a financial investor. In this course we shall encounter manyexamples of both maximisation and minimisation. The same methods can be appliedto both cases.

It is now a good moment to stop and read pages 6-11 from Gravelle and Rees,

Chapter 1, sections A and B.

Note how these authors pay special attention to the assumption of rationality ineconomics. Please read these sections carefully, making notes on the importantpoints as you read. Read also with attention the analysis of the economic and socialframework of choice theory in section B. The structure of an optimisation problem ineconomics is explained by Gravelle and Rees in Appendix A, pages 657–59, and youshould read these pages as well. Note in particular how the set of constraints isdescribed by Gravelle and Rees as the feasible set. Pay special attention also to thedefinitions of choice variables and of the objective function, and to the economicexamples which are provided.

4 Properties of Objective Functions andFeasible Sets

In the previous section, we saw that the general method of microeconomics is tomodel the choice problem as a programme of optimisation under constraints. It istherefore necessary to be able to establish whether the problem we are consideringdoes have a solution, whether the solution is unique, and how the characteristics ofthe solution depend on the parameters of the problem. For instance, when we look atconsumption behaviour, it is important to establish whether there is a combination ofcommodities which maximises the utility of the consumer (existence of the solution),and whether other combinations exist which yield the same level of utility so that the



consumer is indifferent between them (uniqueness of the solution). Even withoutknowing the exact functions involved in the optimisation problem, it is usuallypossible to say something about the solution. The reason for this is that economictheory suggests that the functional forms in a microeconomic decision problem mustsatisfy some given properties, and this in turn could lead to guaranteeing that theproblem has a unique solution with some important additional characteristics.

The main properties we are looking for in the solution to a microeconomic optimisa-tion problem are:

• whether it exists;

• whether it is a global solution;

• whether it is unique.

These properties are explained in the textbook by Gravelle and Rees. Please

stop now and read the relevant pages in the book before continuing with the rest ofthis text. These are pages 660-62 from Appendix B; be sure to make careful notes onthese properties as you read this section.

Existence of a solution is clearly a crucial feature of the optimisation problem, yet itcannot always be taken for granted. Some mathematical problems simply do nothave a solution. Hence, when setting up an economic problem we must always checkwhether a solution exists. The good news is that sometimes the properties of theobjective function and of the constraints do ensure that a solution exists (this will bediscussed further in the next section of this unit).

But even when a solution exists, we cannot always be sure that it is a global solution,i.e. that it achieves a maximum (or a minimum, depending on the problem) over thewhole range of feasible values for the decision variables. Figure B.1 in Gravelle andRees shows an example of a function f(x) which has a local, but not a globaloptimum at x**. When solving an optimisation problem, it is therefore necessary tocheck that the solution is a global, rather than simply a local solution to the problem.

Finally, it is important to establish whether the solution to the optimisation problemis unique, or whether there could exist several choices of the decision variableswhich yield the same value of the objective function, and therefore are equivalent forthe optimising agent. Although there are no theoretical problems in principle withthe latter case, i.e. when there are multiple solutions, there could be difficulties whenit comes to predicting the behaviour of the economic agents. In fact, if the agent isindifferent as between a number of alternative courses of action, it could be impossi-ble to predict with certainty what the outcome of its actions will be. Sometimes it ispossible to anticipate that agents will choose one of these actions over the others –for instance, in problems which involve co-ordination by many agents, there couldbe a focal equilibrium, that is, an action to which all agents co-ordinate theirbehaviour. Thus, when driving a car, keeping to the right side of the road is the focalequilibrium in France, whereas keeping to the left is the focal equilibrium in the UK.In a number of cases, however, it could be quite difficult to predict the action ofagents when there are multiple solutions to the individual optimisation problem.

We are going to look at the issues of existence, unicity, and the global property ofsolutions in the next section. We will see there that, for a large class of problems in

UNIT ONE 9


microeconomics, it is possible to prove that a solution exists, is unique, and is indeeda global optimum. The conditions for this to be true lie in the properties of theobjective function and of the set described by the constraints – the feasible set.

The main properties which an objective function can satisfy are:

• Continuity

• Concavity

• Quasi-concavity

Gravelle and Rees give precise mathematical definitions of the above properties. Anintuitive graphical account can be obtained from the examples of Figures 1.1 – 1.3,shown here:

FIGURE 1.1 CONTINUITY

f(x) g(x)

a b x a b x(a) (b)

Part (a) of Figure 1.1 shows a function f(x) which is continuous, whereas part (b)shows a discontinuous function. In Figure 1.2, part (a) gives an example of aconcave function, whereas part (b) displays a function which is not concave.

FIGURE 1.2 CONCAVITY

f(x) g(x)

(a) (b)o x o x



Finally, part (a) of Figure 1.3 shows a quasi-concave function (note that now wehave two choice variables, x1 and x2), while part (b) displays a function which is not

quasi-concave.

FIGURE 1.3 QUASI-CONCAVITY

x2

x2

x1 x

1(a) (b)

f(x1,x2) g(x

1,x2)

The main properties of the feasible set are:

• Non-emptiness. The feasible set is non-empty if it contains at least one element.This implies that the constraints of the optimisation problem do not contradicteach other.

• Boundedness. The feasible set is bounded if it is not possible to go to infinity,while still remaining in the set.

• Closedness. The feasible set is closed if it contains its boundaries.

• Convexity. The feasible set is convex if, for any two points in the set, the straightline connecting them lies entirely within the set. This implies that the feasible setmust have no ‘holes’.

If the feasible set is both closed and bounded, it is said to be compact. The textbookby Gravelle and Rees contains further explanations and examples of the aboveproperties. It is often possible to verify that these properties are actually satisfied in alarge number of problems analysed in applied microeconomics and managerialeconomics. This turns out to be quite useful, since in these cases we can be moreprecise about the nature of the solutions to the optimisation problem. We shallconsider this point more fully in the next section.

Please now stop and read pages 662–68 from Appendix B of Gravelle and

Rees. It is very useful to go through all the examples in the textbook, and to payparticular attention to the counter-examples and to the intuitive graphical interpreta-tion of the properties of the objective function and of the feasible set. As you readthe textbook, it is very useful to think of examples of functions you already know(for instance, from any previous courses in mathematics), and check whether theysatisfy the properties set out in your textbook

UNIT ONE 11


5 Properties of SolutionsSection 4 has illustrated some of the possible properties of the objective function andthe feasible set. Our reason for considering those properties is that, when they aresatisfied for a particular optimisation problem we can be more precise about thecharacteristics of the solutions of that problem. In particular, we have the followingresults (Gravelle and Rees, Appendices C-E).

Weierstrass Theorem

An optimisation problem always has a solution if:

• the objective function is continuous; and

• the feasible set is non-empty and compact.

Global Optima

A local maximum of an optimisation problem is always a global maximum if:

• the objective function is quasi-concave; and

• the feasible set is convex.

Uniqueness Theorem

Given an optimisation problem in which the feasible set is convex and the objectivefunction is non-constant and quasi-concave, a solution is unique if:

• the feasible set is strictly convex; or

• the objective function is strictly quasi-concave; or

• both.

You can find proofs of the above theorems, together with their intuitive interpreta-tion, in the textbook by Gravelle and Rees. The above results are very important,because they enable us to infer some characteristics of the solutions simply on thebasis of the properties of the objective function and of the feasible set. In particular,it may not be necessary to solve the problem explicitly in order to know that asolution exists, that it is a global optimum, and that it is unique.

It is important to note that the theorems presented above lay out sufficient, but notnecessary conditions for the properties to hold. Thus, an optimisation problem mayhave a solution even if the conditions of the Weierstrass Theorem do not apply, forinstance when the objective function is not continuous. In other words, if theconditions of the theorems are satisfied then we can be certain that the respectiveproperties hold, whereas if the conditions are not satisfied then we cannot saywhether the properties hold or not. In this second case, we must therefore solve theproblem explicitly and check directly whether a solution exists, whether it is a globalrather than a local optimum and whether it is unique.

Fortunately, in a large number of problems in microeconomics and in managerialeconomics all the above properties hold. We can therefore be confident about thesolutions having the desired properties.

This is a good moment to stop and read Gravelle and Rees, Appendices C-F,

pages 670-78. The authors deal with the topics presented above, and present proofsfor the theorems. They also discuss the important issue of interior versus boundary



optima. The latter occur when the optimum lies on the boundary of the feasible set –i.e., on one of the constraints. You should read carefully what Gravelle and Reeshave to say on this. It is very helpful to follow their mathematical proofs, althoughyou are required to reproduce the mathematical proofs of these sections.

6 Constrained Optimisation: the LagrangeMethod

The previous section looked at some of the properties of the solution to a constrainedoptimisation problem, when the objective function and the feasible set satisfy certainproperties. However, we have not yet developed a general method for finding ananalytical solution. This is the object of the present section. We consider a fairlygeneral approach to constrained optimisation, and develop a constructive method forfinding the solution.

Suppose that we have n decision variables: x = (x1,x2,...,xn), and the objective

function is:

f(x) = f(x1,x2,...,xn) (1.1)

where f(x) is a continuous and differentiable function. For instance, the economicagent could be a consumer who has to decide how to allocate his or her incomeamong n alternative consumption commodities, (x1,x2,...,xn) are the quantities

consumed of the commodities where f(x) is the consumer’s utility function.

The optimisation problem is in general subject to a number of constraints. These cantake the form of m<n equality constraints:

g1(x) = g1(x1,x2,...,xn) = b1 (1.2a)

g2(x) = g2(x1,x2,...,xn) = b2 (1.2b)

..........…..............................

gm(x) = gm(x1,x2,...,xn) = bm (1.2c)

where g1(x), g2(x),..., gm(x) are continuous and differentiable functions, and b1, b2,...,

bm are constant coefficients. For instance, in the consumer example g1(x) = b1 could

be the budget constraint where b1 is income, g2(x) = b2 could be a dietary require-

ment, etc. We can define the Lagrange function, or simply the Lagrangean for thisoptimisation problem as:

L(x1,x2,...,xn, 1, 2,... m) = f(x1,x2,...,xn) + j[bj g j (x1,x2, ..., xn)j =1

m

] (1.3)

where 1, 2, ..., m are m additional variables, called the Lagrange multipliers,

associated to the m constraints (1.2a)-(1.2c) as shown in equation (1.3). The first-order necessary conditions for the optimisation problem are:

L

x1=

f

x1j

g j

x1= 0

j=1

m

(1.4a)

UNIT ONE 13


L

x2=

f

x2j

g j

x2= 0

j=1

m

(1.4b)

..........................................................

L

xn

=f

xn

j

g j

xn

= 0j=1

m

(1.4c)

and constraints:

L

1

= b1 g1(x1, x2 ,..., xn ) = 0 (1.5a)

L

2

= b2 g 2(x1, x2 ,..., xn ) = 0 (1.5b)

..........................................................

L

m

= bm g m (x1, x2 ,..., xn ) = 0 (1.5c)

Note that the first-order conditions (1.5a)-(1.5c) simply yield again the constraints(1.2a)-(1.2c). The system (1.4a)-(1.5c) comprises n + m equations in the n + m

unknowns x1,x2,...,xn, 1, 2,... m. If the conditions of the Weierstrass and of the

uniqueness theorems are satisfied (see section 6), then the system has a uniquesolution: (x1*, x2*, ..., xn*, 1*, 2*,... m*).

The optimal choice of the decision variables is therefore given by (x1*, x2*, ..., xn*).

The Lagrange multipliers ( 1*, 2*,... m*) have an important economic interpreta-

tion. We have that

f ( x1*,..., xn*, 1*,..., m )

b1= 1 * (1.6a)

f ( x1*,..., xn*, 1*,..., m )

b2= 2 * (1.6b)

..........................................................

f ( x1*,..., xn*, 1*,..., m )

bm

= m * (1.6c)

Hence, the Lagrange multipliers measure the marginal effect on the objectivefunction of relaxing the respective constraints. Thus, in our consumption example, if

b1 is the income constraint, then 1* measures by how much total utility would

increase, if income were increased by one monetary unit (e.g. 1 US $). We shall relyextensively on this interpretation of the multipliers in the next units of this course.

The method of constrained optimisation based on the Lagrange function is

explained by Gravelle and Rees, pages 679-84. Please read these pages now.



7 Comparative Statics and the EnvelopeTheorem

In microeconomics and managerial economics there are two main questions to beaddressed when we regard economic choice as a problem of constrained optimisa-tion. The first question is:

• what is the solution to our problem?

The second question, closely related to the first one, is:

• how would the solution change, if any of the features which characterise

the economic environment were to change?

For example, we can analyse the choice problem of a household by assuming that thehousehold endeavours to maximise its utility, given a budget constraint. We shallreturn to this problem in Unit 2. The Lagrange multiplier method, illustrated insection 6 of this unit, enables us to find the optimal choices of consumption com-modities for the household, as a function of its income and of the prices of all thecommodities. This answers the first question set out above, to find a solution for ouroptimisation problem.

In the context of our example, the second question is:

• how would the optimal consumption commodities for the household vary, if

household income changes, or if the commodities prices change?

This question is clearly important for our analysis. The present section explains howwe can address this question, for a fairly general problem in microeconomics ormanagerial economics. The remainder of the course will see many applications ofthese methods to a variety of examples.

Suppose that the objective function is:

max y = u(x1, x2) (1.7) (x1, x2)

where the function u(x1,x2) must be maximised by choosing the decision variables x1

and x2. Let the constraint be given by:

p1x1 + p2x2 = m (1.8)

To fix ideas, u(x1,x2) could be the utility function of an individual, x1 and x2 the

quantities consumed of commodities 1 and 2, m could be income, and p1 and p2 the

prices of the commodities. Then equation (1.8) is the budget constraint. TheLagrangean for the optimisation problem is:

L = u(x1, x2 )+ (m p1x1 + p2x2 ) (1.9)

where is the Lagrange multiplier. As explained in section 6, the first-order

conditions are:

Lx1

=u(x1,x 2 )

x1p1 = 0 (1.10a)

UNIT ONE 15


L

x2=

u(x1, x2 )

x2p2 = 0 (1.10b)

L= m p1x1 p2x2 = 0 (1.10c)

The system (1.10a)-(1.10c) has three equations in the three unknowns x1, x2 and . If

there is decreasing marginal utility, then condition (b) of the Uniqueness Theorem ofsection 5 is satisfied (the objective function is strictly quasi-concave) and theproblem has a unique solution (we shall look at this in more detail in Unit 2). Thissolution takes the form:

x1*= x1

*(p1, p2,m) (1.11a)

x2*= x2

*(p1, p2,m) (1.11b)

*= * (p1, p2,m) (1.11c)

that is, the endogenous variables are expressed as a function of the exogenousvariables.

After finding a solution, the question we have to ask ourselves is: what wouldhappen if the exogenous variables change? For instance, suppose that income m were

to increase: how is this going to affect the optimal choice of x1 and x2? This question

is answered by the comparative statics, or sensitivity analysis. We try to measure theeffects of changes in the exogenous variables (prices and income) on the endogenousvariables (the quantities consumed) by computing the following (partial) derivatives:

x1

p1,

x1

p2,

x1

m(1.12a)

x2

p1,

x2

p2,

x2

m(1.12b)

How do we obtain the above derivatives? There are two possible ways to solve thisproblem. The first one is simply to obtain explicit solutions for the endogenousvariables, equations (1.11a) – (1.11c), and then to compute the partial derivatives.There is a second method, however, which does not rely on the explicit solutions(1.11a) – (1.11c). In order to implement this method for comparative statics, wetotally differentiate the first-order conditions (1.10a) – (1.10c) to have:

2u

x12

dx1 +2u

x1 x2dx2 p1d = dp1 (1.13a)

2u

x1 x2dx1 +

2u

x22

dx2 p2d = dp2 (1.13b)

p1dx1 p2dx2 = x1dp1 + x2dp2 dm (1.13c)



The system (1.13a)-(1.13b) above can be solved by substitution to obtain the desiredderivatives. For instance, if we are interested in how the quantities consumed x1 and

x2 change as the price p1 changes, we need to compute

x1

p1 and

x2

p1.

When we look at changes in one exogenous variable, or parameter (in the exampleabove, the price p1), we must leave the other exogenous variables, p2 and m,

unchanged. In other words, when carrying out comparative statics exercises weconsider changes of one parameter only at a time. In order to find these effects, weset dp2 = dm = 0 in equations (1.13a)-(1.13c) and solve the system:

2u

x12

dx1 +2u

x1 x2dx2 p1d = dp1 (1.14a)

2u

x1 x2dx1 +

2u

x22

dx2 p2d = 0 (1.14b)

p1dx1 p2dx2 = x1dp1 (1.14c)

The resulting system (1.14a)-(1.14c) contains the three unknowns dx1 , dx2 and d ,

and the exogenous variable dp1 . It can be solved to give the derivatives in which we

are interested, that is, dx1 / dp1 and dx2 / dp1 .

There is also an alternative method to sensitivity analysis for assessing the effects ofchanges in the exogenous parameters on the economic agent. This method relies onthe envelope theorem, which says that the total derivative of the objective functionwith respect to the parameter is equal to the partial derivative of the Lagrangean,evaluated at the optimal choice.

To understand what this means, note that the Lagrangean function evaluated at theoptimum is:

L* = u(x1*, x2

*) + * (m p1x1* p2x2

*)

= u[x1*( p1, p2 ,m), x2

*( p1, p2 ,m)]+ *[m p1x1

*( p1, p2 ,m)

p2x2*( p1, p2 ,m)]

= u[x1*(p1, p2,m), x2

*(p1, p2,m)]+ *g(x1*, x2

*, p1, p2,m) (1.15)

using (1.11a) – (1.11b), and where we have written the constraint as:

g(x1*, x2

*, p1, p2,m) = m p1x1*(p1, p2,m) p2x2

*(p1, p2,m) (1.16)

The envelope theorem thus says that:

du *

dp1=

u *

p1+ *

g

p1(1.17)

where the partial derivatives are computed at the optimum.

Turn now to Gravelle and Rees, Appendices I, from page 696 to 700, and J,

pages 708-09. Note that these authors make use of matrix algebra in order to derivesome of their results. If you are familiar with matrix algebra, you will see that it is

UNIT ONE 17


convenient for solving large systems of equations. However, it is not necessary tounderstand algebra in order to grasp the underlying economics of the optimisationproblem and the structure of the solution. Note also what Gravelle and Rees have tosay about the second-order condition for unconstrained optimisation, on pages 698-99. Unfortunately, the mathematical second-order conditions for problems ofunconstrained optimisation with many variables or for constrained optimisation arequite complicated without the use of matrix algebra, and we shall not look into them.These are discussed by Gravelle and Rees on pages 700-01, but these are notcompulsory reading.

8 ConclusionsThis unit introduces you to the main methods of microeconomics and managerialeconomics. You should now see how to formulate problems of optimisation underconstraints, which are central to this course. After explaining the language and themain concepts of microeconomics, we looked at the role of optimisation in economicanalysis. We derived some general methods for solving optimisation problems andfor analysing the characteristics of the solutions. The Lagrange method for constraintoptimisation was explained, and we demonstrated the use of sensitivity analysis toexplore how the solution varies with the parameters of the problem. We alsodiscussed the envelope theorem, which shows how the parameters of the problemcan affect the value of the objective function.

The mathematics of this unit could appear to be rather daunting, if you are unfamiliarwith constrained optimisation. However, you should not feel discouraged if you findsome of the concepts or methods obscure. The best way to understand difficultconcepts is to be patient, go over the theory several times, and especially see how thetheory works in practice. We shall certainly see quite a few applications of the theoryin the next units, and I am sure you will feel confident by the end of the course!

Revision ExerciseConsider the problem of maximising the following objective function:

max y = u( x1, x2 ) = Ax1 x2 (1.18)

where A > 0, > 0, > 0, subject to the constraint:

p1x1 + p2x2 =m (1.19)

(a) Find the optimal values for x1 and x2.

(b) Find how the optimal values for x1 and x2 vary with p1, p2 and m.

(c) Find how the maximised value of the objective function varies with p1, p2

and m.



Solution to the Revision Exercise

(a) Write the Lagrangean as:

L = Ax1 x2 + (m p1x1 p2x2 ) (1.20)

The first-order conditions are:

( x1): A x1

1x2 = p1 (1.21a)

( x2): A x1 x2

1= p2 (1.21b)

( ): p1x1 + p2x2 = m (1.21c)

Divide (1.21a) by (1.21b) to eliminate :

x2

x1=

p1

p2(1.22)

Equation (1.22) can be rearranged to give:

x2 =p1

p2x1 (1.23)

If we replace (1.23) into the budget constraint (1.21c) we obtain the optimal choicefor x1:

x1*

=+

m

p1= x1

* ( p1, p2,m) (1.24)

By replacing (1.24) back into the budget constraint (1.21c) we obtain the optimalchoice for x2:

x2*

=+

m

p2= x2

* ( p1, p2,m) (1.25)

Finally, by replacing (1.24) and (1.25) into the first-order condition (1.21a) (or into(1.21b)) we have:

* =A

( + ) + 1

m+ 1

p1 p2

= * ( p1, p2,m) (1.26)

(b) From (1.24) and (1.25) we obtain:

dx1*

dp1=

+

m

p12

< 0,

dx1*

dp2= 0 ,

dx1*

dm=

+

1

p1> 0 (1.27)

dx2*

dp1= 0 ,

dx2*

dp2=

+

m

p22

< 0,

dx2*

dm=

+

1

p2> 0 (1.28)

(c) We can use the envelope theorem. The maximised value of the Lagrangean is:

UNIT ONE 19


L = u( x1* , x2

* ) + (m p1x1*

p2x1* )

= A+

m

p1

+

m

p2

+

A( + ) + 1

m+ 1

p1 p2

m+

m+

m

= A( + ) +

m+

p1 p2

(1.29)

By taking partial derivatives of (1.29) we finally obtain:

u *

p1= A

+1

( + ) +

m+

p1+1

p2

(1.30a)

u *

p2= A

+1

( + ) +

m+

p1 p2+1

(1.30b)

u *

m= A

( + ) + 1

m+ 1

p1 p2

(1.30c)



ReferencesGravelle, Hugh and Ray Rees (2004) Microeconomics, 3rd edition, Prentice-Hall,Harlow.

Milgrom, Paul and John Roberts (1992) Economics, Organization and Management,Prentice-Hall, Inc: Englewood Cliffs (New Jersey).

managerial economics - iicseonline.org · the approach to managerial economics which we study in...

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