Habs Boys: Mathematical and Economic Review -Issue 2
Mathematical
This magazine bears witness to an exiting year of growth for the Haberdashers’ Mathematical Economics Society. A large number of students have enthusiastically grappled with the challenge of applying mathematical ideas to economic theory and the progress they have achieved has been reflected in the quality of the Extended Research Projects they have submitted, the speed and accuracy of their solutions to problems set over the holidays, and above all in the individual reading they have carried out in an array of areas of economics where mathematical reasoning can yield particular dividends. This magazine (so ably edited by Leo Mouskis, Anvita Maloo, Feeza Prajapati, Krish Gupta, and Mark Solomon) assembles some of this work. It begins with an article on the use of envelope theory in economics generously written for the society by its Honorary President, Professor Nicholson of Amherst College. Anvita Maloo introduces another powerful technique in the form of the use of Lagrangian multipliers in optimisation problems. There are articles on the Russian contribution to economics, including an introduction to Linear Programming by Arina Noshchenko. Utility and Production functions are admirably explored by Amogh Agal and Jinying Ouyang (the joint Chairs of the Society), while Alex Browne introduces us to the technicalities of financial economics and Gautam Arun does the same for auction theory. The society’s president Janvi Pankhania helps us appreciate the ubiquity of monopsony in the modern economy through the commercial activities of Spotify, and Kiara Welge uses theory to explain the diffusion of new technologies and products through the economy. As will be apparent, a rich array of themes are covered and the Magazine reflects, too, the genuinely joint efforts of girl and boy school students, testifying to the benefits of collaborative work between the schools in the Sixth Form in general and Economics in particular. I am grateful for all who have helped to make this magazine possible and Chloe Bedford and Jaspreet Rajput for doing so an excellent job in printing this fine edition.
Preface -Ian St. John
The Magic of the Envelope Theorem: Basics
Introduction
This is the first of four short notes about a simple mathematical theorem that has astounding consequences for the study of microeconomics. I first encountered the so-called “Envelope Theorem” about thirty years ago after completing six editions of my advanced micro text. Although the theorem had been around for a much longer time, apparently, I was slow to understand its importance. Once I got the general idea, it caused me to rewrite many sections of my textbook. The result was a much more elegant approach to many topics in mathematical economics. I hope to convey in these notes why I found the theorem so exciting.
By Professor Walter Nicholson Honorary President of the Haberdashers’ Mathematical Economics Society
not too far-fetched to suggest that virtually any economic relationship that results from an optimization problem can be derived by differentiating the appropriate value function. The key problem, then, is how to decide what these derivatives represent. The ET provides the route to doing this. The theorem says that the derivatives of a value function can be interpreted by returning to the original optimization problem and partially differentiating it with respect to the exogenous variable of interest while holding all of the endogenous variables constant at their optimal values. Of course, this is quite a bit of verbiage to swallow. We could provide a mathematical proof (and you should eventually look one up). But, for the moment a very simple example should suffice.
The Structure of Economic Models The Supply Curve for a Profitmaximizing Firm
To understand the Envelope Theorem (hereafter ET), let’s start by reviewing the structure of any economic model. In such a model, the crucial distinction is between exogenous variables whose values are set outside the model and endogenous variables whose values are determined by the model. Usually, the model is built round some sort of optimization principle. In consumer theory, for example, the exogenous variables are the prices of goods and the consumer’s income. The endogenous variables are the quantities of each good chosen to maximize utility. In the theory of the firm, the exogenous variables are prices of both output and inputs and the endogenous variables are the quantity of output produced, and the quantities of inputs hired to maximize profits.
Value Functions
The final value of what is being optimized in a problem will ultimately depend only on the values of the exogenous variables – the optimization process will “solve out” the values of the endogenous variables. This functional relation is called a “value function”. For example, in consumer theory, the utility attainable by a utility-maximizing individual will ultimately depend only on the prices he or she faces and the income they have. This value function is called an “indirect utility function”. In the theory of the firm, the value function yields the firm’s profits as a function of the prices the firm faces. Naturally enough, this value function is called a “profit function”.
The Envelope Theorem
Once one has a value function in hand, it is natural to think about differentiating it with respect to its (exogenous) variables. This is where the magic of the ET comes in. It turns out that these derivatives almost always have interesting economic interpretations. Indeed, it is
Suppose a firm can sell its output (q) at an exogenously determined price (p) and that its costs are given by: C(q) = 0.1q2
The firm’s profits are therefore given by:
π(q) = pq - C(q) or q = 5p
The first order condition for a maximum requires:
and this is the supply function for this firm.
Now let’s compute the value function for this firm. Substituting the (optimized) supply function into the expression for the firm’s profits yields:
π(p) = p(5p) - 0.1(5p)2 = 2.5p2
This is the value (profit) function for this problem. It shows how the firm’s profits depend only on the price of its output. The process of maximizing profits has been subsumed into the creation of this function.
There seems to be only one useful derivative one could take in this situation: dπ(p)/dp = 5p. Surprisingly this function is identical to the firm’s supply function. We can show that this interpretation is correct by partially differentiating the original profit function, treating all of the endogenous variables as being held constant at their optimal values: ∂π(p)/∂p = q*, that is, this derivative yields the optimal supply function – just what differentiation of the profit function showed explicitly.
Of course, this example is very much overkill for the problem at hand. It is much easier to get the firm’s supply function by using the price equals marginal cost result from introductory microeconomics. But in subsequent notes we will show that most of the crucial results from both demand and supply theory can be derived in the same way. Stay tuned.
Let Linear Programming, Make the Decision
Have you ever wondered how to make the most out of the limited resources you have, whether it is time, money or ingredients in your kitchen? Imagine you run a bakery, and you have got just enough flour and sugar to bake either cakes or cookies, but not enough to make as many as you want of both. Cakes bring in more profit per unit, but they also need more ingredients. Cookies are quicker and use fewer resources but earn you lesst. Of course, you aim to have as much profit as you can, but how do you decide what to bake to earn the most? That’s where linear programming comes in handy!
In 1938 Leonid Kantorovich, a USSR mathematician, who is a pioneer founder of Linear Programming, introduced a mathematical method of using linear equations to find the most optimal way of allocating scarce resources to maximise the output. His invention started from a “Plywood Trust” problem. In the period of rapid industrialisation in USSR, an organisation asked L.V. Kantorovich to find the way to allocate the work of cutting wood to several kinds of machines with different productivity to obtain plywood, a vital material, which was even used in aircraft construction. Therefore, the engineers asked the mathematician to help calculate the optimal distribution of different types of wood among the machines to maximise the productivity i.e. output per unit of time. “I discovered that a whole range of problems of the most diverse character relating to the scientific organisation of production (questions of the optimum distribution of the work of machines and mechanisms, the minimisation of waste, the best utilisation of raw materials and local materials, fuel, transportation, etc.) lead to the formulation of a single group of mathematical problems (extremal problems)”,he said. He published his idea in his book in 1942 and was awarded the Nobel Prize in Economics.
Let’s look closer into the Linear Programming approach:
Imagine we have a function, that we want to maximise. But it is a complex function with many constraints. In this case, taking the derivative is not enough, so, for this problem, approaches like the simplex method are used to test the vertices of the feasible region and find the optimal value. Until then, no one had managed to produce such a large volume of products.
By Arina Noshchenko
Let's have a look at a simple real-life problem that could be solved using Linear Programming:
Imagine you have 60 cups of flour and 40 cups of sugar available. To make one cake, you need 5 cups of flour and 3 cups of sugar. To make one pack of cookies, you need 2 cups of flour and 3 cups of sugar. Each cake brings a profit of £50, and each pack of cookies brings a profit of £10. Let x be the number of cakes you produce and y- the number of cookie packs.
We can now form two inequalities based on the resource constraints f(f) and f(s) and an objective profit function, where we aim to maximise profit: P = 50x + 10y. We will construct 2 possibilities of profit to see the trend of the function.
To find the most optimal solution, you sketch the inequalities and find the feasible region (a region where the solutions satisfy all the constraints). From this region you determine which point gives the highest profit when plugged into the profit function. It can be found from the feasible region’s vertices that is formed. The most optimal solution in this case will be where f(f) and f(s) meet, because when we observe the trend of the profit function, we can see how it shifts outwards, as profit increases. If we draw a line of the shift and assume this line is the direction of maximising profit, then if we draw this line on the first graph, we will see that the furthest point in the feasible solution is the point where f(f) and f(s) meet. It is important to note, that in this example we assume that the solutions can be any real number. Of course, this is one of the easiest examples where Linear Programming can be used. There can be a lot more constraints, and in this case, you would need to use more advanced methods to find the solution- simplex method.
Bond Pricing and Yields
By Alex Browne
Given the recent volatility with bond pricings and yields in the US and further back during 2022 in the UK, the bond market is becoming ever more of an interesting area of economics as a whole. You may remember Kwasi Kwarteng’s mini-Budget, where a crash in the bond market occurred as bond prices dropped, and bond yields increased, which was overall caused by unfunded tax cuts and a resulting lack of confidence in the UK government. What happened in actuality was more nuanced, but the resultant impact on the UK economy was still devastating, despite the Bank of England stepping in, showing the importance and significance of bond markets in today’s world.
A government bond, or gilt, is essentially an IOU. If a person buys a bond, they are effectively loaning the government money (a principal), and the government promises to pay the principal back at some point in time (the maturity date), plus an additional rate of interest each year (called coupons). In the US, UK, and Japan, these coupons are actually typically paid out twice a year. The ‘bullet’ bond is the most conventional type of bond and is the subject of this essay, where there is a regular fixed interest rate over a fixed period of time. In this essay, bonds are also assumed to be default free, meaning it is assumed that the government will always be able to repay the principal and interest payments.
The principal of the bond multiplied with the coupon rate provides the cash amount of the coupon. For example, a
bond with a principal of £100,000 and a coupon rate of 5% will pay an annual interest of £5,000.
Bond prices are also expressed ‘per 100 nominal’, e.g. if a bond price was quoted as 98.00, a buyer would pay £98 for every £100 of the bond’s face value.
The interest rate is used to figure out current cash flows, otherwise known as the discount rate. For example, if a theoretical bond that matures next year has a face value of £1000, a coupon rate of 6%, and a discount rate of 5%, the expected return after a year would be £1,060. However, after discounting 5% (1060/1.05) , the price of the bond would be £1,009. The bond is said to trade above par (£1,000), because the coupon rate is higher than the discount rate. The nature of the issuer, maturity date, the coupon and currency in which it was issued all influence the bond’s discount rate.
The idea that money has a ‘time value’ is because you can invest it with rates of interest. The future value of an investment with simple interest is shown:
FV = PV (1 + r)n
Where:
FV = Future value
PV = Initial value
r = Interest rate
n = number of periods for which the principal is invested (e.g. number of years)
This assumes that coupons collected are reinvested at an interest rate equal to the first year’s rate.
More frequent compounding results in higher total returns - in general, if compounding takes place m times per year, then at the end of n years, mn interest payments will have been made, and the future value of the principle is:
FV = PV(1 + r/m)mn
A discount factor for any term is given by:
d n = 1/(1+r)n
Where
n = period of discount
Therefore, the present value can be calculated:
PV = FV/(1+r)n = FV * d n
The price of a bond is the sum of all the coupon payments and the principal repayment:
Where
P = Bond price
C = Annual coupon payment
R = Discount rate
N = Number of years to maturity
M = Maturity payment (Bond redemption)
From the bond price formula it can be seen that a bond’s yield and its price are inversely related. This is because a bond’s price is the net present value of its cash flows. If the discount rate or yield required by investors increases, the present values of the cash flows decrease, so the price decreases. This means that if bond prices were decreased, yields are required by investors to be high, meaning investors could believe they are taking greater risk by buying the bond, or are uncertain about the stability of the economy in the future for example.
Source: [Fig.1 + general equations] Google Books. (2025). An Introduction to Bond Markets.
The simplicity of bond pricing comes from the lack of an uncertain dividend, since we know the return will be constant – that is why bonds and related securities are referred to as ‘fixed income’. However, bond prices can and do vary significant amounts and not always with concrete explanations why. Whilst it is the job of economists to try and explain changes in these markets, these explanations can sometimes be merely educated guesses and not actually the truth. Information asymmetry is at its most potent in these settings, as there are so many variables to affect whole markets such as the bond market, which is why it is modelled as a stochastic model – this is where potential outcomes are estimated where randomness or uncertainty is present.
Preference and Utility Function
By Amogh Agal
In economics, we often use the assumption that consumers act to maximise their utility. In other words, consumers maximise their happiness. But what does this mean? We assign actions that consumers take a utility function, and this mathematical representation of what is in essence a very normative topic allows for economists to analyse the choices that we make. The utility function is often described as a tool for “preference relation,” as in essence it is comparing what a consumer prefers between their options.
We call the objects of consumer consumption “consumption bundles”, and these show all the goods and services included within a choice. In order to make our models align with the real world, we often take one of our variables to be defined as everything else. For example, a consumptions bundle of goods x1 and x2 can be shown as (x1, x2), and then subsequently compared with other bundles of goods.
Notation:
means X is strictly preferred to Y
means X is weakly preferred to Y
means consumer is indifferent between X and Y
This notation allows us to bring mathematical logic into our comparisons. For example:
The examples above show how mathematical logic can be used to aid economic reasoning, but for this system to work there must exist some rules to our logic, called axioms. Axioms are statements which are taken to be true, which serve as a premise for further arguments and reasoning. In the study of utility, there are 3 main axioms, which are called the axioms of rational choice, and are shown below.
Given these axioms, we then follow in the footsteps of Jeremey Bentham through ranking utility. More desirable situations will give higher utility than less desirable ones, and hence if situation A is preferable to situation B, then the numerical value attached to U(A) would be larger than that assigned to U(B). (Where U represents a utility function).
Indifference Curves
Figure 1: An example of an indifference curve, showing all the bundles that are indifferent to (a,b)
This means that a consumer would be equally happy consuming all the different combinations of x1 and x2 that lie on this curve.
The gradient of an indifference curve is called the marginal rate of substitution and can be calculated by finding the gradient of a tangent at a particular point. The MRS is usually negative and can be found using the following formula. In words, the MRS shows the rate at which a consumer will give up one good in return for another.
Another important thing to note before our final observation is that within an x1, x2 pane, there lie an infinite number of indifferent curves, each at a different level of utility. In a usual indifference curve, moving to the Northeast increases our utility. Following this, we meet an interesting observation.
Suppose we meet a consumer, and we offer him a trade of good 1 for good 2, at the exchange rate of E. This means that Δ21 leads to EΔ21 of 21 We can think of this graphically, as if a line with the gradient -E crosses the curve, there will be points preferred to the current level of utility, and hence they should trade. For no movement, the slope with gradient -E must be tangental to the indifference curve. At any E that is not the MRS, the consumer wants to switch, but If E= MRS the consumer doesn’t switch.
Figure 2: if the gradient -E is equal to the MRS, there is no switching
Properties of the Cobb-Douglas Production Function
By Jinying Ouyang
Introduction
The Cobb-Douglas production function was invented by mathematician Charles Cobb and economist Paul Douglas and first introduced in their 1928 paper ‘A Theory of Production’. Douglas was interested in understanding the relationship between the inputs of labour and capital and the total output of the US manufacturing sector. Using Cobb’s mathematical expertise, the two men created a mathematical model which expressed total output Q as a function of labour L and capital K.
The Cobb-Douglas function can be written as:
where:
Q = total output
L = quantity of labour
K = quantity of capital
A = total factor productivity
α = output elasticity of labour
β = output elasticity of capital
0 < α< 1
0 < β < 1
The main aim of this function was to estimate how much of output growth could be attributed to changes in labour and capital, instead of improvements in total factor productivity A (which is expected to increase over time) . It was also designed to test the neoclassical idea that each factor of production is rewarded according to its marginal contribution to output.
Marginal Products of Labour and Capital and Diminishing Returns
One important aspect of the Cobb-Douglas production function is how the marginal products of labour and capital may vary. The marginal product of labour (MPL) measures the change in total output when one additional unit of labour is added (and capital is held constant). Similarly, the marginal product of capital (MPK) measures the change in total output when one additional unit of capital is added. We can find MPL and MPK using partial differentiation.
As all variables are positive, MPL and MPK are also positive, and one additional unit of either labour or capital alone will yield an increase in total output. However, although total output will increase overall, the rate of increase will decrease due to the law of diminishing returns. We can use this by again using partial differentiation to find the second derivatives of labour and capital.
As both α < 1 and β < 1, α – 1 < 0 and β – 1 < 0. Therefore, both ∂MPL/∂L < 0 and ∂MPK/∂K < 0. If only capital increases while labour is held constant (or vice versa), output increases at a diminishing rate. This suggests that economic growth from increasing labour or capital alone is unsustainable, unless there is an improvement in total factor productivity A (eg. improved education for labour force / improved technology).
Returns to Scale
Although increasing labour and capital in isolation will produce diminishing returns for total output, increasing both L and K together gives a different result. Returns to scale describe how output changes when all inputs are increased by the same proportion.
The sum of the output elasticities determines the type of returns to scale:
• If α + β = 1, the function has constant returns to scale. Doubling both labour and capital will exactly double output.
• If α tβ > 1, there are increasing returns to scale. Doubling inputs leads to more than double the output.
• If α + β < 1, there are decreasing returns to scale. Doubling inputs results in less than double the output.
The proof for constant returns to scale can be shown below:
In the Cobb-Douglas production function:
Q=ALα Kβ=ALα K1-a
Assuming that both L and K are increased by a common factor λ:
Q=A(λL)α (λK)1-a
Q=Aλa La λ1-a K1-a
Q=Aλa+1-a La K1-a
Q=AλLa K1-a=λ(ALa K1-a)
Output Elasticities and Factor Shares
Output elasticity measures the percentage change in output resulting from a 1% increase in an input. It can be calculated using:
In the Cobb-Douglas production function, α and β represent the output elasticities of labour and capital respectively. A key feature of the Cobb-Douglas production function is that the parameters α and β can represent factor income shares, assuming competitive markets.
In a perfectly competitive market, firms are price takers in input markets, which means they cannot pay below the market wage or capital rate without being outcompeted by another firm. Under perfect competition, this means that firms have to pay each input its marginal product. The share of total output an input increases must equal its output elasticity.
This means that income is distributed based on the MPL of an input (or, in other terms, the productivity of an input). α corresponds to the share of income paid to labour, and β to the share paid to capital. In real-world economies, labour typically receives around 70% of national income and capital around 30%.
Elasticity of Substitution
Elasticity of substitution measures the ease at which one input can be substituted for another in production. It can be calculated using the formula:
For simplicity, we can rewrite MPL/MPK as the marginal rate of technical substitution MRTS.
Therefore:
Using partial differentiation to find the values of MPL and MPK, we can calculate MRTS ( MPL/MPK) . The values of MPL and MPK can also be found in the above section Marginal Product of Labour, Marginal Product of Capital, and Diminishing Returns.
Substituting this into the equation for σ:
Thus, it can be shown that the Cobb-Douglas function assumes that the elasticity of substitution between capital and labour is equal to one. Capital and labour can be substituted at a constant rate – a percentage change in the ratio of capital to labour will lead to a proportional change in the marginal rate of technical substitution.
In reality, however, this assumption may be restrictive and the effect of substituting capital for labour (or vice versa) may not be the case in every industry. Industries with an elasticity of substitution < 1 where capital cannot easily replace labour include healthcare and education. On the other hand, an example of an industry with an elasticity of substitution > 1 is manufacturing, where machines can easily replace workers.
Conclusion
The Cobb-Douglas production function was invented almost a century ago, but it is still a simple yet powerful tool for economists. Despite making some assumptions (most notably perfect competition) which may limit the accuracy of the function in its real-world application, the Cobb-Douglas production function can explain the inputs of labour and capital are linked to total output, and has since been adopted by economists such as Paul Samuelson and Robert Solow in further modelling economic growth.
Spotify and the Rise of Monopsony Power in the Music Industry
By Janvi Pankhania
Due to today’s digital age, streaming music has become a second nature to us – a few clicks on a screen and we can stream any song we like. It is easy to forget or even imagine a world not so long ago where people had to physically go to a store to buy CDs, DVDs, or vinyl records. Music actually came with a price tag and a plastic case. This meant that accessing music was participating directly in a free market as each music sale was a clear market transaction: consumers paid a set price, artists and labels earned revenue and supply chains moved tangible goods. The music industry functioned on the traditional economic principles of supply and demand. However, today this model has dramatically changed as streaming platforms such as Spotify have shifted music consumption from ownership to access, and essentially created a digital ecosystem where millions of songs are available instantly – often for free or at a low subscription cost. As consumers, we appreciate this convenience and variety but this has concentrated market power in the hands of Spotify, making it into a monopsony.
A monopsony is a market situation where there is only one buyer for a particular good or service, giving that buyer significant power over suppliers or workers. Due to no competition among buyers, the monopsonist can often push prices lower than in a competitive market. Spotify is a key example of having monopsony-like power in certain parts of the music industry, especially how it interacts with artists, labels and rights holders which leads to concerns about equitable compensation and the overall health of the music industry. Although Spotify is not the only music streaming platform (Apple music, YouTube music, Amazon music etc) it is the dominant player in music streaming globally with over 600 million active users and holds disproportionate bargaining power when negotiating with artists, labels and rights organisations. Spotify holds a significant market share in the global music streaming industry, estimated to be around 31.7% (as of 2025). For most artists, especially independent of unsigned artists, being on Spotify is not optional since it is a digital age where everyone’s performing. Artists have no choice but to sell their music on a streaming platform for fractions of pennies to obtain any sustained success or relevance.
How does Spotify act as a Monopsony?
We assume Spotify is a monopolist and sells all the music that is sold in the market to consumers. We also assume that Spotify is a monopsonist in the purchase of music from artists. So, Spotify buys the entire supply of music in the market and then sells this music to consumers.
Spotify faces a downward sloping market demand curve for music and the price it receives is an inverse function of quantity sold. Pd is the price which music is sold to consumers.
We assume Spotify sets their marginal cost = marginal revenue in the context of the downwards sloping line. We assume that Spotify practices perfect price discrimination. This means that Spotify charges each consumer exactly the maximum price they are prepared to pay for music. Each consumer thus pays a different price according to the maximum price they are prepared to pay for music. This maximum price is represented by the demand line. The effect of this makes price and marginal revenue the same (to model Spotify more like a monopsony rather than a monopoly). The extra revenue from selling one more unit of music is identical to the price of the last unit of music so: Pd = MR. The firm's MR function is thus the demand function for music as a whole.
If Spotify wishes to purchase more music from artists, it moves up the supply curve for music, thus raising the supply price for music to artists (Ps). This causes the marginal cost of music to Spotify to be greater than the price of music to artists because the extra cost is the price of the new music it buys plus the increased price it has to pay for the music it is already buying. Assuming Spotify is a profit maximiser, it buys music until Pd = MC and this is a lower output (Qm) than the free market quantity (Qc). This lower quantity purchased yields a lower price to suppliers of music. This is the point PmQm. Thus, this explains why artists get a lower price and the quantity of music bought and sold declines and there is a deadweight welfare loss. Society and musicians are worse off than world without Spotify.
Optimal Bidding Strategy in First-Price Auctions
In a first-price sealed-bid auction, bidders each have a private value for the item being sold—essentially, what it is worth to them. However, the challenge lies in the fact that the winning bidder pays exactly what they bid. To make a profit, a bidder needs to bid lower than their true value, but not so low that they lose out to another bidder. Striking the right balance is key. The Bidding Strategy In a first-price auction with n bidders, each bidder’s private value s is assumed to be randomly drawn from a uniform distribution between 0 and 1. This means that any value within this range is equally likely to occur. The bidding strategy for each bidder is denoted as b(s), where b(s) represents the bid placed by a bidder whose private value is s. To win, a bidder must submit a bid that is higher than all the other bidders. The probability that a single opponent bids lower than the bidder is simply the probability that their value is lower than the bidder’s. Since all bidders are assumed to follow the same strategy, this probability is given by:
Since there are n−1 other bidders, the probability that all of them bid lower than the bidder is:
Expected Profit
The expected profit, or expected payoff, of a bidder is given by the difference between their value and their bid, multiplied by the probability of winning. This can be expressed as:
To determine the optimal bidding strategy, we must maximize the expected profit by choosing the appropriate bid b(s). This can be done using calculus.
Maximising the Profit: Differentiation
We begin by taking the derivative of the expected profit with respect to b:
Apply the product rule:
By Gautham Arun
Since sn-1 is independent of b, the derivative simplifies to:
We set this equal to zero and solve for b(s):
Interpreting the Formula
This equation tells us that the optimal bid is a fraction of the true value of the bidder. Specifically, it is (n-1)/n of the bidder’s value s. To illustrate this with some examples:
• For 2 bidders: b(s) = 1/2 s. The bidder will bid half their value.
• For 3 bidders: b(s) = 2/3 s. The bidder will bid two thirds of their value.
• For 10 bidders: b(s) = 9/10 s. The bidder will bid 90% of their value.
As the number of bidders increases, the optimal bid approaches the true value of the bidder, as more competition requires higher bids to remain competitive.
Example: Calculating the probability of winning
Let’s consider an example where there are 4 bidders and the private value of the bidder is 0.8. Using the optimal bidding strategy formula:
The probability of winning with this bid is:
Thus, with a bid of 0.6, the bidder has a 51.2% chance of winning the auction. Hence, using the optimal bidding strategy formula above, we can calculate the best bids given the constraints with an associated probability.
Lagrangian Multipliers and their Uses in Economics
Lagrangian multipliers are a mathematical method used to determine optimization subject to constraints –problems which are common in Economics. This can either minimising or maximising utility or profit subject to at least one constraint – for example, resources available or budgets. This method was developed by Joseph-Louis Lagrange in the late 18th century, and his aim was to simplify optimisation problems by using constraints directly into the optimised function. His method became increasingly important in calculus of variations and has applications in Economics, Physics and Engineering. In Economics in particular, Lagrangian multipliers can be used to configure profit maximisation, cost minimisation, and utility maximisation.
In order to build a Lagrangian multiplier, we first need an objective and a constraint. For example:
f(x1,x2) as the objective, and g(x1,x2)=c, where c is the constraint
The Lagrangian function, combining the two, is then defined as:
L(x1,x2,λ)=f(x1,x2)-λ[g(x1,x2)-c]
The function here equates to the objective function minus λ, the Lagrangian multiplier, multiplied with the rewritten constraint such that the right-hand side equals zero.
Since the Lagrangian function has three variables (classified here as x1, x2 and λ) we can find the optimisation conditions for each problem using partial derivatives with respect to each of the variables.
v
Partially differentiating the Lagrangian function with respect to x1:
Partially differentiating the Lagrangian function with respect to x2:
Partially differentiating the Lagrangian function with respect to λ:
By Anvita Maloo
Now, we have three equations which we can therefore solve simultaneously to find values for x1, x2 and λ. These are the optimal solutions for the first order conditions.
The first order conditions are the required conditions for the solutions to be a type of stationary point (i.e. a maxima, minima or saddle point). However, to classify which type of the three points the function is, we need to look at the second order conditions.
To find second order conditions, we utilise a Bordered Hessian Matrix. This matrix increases in size depending on the number of constraints we use. For example, using one constraint we get the following:
We then need to find the determinant of this 3 x 3 Bordered Hessian Matrix. If the determinant is positive, then the function is a local minimum; if the determinant is negative, then the function is a local maximum; and if the determinant is 0 (or unclear), then the function is a saddle point. The Bordered Hessian Matrix makes is viable for classifying the stationary point as it makes use of the curvature in the function.
Overall, the Lagrangian multiplier offers an elegant approach to solving optimisation problems subject to constraints in Economics. A powerful tool of the function is that the underlying mathematical method doesn’t change based on what the constraints or objectives are. The method allows us to delve into how efficiently resources can be allocated under limitations so we can explore trade-offs more numerically. Although mathematically derived solutions may not be what’s best for some real-life situations, Lagrangian multipliers can still play a large role in effective decision and policy making for firms, consumers, and the government due to their theoretical accuracy.
The Bass Diffusion of Innovation Model
The Bass Diffusion Model is a mathematical model, developed by Frank Bass in 1969, that is used to describe the adoption of new products and technologies in a population, and therefore predict consumer behaviour. The model is built on the premise that the adoption of a new product follows a predictable pattern driven by two forces: innovation and imitation. These are represented mathematically by the parameters p (the coefficient of innovation) and q (the coefficient of imitation). Innovation refers to customers choosing to purchase based on their own knowledge, independent of social influence. On the other hand, imitation refers to when customers choose to purchase based on the experience of other purchasers. If imitation exceeds innovation, the new product will take a while to be popular, but rate of adoption will be greater.
The Mathematical Model
The Bass Diffusion Model formula is derived from a few assumptions about how innovations spread in a population. It is assumed that:
· There is constant social influence
· There is a fixed population of eventual adopters, also known as market potential, denoted by m.
· The cumulative number of people who have adopted the product by time t is N(t), so the remaining potential adopters are m - N(t)
· The probability that a potential adopter adopts at time t is only influenced by external influence (innovation) with probability p, and internal influence (imitation) with probability q, proportional to the fraction of previous adopters which is N(t)/m.
The derivation results in the differential equation that captures the rate at which people adopt a new product over time. This is a product of the remaining potential adopters and the probability that any one of them adopts at time t:
The equation generates an S-shaped curve diffusion curve:
Growth in the early stage of a new process, product, or technology is slow as adoption is mainly driven by innovators after the product is initially launched. So at this
By Kiara Welge
point the rate of adoption depends chiefly on the parameter p. As more people adopt and N(t) rises relative to m then the imitation coefficient q creates a positive feedback loop, which increases number of adopters until the peak adoption rate. Eventually, only a few potential adopters are left as m-N(t) decreases, so the growth rate declines and the curve flattens, indicating market saturation.
Use of the Bass Diffusion Model
The process of calibrating The Bass Diffusion model makes sure that the model's predictions accurately reflect adoption patterns in the real world. It involves adjusting the model's parameters, which are estimated through non-linear regression against sales data, or maximum likelihood estimation. Firms can then evaluate market potential, consider the likely circulation of innovations within their target market, and therefore determine the best ways to introduce new products. The model mostly applies to new products, often technology and consumer durables. This is because they are one time purchases, so adoption of the product is more important to firms than how often it is used.
Limitations
The model does not take into account repeat purchases which have more complex influences, such as brand loyalty and habitual behaviour. It is assumed there is a homogeneous market where all adopters have the same traits and behaviours, which ignores diversity in preferences, income, or access. Another significant barrier of the Bass Model is that it assumes consistent parameters of p, q and m over time, which in reality frequently change.
Ultimately, the Bass Diffusion Model is useful because it helps predict how quickly and widely a new product will be adopted. It captures the way people are influenced by both marketing and word of mouth, which proves to be helpful in planning launches, estimating demand, and guiding marketing strategy.
Frank Bass (1926 - 2006)
The Slutsky Equation By Dr Ian St John
When the price of a good changes there are two effects operating to change demand for the good: first, the substitution effect, arising from the fact that the price of the good has changed relative to other goods; and second, the income effect, which occurs since when the price of a good changes the consumer’s real income (or purchasing power) changes. One way to isolate the substitution effect from the income effect when using indifference curve analysis is to hold the consumer to the same indifference curve before and after the price change. This means that the consumer’s total utility from consuming two goods doesn’t change. This is the convention established by John Hicks. However, the Russian economist Eugen Slutsky had earlier, in 1915, put forward an alternative method for distinguishing between price and income effects of a price change and this yields the now well-known Slutsky Equation. This equation has the advantage over the Hicksian method in that it does not involve holding utility constant – which is an advantage since we cannot measure utility or observe an actual Indifference Curve. It deals, in other words, with observable phenomena.
Figure 1. Analysing the Effect of a Change in Price of X using the Slutsky Method
To understand Slutsky’s method, consider Figure 1. The consumer faces a choice between two goods, Y and X. The price of Y is Py and the price if X is Px. The consumer has a fixed money income (I) to spend on the two goods. We call this the Budget Constraint. This is represented by the lower blue Budget Line in Figure 1. If the consumer spends all their income on good Y then the maximum amount of Y they can buy is I/Py and we see that the Budget Line intercepts the Y-axis at this point. And if they spend all their income on X then they can consume I/Px of X the Budget Line intercepts the X-axis at this point. The consumer maximises their utility from their fixed income when they attain their highest possible indifference curve at this income. This occurs at point A, where IC1 is tangential to the Budget Line, meaning that the consumer purchases the bundle (X1, Y1) of the two goods. Now assume the price of X falls to Px1. As a result, they can now buy more of X for a given money income I than before and the Budget Line pivots out to I/Px1. In the diagram the fall in price of X causes the consumer to move to the new tangency point B, consuming the bundle (X2, Y2). Consumption of X has increased from X1 to X2 and consumption of Y has also increased slightly, from Y1 to Y2. This is the total effect of the fall in the price of X and reflects both the substitution effect of the change in the relative prices of X and Y and the income effect of the fact that, due to the fall in the price of X, the consumer is better off in real terms given the money income I (which explains why the demand for Y has increased also).
Slutsky broke down the total price effect into its two components in the following way. What he did was reduce the consumer’s money income I to take account of the fall in the price of X. He reduced their money income I to I1, such that, at the new set of prices (Px1, Py), the consumer could just buy their initial consumption bundle (X1, Y1). In Figure 7 we represent this by moving the new outer Budget Line inwards until it runs through the initial point A. In this way, although the price of X has fallen, the compensating fall in money income (from I to I1) means the consumer would still have to use all their money income (I1) to purchase the original combination (X1, Y1). However, the new ‘compensated’ Budget Line has still got the lower gradient reflecting that the price of X has fallen relative to Y. Hence, the consumer will no longer choose to consume bundle A, but will rather choose bundle C, where the compensated Budget Line is tangential to Indifference Curve IC3. This combination of X and Y is one previously unavailable to the consumer at the old income and prices; the fact that the consumer prefers C to A shows that the consumer is still better off in real terms after the fall in the price of X. This is the difference with the Hicks adjustment: Hicks reduced money income until the consumer was forced back onto their initial Indifference Curve IC1, thereby keeping the consumer’s total utility constant. With Slutsky’s adjustment, the consumer’s total utility has increased from IC1 to IC3. The problem is that Hicks’s adjustment is theoretical only (since we cannot actually locate a consumer’s Indifference Curve), whereas Slutsky’s can be performed in reality since we could (in theory) reduce the consumer’s money income to I1 and observe what combination of X and Y they choose to consume. The movement from A to C (or X1X3) Slutsky called the substitution effect of the price change, reflecting the changed gradient of the Budget Line due to the fall in the price of X from Px to Px1, while the movement C to B (or X3X2) he
For this section, see Snyder and Nicholson, Microeconomic Theory, pp. 148-150 and M. Friedman, Price Theory: A Provisional Text (Frank Cass, London, 1962), pp. 48-55.
called the income effect, because, holding actual money income constant at I the consumer is better off in real terms from the fall in the price of X. In algebraic terms, the money income I was initially just sufficient to buy the bundle (X1, Y1). Hence I = PxX1 + PyY1. Because the price of X has fallen from Px to Px1, then to keep the consumer’s real income constant in the sense of just being able to buy the initial combination (X1, Y1), their money income must be reduced by I – X1(ΔPx), where the minus sign is because ΔPx is negative, since the price of X has fallen. So I1 = Px1X1 + PyY1. Slutsky summed up his analysis of the overall effect of a change in the price of a good on demand in his equation:
We can derive Slutsky’s equation as follows.
Consider a consumer who spends their money income I on two goods, X and Y. Their demand function for X is:
X = X(Px, Py, I)
which means that the demand for X is a function of the price of X (Px), the price of Y (Py), and their money income I. We hold the price of Y constant and focus on change in Px alone. As we have seen, if the price of X falls, then with a given income (I) the consumer’s real income will rise, yielding both the substitution and income effects of a price change. To isolate the real income effect of a fall in the price of X we adjusted money income to keep real income constant at the level just required to purchase the initial bundle of X and Y, which we identified as (X1, Y1). As we noted, if the price of X fell by ΔP then the downward adjustment in income required to keep real income constant was X1(ΔPx). The demand function which incorporates these changes in money income to keep real income constant we call the compensated demand function Xc, which is function of the prices of X and Y and real income (RI). Thus:
Xc = Xc(Px, Py, (RI) )
Since (RI) is income I adjusted for the change in money income necessary to keep real income constant, we can write I as E, where E is the level of money income which keeps real income constant, i.e. E = E(Px, Py, (RI) ). So if Px rises and Py and RI remain constant, E must rise by X1(ΔPx), and if Px falls with Py and RI constant, E must fall by X1(ΔPx) – which is, of course, the example we are considering. Hence we can say that:
Xc(Px, Py, (RI) ) = X[Px, Py, E(Px, Py, (RI) )]
This states that the compensated demand for X, namely Xc, equals the demand for X when E adjusts in response
to any change in the price of X or Y to keep real income (RI) constant. If Px falls then E must fall if the consumer is to stay at the initial level of real income. So E is the expenditure necessary to keep the consumer at the initial level of real income when they purchased the bundle (X1, Y1), which is point A in Figure 7.
Differentiating the expression:
Xc(Px, Py, (RI) ) = X[Px, Py, E(Px, Py, (RI) )]
by Px (holding Py constant) using the product and chain rules we get:
The left-hand side of this equation is the change in demand for X when the price of X falls and when the consumer’s income is adjusted to compensate for the real income effect of a price fall. It is thus Slutsky’s substitution effect of a price change. It is a negative number since a fall in the price of X causes demand for X to increase and vice versa. The partial derivative ∂X/∂Px is the overall effect of a change in Px on demand for X with no adjustment for the income effect. In our example it is the overall increase in demand for X when the price of X falls. It, too, is a negative number. However, in calculating the compensated demand effect of a fall in price, the increased real income effect of a price fall must be removed by subtracting money income (spending E) to offset the effect of the price fall (i.e. we must subtract X1(ΔP), where ΔP is negative). Thus, the fall in Px leads to a fall in E, so ∂E/∂Px is positive. If X is a normal good, then ∂X/∂E is also positive – the fall in E reduces demand for X. Hence the expression (∂X/∂E)(∂/∂Px) is positive. Given that ∂X/∂Px is negative, then adding a positive to a negative reduces the negative, which means that the increase in compensated demand Xc for a price fall is less than the uncompensated overall demand effect ∂X/∂Px, which is to say that ∂Xc/∂Px is less negative than ∂X/∂Px. The effect on demand for X of a fall in the price of X due to the substitution effect alone (holding real income constant) is less than the effect on demand for X when Px falls and real income is allowed to increase also.
Rearranging this equation to express it in terms of the overall change in demand from a change in price we get:
Δ Demand for X = Substitution Effect – Income Effect
Now we have an expression for a regular demand curve.
It states that the overall change in demand for X due to change in the price of X is equal to the compensated change in demand for X holding real income constant (substitution effect) minus the change in demand for X due to the change in money income necessary to hold real income constant (income effect). In other words, to derive the actual change in demand for X we must remove from the substitution effect with real income constant the change in money income necessary to yield constant real income, which is the income effect. For example, if Px falls then overall demand rises: ∂X/∂Px < 0. Due to the substitution effect, this fall in Px causes Xc to increase. So ∂Xc/∂Px < 0. But, if X is a normal good, in terms of absolute values ∂Xc < ∂X because, to isolate the substitution effect, the consumer’s money income has been reduced to hold their real income constant in the sense of just allowing the consumer to purchase their initial preferred combination of X and Y (X1, Y1). Hence, to arrive at the overall change in demand for X given the fall in Px we must restore the real income effect of the fall in the price of Px by removing the compensating change in money income used to generate Xc. As we have seen, the expression (∂X/∂E)(∂/∂Px) is positive since the fall in Px necessitates a fall in E, and a fall in E causes (in the case of a normal good) a fall in demand for X. Deducting a positive term from a negative number increases the absolute value of the negative number. Thus, the increase in demand for X due to the substitution effect is magnified by the addition of the income effect so that:
where both partial differentials are negative numbers.
This is exactly the result we previously illustrated in Figure 1, where the total demand effect of a fall in the price of X (i.e. X2 – X1), was compounded out of the substitution effect holding real income constant (X3 – X1) and the real income effect of a price fall, (X2 – X3). We saw above that the change in expenditure (E) or money income required to keep real oncome constant when the price of X changes is ΔE = X1(ΔPx), which is the change in the price of X multiplied by the initial quantity of X chosen, X1. Hence:
Thus, the Slutsky Equation is generally written as:
If X were an inferior good then the rise in real income from the fall in the price of X (∂X/∂E) would be
negative and the overall increase in demand for X would be less than the compensated change in demand. This income effect will almost certainly be smaller than the substitution effect and demand for X will still increase if the price of X falls. If the income effect were larger than the substitution effect then the overall demand for X would fall if the price of X falls – this is the case of the Giffen Good.
Note From Editors
We would like to give a huge thank you to everyone who submitted articles to the publication this term, as well as to Dr St John for taking the time to run the Mathematical Economics Society. We encourage all readers and those interested in pursuing Economics further to contribute to the magazine and for those contributing to email their articles (no more than roughly 1500 words) to istjohn@habselstree.org.uk
