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MATH60013/96011 – Mathematics of Business and Economics Imperial College London 1 MATH96011 – Mathematics of Business and Economics Dr Ioanna Papatsouma Spring 2022 Introduction This course aims to provide a broad mathematical introduction to economics and its application in a business setting. In this introduction, I provide some initial definitions relevant to the content of the course, in addition to detailing the course structure and specifying some course objectives. What is an economy An economy is an ecosystem, in which governments, markets, firms and individual consumers all interact with the aim of enabling the provision of goods and services in return for payment. Broadly speaking, there are four groups of agents that enable an economy to function: – individuals and households, who act as consumers in obtaining goods and services from producers, and as suppliers in providing labour to producers; – firms, who provide goods and services to consumers, and who also employ individuals from the first group; firms also act as consumers for other firms; – governments and regulatory bodies, who provide oversight, regulation and intervention in order that their economies function smoothly and in service of particular goals; – and trade partners external to the economy, who also interact with these agents to influence production and consumption within the economy. The above definition of an economy indicates that economic analysis can be carried out at a number of different scales. Households and firms come together to trade goods and services within (conceptual) markets; the analysis of such interactions and the behaviour of these markets is the principal focus of microeconomic theory. In contrast, the actions of governments that influence the operation of markets, as well as the interaction of markets with external trading partners, falls under the heading of macroeconomic analysis. This course will focus on the interaction of economic agents within a market economy, i.e. one where production and trade are private enterprises. As such, we consider an ecosystem whose constituent parts are each controlled by their own decision-making processes; the study of economics is therefore important to understand how these processes affect one another and how each agent should operate in order to satisfy their particular goals whilst MATH60013/96011 – Mathematics of Business and Economics Imperial College London 2 acknowledging the behaviour of other agents and changes to the economic environment in general. Course Structure The structure of this course reflects this division of analyses. In Parts 1 and 2, we focus on microeconomics, before taking a macroeconomic viewpoint in Part 3. – In Part 1, we will analyse the aims and objectives of both firms and consumers, and we will use mathematical arguments to show how these objectives lead to the observed behaviour in a competitive market environment. – In Part 2, we will consider how the behaviour of firms and consumers is affected by the properties of the market in which they operate, and how their behaviour is affected by changes to the market environment. – In Part 3, we analyse the macroeconomic environment; in particular, we explore aggregated concepts of supply and demand, we look at the circular flow of income and discuss the Gross Domestic Product (GDP). Syllabus: Theory of the firm Profit maximisation for a competitive firm Cost minimisation. Geometry of costs Profit maximisation for a non-competitive firm Theory of the consumer Consumer preferences and utility maximisation The Slutsky equation Levels of competition in a market Consumers’ and Producers’ surplus Deadweight loss Macroeconomic theory Circular flow of income Cross Domestic Product Social welfare and allocation of income Mathematical Methods: (Constraint) Optimisation. Quasi-concavity. Preferences relations and orders. Course Objectives This course provides an introduction to the fundamental aspects of both microeconomics and macroeconomics, using a mostly rigorous mathematical approach to both the exposition and demonstration of these subjects. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 3 In a business context, this course will provide you with the tools required to analyse the goals of a firm and the decisions that a firm may make in the context of their particular market. At the end of the course, you should understand the effects that these decisions have on the firm itself, on the various connected individuals and firms, and on the economy as a whole. Solving problems in this course will require both an economic understanding of the concepts as well as a sound mathematical derivation. That means that you should be prepared to come up with mathematical proofs as well as to explain notions in form of (very) short essays. Additional Course Information Lectures: Pre-recorded lectures Office Hour: Doodle poll Problem Classes: Bi-weekly; see weekly plan The problem sheets will be available on Blackboard and the solutions to the problem sheets will be uploaded after the problem classes. If anybody wants to have some feedback on their un-assessed problem sheets, you can give me your solutions and I will have a look at it. Course Rep: You should agree on a course rep in the first week. Lecture notes: The lecture notes are available on Blackboard. They have gaps and we will fill these gaps during the lectures. Textbooks: All the material used in this module can be found in various textbooks. Gillespie, A. (2013) Business Economics (2nd Edition). Oxford University Press. Varian, H. R. (1992) Microeconomic Analysis (3rd Edition). W. W. Norton & Co. Varian, H. R. (2014) Intermediate Microeconomics (9th Edition). W. W. Norton & Co. These books can be found (some electronically) in the college library. However, the course will be self-sufficient. Assessment: 1 piece of assessed coursework, worth 10% (Probably from 28/02 till 14/03) 1 two-hour final exam, worth 90% MATH60013/96011 – Mathematics of Business and Economics Imperial College London 4 Part 1 – Microeconomics Supply and Demand – an introduction Principal questions to be addressed by economics – what price should we be paying for goods, what price should a vendor be selling their goods for Do the different motivations of the different parties give different answers to this question Supply refers to the quantity of a product that a vendor (or vendors) is willing and able to sell, at a given price in a given period of time. Correspondingly, demand refers to the quantity of the product that the buyer (or society at large) is willing and able to purchase at a given price in a given period of time – Willingness and ability both important, e.g. pint of beer, flat in South Kensington – ‘…in a given period of time’ also important The good’s price is not the only determinant – Demand can also depend on… number and price of substitute goods e.g. tea/coffee, bus/tube number and price of complementary goods e.g. cars-gas, printers-ink cartridges level and distribution of income, e.g. flat in South Kensington consumer’s expectation with regards to the future consumer’s tastes and habits, e.g. chocolates, tobacco – Alternative determinants for supply: Changes to the overall cost of production Change in prices of inputs to production Change in technological capabilities Organisational change – Both supply and demand may also change over time Demand: e.g. tapes-> CDs -> Minidiscs -> MP3s Supply: e.g. cheaper production due to tech advances Law of Demand: Ceteris paribus (everything else being equal), an increase in price will usually lead to a dropp in demand. – This is often linked to either the income effect or the substitution effect: A rise in price results in a decrease in the consumer’s purchasing power: their income no longer covers the same quantity of the good in question. (income effect) A rise in a good’s price may result in consumers substituting it for a similar, less expensive good. (substitution effect) MATH60013/96011 – Mathematics of Business and Economics Imperial College London 5 Law of Supply: Ceteris paribus, an increase in price will lead to an increase in supply. – This is because higher price (per unit) will incentivise greater production. It may be, e.g., that a manufacturer produces more than one product, and in order to optimise their profit, they have to split resources according to the revenue earned by each product. If this revenue split changes, so must the resource split, and therefore the amounts being produced. In a market economy, the price of a good is determined according to its supply and demand, through the price mechanism: – If supply exceeds demand, price drops as the producers compete to sell the good. This acts to encourage demand via the Law of Demand. (addressing the good’s surplus) – If demand exceeds supply, the price increases as consumers compete with each other to obtain the good. This acts to incentivise production, as per the Law of Supply. (addressing the shortage of the good) This interaction of the price mechanism with the Laws of Supply and Demand means that changes in both supply and demand will both cause and be caused by changes in the price of the good. This relationship determines the equilibrium price of the product; where supply and demand are equal. Supply and demand curves In mathematical terms, the demand and supply can be considered as functions and mapping a price ∈ [0,∞) to some level of demand () or supply (). It depends on the good of interest if these functions are integer-valued (discrete) or real-valued (continuous). If they can be inverted, their inverses are referred to as inverse demand and inverse supply. As such, they map from or to [0,∞). Note that we ignore the fact that prices are also reported in discrete units and treat the price variable as a continuous quantity. It is often convenient and revealing to analyse the supply and demand function graphically. For historic reasons (due to the economist Alfred Marshall) we use the convention that prices are depicted on the vertical axis and quantities on the horizontal axis. For trained mathematicians, this praxis is rather counter-intuitive. However, since the convention to do so is pervasive in the economic literature, we shall stick to it in this course. The graphs of the supply and demand functions are referred to as supply and demand curves. If one is to analyse stylised facts rather than precise quantitative results, one commonly uses linear functions for supply and demand for the sake of simplicity. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 6 Examples of supply and demand curves: MATH60013/96011 – Mathematics of Business and Economics Imperial College London 7 A good with more close subs will have higher |!| The higher the proportion of income spent on a good, the greater the elasticity Analysing the supply and demand curves: It is of interest to economists to characterise the sensitivity of a product’s supply or demand to shifts in its price; the measure of such sensitivity is known as the price elasticity of supply/demand. Example: a) Change in demand for a pint of bear if price changes from ￡3 to ￡4. b) Change in demand for a car if price changes from ￡30000 to ￡30001. In general, the elasticity of a quantity refers to the relative magnitude of its reaction to a change in any variables on which it depends. Consider the demand for a product, which depends on its price: – if the demand for a product is fairly resilient and robust to price changes, then it is inelastic: demand changes relatively little for a given change in price – Conversely, if the demand is particularly sensitive to changes in the price, then it is said to be elastic. More rigorously, we denote and define the price elasticity of demand to be ! = !() = % (or proportional)change in quantity demanded% (or proportional)change in price = () /()1/ = () () = ln(exp(ln())) ln Note that generally ! < 0, as demand is decreasing in . - For a good with elastic demand at price , if |!()| > 1 – For a good with inelastic demand at price , |!()| < 1 - If |!()| = 1, then the good is said to have unit elastic demand at price . Determinants of ε!: - Number and closeness of substitute goods - Proportion of income spent on the good - Time period We define the price elasticity of supply " similarly; this measures the sensitivity of a good’s supply function to changes in the good’s price. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 8 In addition, we can consider: - the income elasticity of demand (supply), which measures the sensitivity of a good’s demand (supply) function to changes in the consumer’s income; and - the cross-price elasticity of demand (supply), which measures the sensitivity of one good’s demand (supply) function to changes in the price of another good. Demand, Price and Revenue The revenue generated by a particular good is simply defined as the product of its price and quantity demanded () = () . Since the demanded quantity of a good is inversely related to the good’s price, an increase in price will not necessarily increase the revenue generated. When will an increase in price result in an increase in revenue How is this linked to the price elasticity Consider the derivative of with respect to : () = () + () > 0 () () < 1 !() < 1 So an increase in price will lead to an increase in revenue when demand is inelastic. When demand for a good is elastic, revenue is increased by decreasing the price. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 9 Theory of the Firm Production Functions, Cost, Revenue and Profit The principle aim of a firm is to turn various inputs, such as raw materials and labour, into output that can then be sold, ideally for profit. Inputs to the production process are referred to as factors of production, and are broadly split into four categories: Labour Land Raw Materials Capital The first three are self-explanatory; capital requires some explanation: - Broadly speaking, capital refers to those inputs to production that may be consumed now, but that will deliver greater overall value to the firm if consumption is deferred. - Capital goods are those inputs to production that are themselves produced goods, and which are durable, such as machinery. - Capital finance refers to the financial assets of a firm that are themselves used to generate wealth; it differs from ‘money’ in general in that it is not used to purchase consumable goods and services. We start by considering constraints that might be placed upon a firm’s production capabilities; only certain combinations of input and output quantities may be technologically feasible, and so these are referred to as technological restraints, and the set of all inputs and outputs that satisfy such restraints are referred to as the production set. We will denote the vector of input factor quantities as ∈ #$% and the vector of output quantities as ∈ #$& ; thus the production set is the collection of vectors U, V ∈ #$% × #$& such that ≤ (), for some production function . The production function, also known as the technology of the firm, prescribes the maximum level of output for a given level of input . For given ∈ #$& , the set of all points ∈ #$% , such that (', (, … , %) = is known as an isoquant; this is the combination of inputs that can produce the given . Mathematically speaking, this is the pre-image )'({}). We will mostly consider single-output cases, = 1, but the methodology extends to ≥2. Similarly, we will usually consider ∈ (, but mostly for illustrative convenience. When ∈

#$, the input requirement set is the set of all vectors that produce at least , that is )'([,∞)). MATH60013/96011 – Mathematics of Business and Economics Imperial College London 10 We now consider three examples of production functions that are often used in microeconomic analysis. Leontief Technology (aka fixed proportions/perfect complements) Suppose we have two inputs ‘ and (: the Leontief production function takes the form (‘, () = min(‘, () , , ≥ 0 Example – Output: car, inputs: engine, wheels Perfect Substitutes: (perfect complements) Suppose, in contrast, we have inputs to production that can be easily substituted for one another without affecting the level of output. (‘, () = ‘ + (, , ≥ 0 Example – Output: coffee, input: milk or soya milk Cobb-Douglas Technology: Suppose we have two inputs, ‘ and (: the Cobb-Douglas production function takes the form (‘, () = ‘*(+ , , , ≥ 0 MATH60013/96011 – Mathematics of Business and Economics Imperial College London 11 Properties of production functions / the input requirement sets: Monotonicity: if some input is increased, then the maximum output will be at least as great as it was before ≥ (‘ , … , % ) ≥ (‘, … , %) This reflects the property of free disposal: if a company can dispose of an input without it costing anything, then increasing the inputs available can only be a benefit. Convexity of the input requirement set: If for some , (‘, () and (′(, ′() are both in the input requirement set, so will be their weighted average. This allows a producer to mix production techniques. The convexity of the input requirement set is equivalent to the quasi-concavity of the production function . Long-run and Short-run Broadly speaking, in the economic literature, analysis is split into two scenarios, considering the behaviour of the firm or individual in either the short-run or the long-run. These are inexact periods of time, defined implicitly by the number of production inputs ‘, … , % that may vary within such a timeframe: in the long-run, all inputs may vary, whereas in the short run, at least one input will be held constant. The Marginal Product Question: how much can we increase output by varying the input factors Suppose we are operating with an element (‘, (, ) in the production set of and we wish to obtain a level of output – > by increasing ‘… – The marginal product of factor is defined as .(‘, () = (‘, (). , = 1,2 There is a slight issue with the marginal product – it is dependent on the units used to measure the input quantities and also on the unit of the output quantities. In order to measure the marginal effect of increasing each input independently of its units, we turn to the output elasticity with respect to each input. For a production function : #$% → #$, the output elasticity with respect to input . is defined
as the ratio of the relative change in output to the relative change in
input: .(‘, () = (‘, (). .(‘, () (= ln (‘, () ln .
= 1,2) MATH60013/96011 – Mathematics of Business and Economics Imperial College London 12 Substitution How about if we wish to keep the same level of output, but use less of a particular input factor Graphically, this is simply the act of moving along an isoquant. We are interested in the rate of change of input ( with respect to ‘, in order to keep the output constant. This is the Marginal Rate of Technical Substitution (MRTS), or the technical rate of substitution. Mathematically
speaking, for a fixed ∈ , we are interested in the derivative of a
function /, where / is implicitly defined as /(‘) = ( (‘, () = The implicit function theorem asserts that such a / exists at least locally. Moreover, if is a ‘-function, then also / is a ‘-function. Then we obtain U’, /(‘)V = Differentiation with respect to ‘ yields ‘ U’, /(‘)V + ((‘, /(‘)) /′(‘) = 0. This implies that, assuming ( U’, /(‘)V ≠ 0, /- (‘) = ‘ U’, /(‘)V( U’, /(‘)V. We define the MRTS of a production function to be (‘, () = ‘(‘, ()((‘, (). Reasoning: Start with the pair (‘, () and determine (‘, () = . Then proceed with the rationale from above, choosing / . MATH60013/96011 – Mathematics of Business and Economics Imperial College London 13 Consider the behaviour of the marginal product as a function of ‘, … , %: since the production function is nondecreasing, we have that the marginal product is nonnegative, and it is common to assume that it is nonincreasing. In other words, increasing an input ‘ from 100 to 101 is likely to result in a smaller increase in production than if we were increasing ‘ from 1 to 2. This is known as the law of diminishing marginal productivity, and holds ceteris paribus. This is also referred to as the law of diminishing returns, not to be confused with decreasing returns to scale (later). Similarly, it is common to assume that a firm’s production has diminishing marginal rate of technical substitution: if we consider substituting factor ( for factor ‘ (i.e. decreasing ‘ and increasing ( such that the output is fixed), the larger the value for ‘ (before substitution), the smaller the absolute value of the MRTS: in other words, the absolute value of the slope of the isoquant must decrease as ‘ increases. Example: (‘, () = √'(. Then we get (‘, () = 0!0″ Returns to Scale: We have considered the effects on the production function of increasing individual factors whilst keeping others fixed, and we have considered the effect of substituting one factor for another whilst keeping the output level fixed. We now consider the effect on the production function of scaling all variable input factors by the same constant, keeping the ratio of inputs fixed. That means for some production function : #$% → #$ and ∈ #$% you should consider the behaviour of the function #$ → rs. – The most common scenario is that of constant returns to scale, and this is where an increase in all inputs results in a proportional increase in the output: rs = rs > 0 ∈ #$% This is considered to be the most common scenario as, in most scenarios, the production process can simply be replicated: if we double all input factors (land, capital, labour and raw materials) then the production process can simply be duplicated. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 14 – In some scenarios, increasing returns to scale may be observed; this is where rs > rs > 1 ∈ #$% – Decreasing returns to scale refers to the case where rs < rs > 1 ∈ #$% Decreasing returns to scale can only happen when one or more inputs cannot be varied. In reality, all inputs are variable in the long run – so decreasing returns to scale is a short-run phenomenon Example – returns to scale of the Cobb-Douglas production function. In order to fully characterise the scalability of a firm’s production process, we wish to find a quantitative measure of the returns to scale; we turn again to the use of an elasticity measure. () = () We are interested in the reaction of to percentage changes in when we start with = 1. If is differentiable, we can define its elasticity of scale at as rs = () ()y12′ = rs ()y12′ = rs y12′() = zrs, |rs That means rs is a local measure of the scale behaviour. If is continuously differentiable, one can show that (i) rs ≥ 1 for all ∈ #$% if and only if rs ≥ rs ≥ 1 ∈ #$% . (ii) rs ≤ 1 for all ∈ #$% if and only if rs ≤ rs ≥ 1 ∈ #$% . (iii) rs = 1 for all ∈ #$% if and only if rs = rs > 0 ∈ #$% . The idea of the proof is to consider for ∈ #$% the function 0: (0,∞) → , 0() = rsrs. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 15 Additional potential properties of the production function – Homogeneity: : #$% → is positively homogeneous of degree ∈ if rs = 3rs > 0 ∈ #$% . This has obvious links to the returns to scale: e.g. displays increasing returns to scale if and only if is homogeneous of degree > 1. – Homotheticity: : #$% → is homothetic if there is a positively homogeneous function : #$% → and a strictly increasing function : → such that rs = U rsV ∈ #$% . Homogeneous and homothetic production functions are useful modelling scenarios as they prescribe isoquants that vary simply for differing levels of output. A particularly useful property of such technologies is that, for each, the corresponding MRTS is independent of the scale of production. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 16 Profit and profit maximisation So far, we have only considered production functions and the influences of different factors to the production. However, what is the ultimate reason and motivation for a firm to produce products at all Most microeconomic analyses assume that the firm is profit driven – we will too! The profit of a firm is simply its revenue minus its costs, where all costs of the firm are taken into account. It is often easy to overlook costs (e.g. labour costs for a self-employed person). In application, the decision of how to maximise profit comes down to deciding how much output to produce and at what price, or how much input to buy and at what price. We therefore construct the profit function in terms of these variables: (, , ) = rs4 4 (, , ) = rs 4 (single-output case) We will assume the conditions of a purely competitive market, i.e. where all firms are assumed to be price-takers (their actions have a negligible effect on the prices). In such a scenario, one need only determine the quantities of inputs/outputs in order to maximise profits. Treating this as an unconstrained optimisation problem then, the first-order conditions for to be maximised can be found straightforwardly: = 1,… , ∶ rs. = 0 5 5rs.&52′ = . This first-order condition is often referred to as the fundamental condition of profit maximization: – Profit is maximised if marginal revenue is equal to marginal cost – Also stated as: ‘the value of the marginal product wrt a factor is equal to its price’. For fixed U, V, this condition is easily solved to provide a necessary condition for the profit-maximising input vector , which can then be used to establish the profit- maximising output. So
for the single-input case, the second-order condition for profit
maximisation is that –( ) ≤ 0 (necessary) or –( ) < 0 (sufficient). For ∈ #$% , > 1, the corresponding condition is that the Hessian of rs4 is negative semidefinite (necessary) or negative definite (sufficient) at . Quite often, this (local) concavity of the production function is part of our assumptions. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 17 There are some caveats with this canonical procedure: 1. The production function might not be differentiable, e.g., in a Leontief technology. We need to be careful about that. 2. Some of the input variables might be discrete and not continuous. So that means we need a discrete minimisation here. 3. We might have boundary solutions (usually at 0 meaning that it is optimal not to use a certain factor in the production at all). Example: () = and > . 4. A best strategy might not exist (e.g. because the profit function is not bounded). Example: () = and < . 5. The optimal strategy might exist, but is not unique. Example: () = and = . For a specific production function : #$% → #$& we introduce the map ∶ #$& × #$% → 2 #$% yielding the / an optimal specification of input quantities, given a price ∈ #$& and prices for input factors ∈ #$% . That is U, V = argmax0∈ #$% (, , ) = argmax0∈ #$% rs4 4. In the light of the caveats mentioned above, our standard assumption is that the optimal input specification exists and is unique. Under this assumption, the values of are singletons and we can identify the singletons with their unique element. Consequently, we can consider as a function – the factor demand function – being a map ∶ #$& × #$% → #$% . Recall that we are in the setting of a competitive firm such that the firms are price takers and r, s can be considered as exogenous variables. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 18 Under this assumption we can also define the output supply as the function ∶ #$& × #$% → #$& U, V U, V = U, V Moreover, we can define the profit function ∶ #$& × #$% → U, V U, V = U U, V , , V = max0∈ #$% (, , ) = max0∈ #$% rs4 4 Properties: 1. The factor demand function is positively homogeneous of degree 0. U, V = U, V > 0 U, V ∈ #$& × #$% 2. The profit function is positively homogeneous of degree 1. 3. The profit function is non-decreasing in and non-increasing in . 4. The profit function is convex. 5. Under some regularity assumptions, the profit function is continuous. Proof: 5. Follows with the Berge Maximum Theorem. The other assertions are exercises. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 19 Hotelling’s Lemma (special case of the Envelope Theorem) The output supply and factor demand functions can be obtained directly from the maximised profit function through partial differentiation with respect to the price vector. Recall U, V = U, V 4 U, V4
U, V5 = 5 U, V + 3&32′ .3 U, V . U, V5%.2′ . . U,
V5%.2′ = 5 U, V + %.2′ 3.3 U, V &32′
. $ . U, V5 = 5 U, V U, V5 = 3&32′ .3 U, V . U, V5%.2′ 5 U, V 3&32′ 3 U, V5 r s. . 5%.2′ 5 . . 5%.2′ = 5 U, V + %.2′ 3.3 U, V &32′ . $ . U, V5 = 5 U, V One convenient corollary of Hotelling’s Lemma is that each row of the maximised profit function Hessian (wrt ) is equivalent to the gradient of the firm’s supply function for the corresponding output. This makes it easier to derive some properties of profit-maximizing firms, such as… The LeChatelier principle: The long-run supply response to a change in price is at least as large as the short-run supply response. Essentially, it states that . U, V. ≥ 0. MATH60013/96011 – Mathematics of Business and Economics Imperial College London 20 Weak Axiom of Profit Maximization (WAPM): So far, we have started with a given production function and we have derived some results about the factor demand function or the output supply function. Now, we turn the perspective and assume that we can observe a firm’s ‘behaviour’. The Weak Axiom of Profit Maximization (WAPM) is a necessary condition for the rational, i.e. profit maximizing behaviour of that company. In a first step, we can use the WAPM to check if a company is profit maximizing by checking if the observed dataset satisfies the WAPM. In a second step, we can even consider some attempts of statistical inference for the production function using a dataset that satisfies WAPM. The details for the second step will be the content of some assignment. Suppose we observe the net output vectors 1 = U1 , 1V ∈ #$& × #$% and their corresponding price vectors 1 = U1 , 1V ∈ #$& × #$% for some firm at discrete time points = 1,… , . Assuming that the firm is acting to maximise profits, we can deduce that 1r1s4 ≥ 1r9s4 , = 1,… , . Where we index via superscripts as subscripts are reserved for indexing vectors This is the Weak Axiom of Profit Maximisation. 1r1 9s4 ≥ 0 9r1 9s4 ≥ 0 Writing WAPM with indic