10 способов вывести компанию из бедственного положения

 

Дмитрий Трепольский

Как управленцу вовремя распознать, что в бизнесе начались критические проблемы и предотвратить развитие негативного сценария?

Даже хорошие руководители могут не заметить первые признаки того, что компания находится в бедственном положении. Дело не в компетенции – просто управляющие часто продолжают по инерции работать в рамках набора привычных парадигм, даже когда они теряют свою актуальность. А если не распознать вовремя проблему, легко упустить время для оптимального решения.

25 признаков критического положения в бизнесе

Оборотный капитал/ликвидные средства:

• Снижающийся или отрицательный поток свободных денежных средств.
• Огромные условные задолженности.
• Непогашенные краткосрочные задолженности.
• Проблемы с возобновляемыми кредитами.
• Ужесточение условий работы с поставщиками.
• Увеличение сроков погашения дебиторской задолженности.
• Рост непогашенных кредиторских задолженностей.

Финансы:

• Снижение цен на бирже.
• Снижение цен облигаций.
• Невозможность придерживаться обязательств по соблюдению установленных финансовых показателей.
• Увольнение ключевых членов финансового департамента.
• Снижение ликвидности.
• Постоянный пересмотр договоренностей с банками.
• Снижение кредитного рейтинга.
• Повторяющийся пересчет статей бухгалтерского баланса.
• Неспособность предоставить финансовую отчетность.

Прибыльность и перспективы в отрасли:

• Снижение EBIDTA (прибыль до уплаты процентов, налогов и начисления амортизации и износа).
• Уменьшение инвестиционных программ.
• Неспособность компании продолжать непрерывную деятельность в течение разумного периода времени. Отсутствие сбалансированного соотношения финансового состояния, персонала, производственных или сбытовых мощностей и ясных рыночных перспектив.
• Ухудшение отраслевых основ.
• Неблагоприятная среда нормативно-правового регулирования.
• Запросы со стороны регулятора.

Персонал:

• Крупные или незапланированные сокращения рабочей силы.
• Текучесть кадров среди управляющих.
• Сбои в единстве командной работы.

Как исправить критическую ситуацию в компании

1. Пересмотрите свои представления о том, что такое «бедственное положение»

Практически невозможно сформулировать единое понятие для «компании в бедственном положении». И опасно думать, что у вас есть готовое определение конкретно для вашего бизнеса. В зависимости от ситуации могут проявить себя 25 признаков надвигающегося кризиса, перечисленные выше. Но проблема редко состоит из одного-двух компонентов, обычно речь идет о сочетании множества взаимосвязанных факторов.

2. Пересматривайте собственные планы

Главное, что можно сделать, чтобы избежать кризиса, – это периодически пересматривать свои бизнес-планы. Впервые их формируя, продумайте несколько триггерных точек. Это может быть и простое напоминание, например: «Если у нас не будет N-ной производительности или мы не сделаем следующие 10 вещей к определенной дате, нужно сделать шаг назад и подумать о том, правильный ли мы выбрали путь».

Такие триггерные точки должны быть ориентированы на операционные, рыночные и финансовые показатели, а также учитывать движение денежных средств. Посмотрите, в каком состоянии находится компания относительно основных запланированных финансовых и денежных показателей. Затем оцените положение бизнеса в индустрии, относительно конкурентов. Если вы не двигаетесь вместе с остальной отраслью (или не опережаете ее, если отрасль испытывает трудности), возможно, ваш план устарел. И не забудьте вспомнить свою производительность за прошлые периоды, чтобы выявить какие-либо тенденции. Если на протяжении какого-то времени вы не достигаете целевых показателей эффективности, спросите себя, почему.

3. Повысьте свои ожидания от совета директоров

Преимущество совета директоров в том, что его члены достаточно отдалены от дел компании, чтобы смотреть на вещи объективно. Именно на совет директоров ложится ответственность честно сказать генеральному или финансовому директору: «Нам нравится ваш план, но давайте подумаем о том, чтобы сократить расходы не на 3, а на 20%. И давайте обсудим все, что может пойти не так – все риски для бизнеса». Топ-менеджмент должен вести списки потенциальных рисков для бизнеса и ежеквартально рассматривать его вместе с советом директоров, чтобы все оставались в курсе.

Конечно, случаются события, которые невозможно предвидеть. Но также часто речь идет о полутора-двух годах не самой эффективной работы, когда совет директоров не знал о проблемах или не задавал нужные вопросы. По-настоящему независимые члены правления могут многое сделать для успеха бизнеса.

4. Сфокусируйтесь на движении денежных средств

Успешный выход из кризиса сводится к одной вещи – акценте на денежных средствах и доходах. Основа любого бизнеса: он должен приносить деньги, а не «сжигать» их.

Иногда полезно сфокусироваться на простых вещах. Представьте, что управляете небольшим хозяйственным магазином. Вы будете задаваться такими вопросами: «Достаточно ли в кассе денег для оплаты счетов или для оплаты новой партии товаров, которая прибудет на следующей неделе?» или «Сколько я смогу заработать, если вложусь в покупку нового грузовика для доставки?». Возвращаясь к таким базовым понятиям, проще понять, в каком направлении двигаться дальше.

Отслеживание денежных средств – это не только просмотр банковского баланса. Чтобы избежать неожиданностей, компаниям также важно уметь строить прогнозы, учитывающие среднесрочную и долгосрочную перспективу. Например, игнорирование денежной составляющей капиталовложений обычно приводит к проблемам. Чистая текущая стоимость проекта может выглядеть одинаково, независимо от того, начнет ли он постепенно приносить доход на втором году реализации или совершит скачок через пять лет. Если не учесть, что в течение этих пяти лет инвестированных денег у вас не будет, можно внезапно обнаружить, что средств на ведение бизнеса почти не осталось.

5. Придумайте вовлекающую историю изменений

Компании, которые находятся в бедственном положении, обычно недостаточно времени уделяют проработке понятного всем нарратива изменений. Например, в результате реструктуризации компании увеличивается маржинальность и прибыльность, но падает цена на товар. Из-за этого у совета директоров может возникнуть беспокойство: будет ли достаточно свободного денежного потока для покрытия потребностей бизнеса?

Решением станет создание истории изменений, которая гласит, к примеру: «Да, мы прибыльны. И весь смысл прибыльности заключается в том, чтобы генерировать достаточно денежных средств для расширения, роста и поддержания операций. Если мы не сможем этого сделать, нас ждет долгий и медленный спад, когда снижение производства становится новой реальностью».

Важно, чтобы история была лаконичной – чтобы ее мог понять каждый рядовой сотрудник. Часто простое сообщение приносит куда больше пользы, чем сложные метрики.

6. Называйте вещи своими именами: кризис – это кризис

Если у членов команды не развито антикризисное мышление, реакция на любые изменения будет заторможенной. Главной целью будет избегание рисков, а в принятии решений будет преобладать инкрементализм. Работников будут просить делать больше (или то же самое) меньшими средствами. Более решительные идеи будут обсуждаться болезненно долго, а их реализация будет излишне медленной.

Кризис требует серьезных действий прямо сейчас. Менеджеры должны использовать слова вроде «кризис» и «срочность», как только осознают необходимость кардинальных изменений. Компания, переживающая настоящий кризис, будет готова попробовать решения, которые обычно не рассматриваются. Именно эти смелые действия зачастую меняют траекторию развития компании. Кризис побуждает людей к действию, побуждает менеджеров рассматривать весь спектр вариантов.

7. Способствуйте переменам с помощью быстрых побед

Менеджеры в большинстве своем склонны фокусировать свое внимание и ресурсы на 3-4 крупных решениях, которые должны изменить ситуацию. Хотя иногда действительно нужны крупные перемены, они обычно требуют усилий и времени, которых может просто не быть. Важно дополнять крупные «ставки» быстрыми победами, которые позволят запустить процесс изменений внутри организации. Например, сокращение расходов на ненужные внешние услуги или политика в отношении командировочных расходов.

Такие решения не только улучшают итоговый результат, но и позволяют добиться поддержки среди сотрудников. В целом, почти в любой компании есть 20% сотрудников, которые поддержат руководство несмотря ни на что. Есть еще 20%, которые активно сопротивляются изменениям и ищут способы их избежать. И есть еще 60% – это наблюдатели, которые настроены на действие, но для этого нужно дать им повод. Предпринимая реальные действия для быстрых побед, вы привлекаете на свою сторону тех, кто обычно занимает нейтральную позицию.

8. Откажитесь от старой системы поощрений

Важность бонусов для руководства в процессе вывода бизнеса из кризиса часто недооценивают. Во многих компаниях краткосрочные планы поощрения могут представлять собой сложный набор целей, связанных с безопасностью, финансовыми и операционными показателями и личным развитием. Многие из них настолько сложны, что менеджеры и сами не знают, как они устроены.

В период вывода компании из кризисного состояния лучше предложить менеджерам поощрения, четко привязанные к целям, которых вы стремитесь достичь. Если нужно увеличить выручку на 10 миллионов за счет ценообразования, пусть это станет задачей специалистов по продажам. Нужен дополнительный доход от поставок? Поставьте задачу своему директору по закупкам. Будьте готовы лишить премий тех, кто не достигает своей цели на 100%, и щедро платить тем, чьи результаты превосходят ожидания.

9. Замените топ-менеджера или двух

Как показывает практика, успешный вывод бизнеса из кризиса обычно подразумевает замену одного-двух членов топ-менеджмента. Дело не в том, что они «плохие» управляющие. Но реальность такова, что кто-то обычно ответственен за происходящее. Чаще всего это сотрудники, которые не способны изменить образ мышления, которым руководствовались долгие годы. Понимают они это или нет, но они тормозят изменения, потому что стремятся защитить то, что кажется им правильным. Это сложное решение, но оно позволит показать стейкхолдерам, что вы настроены решительно.

10. Ищите и удерживайте талантливых сотрудников

Также стоит обратиться еще к двум типам людей. Во-первых, к тем, кто знает все тонкости работы компании. Часто они недовольны положением дел, но мнение людей, которые готовы говорить неудобную правду, очень важно. Во-вторых, стоит рассматривать кризис как возможность раскрыть в людях новые таланты. Бывает, что на нижних уровнях работают потрясающие руководители, которые просто ждали удобного случая. И предоставление им возможности почувствовать себя частью чего-то большего, спасти компанию поможет надолго заручиться их поддержкой. В обоих случаях важно понимать, что удержание не всегда связано с деньгами и бонусами. Необходимо учитывать индивидуальные потребности каждого сотрудника.

https://bit.ly/3TUFhNP

Market equilibrium

 

• Definition of market equilibrium – A situation where for a particular good supply = demand. When the market is in equilibrium, there is no tendency for prices to change. We say the market-clearing price has been achieved.
• A market occurs where buyers and sellers meet to exchange money for goods.
• The price mechanism refers to how supply and demand interact to set the market price and amount of goods sold.
• At most prices, planned demand does not equal planned supply. This is a state of disequilibrium because there is either a shortage or surplus and firms have an incentive to change the price.

Market equilibrium

Market equilibrium can be shown using supply and demand diagrams

In the diagram below, the equilibrium price is P1. The equilibrium quantity is Q1.

If price is below the equilibrium

• In the above diagram, price (P2) is below the equilibrium. At this price, demand would be greater than the supply. Therefore there is a shortage of (Q2 – Q1)
• If there is a shortage, firms will put up prices and supply more. As price rises, there will be a movement along the demand curve and less will be demanded.
• Therefore the price will rise to P1 until there is no shortage and supply = demand.

If price is above the equilibrium

• If price was at P2, this is above the equilibrium of P1. At the price of P2, then supply (Q2) would be greater than demand (Q1) and therefore there is too much supply. There is a surplus. (Q2-Q1)
• Therefore firms would reduce price and supply less. This would encourage more demand and therefore the surplus will be eliminated. The new market equilibrium will be at Q3 and P1.

Movements to a new equilibrium

1. Increase in demand

If there was an increase in income the demand curve would shift to the right (D1 to D2). Initially, there would be a shortage of the good. Therefore the price and quantity supplied will increase leading to a new equilibrium at Q2, P2.

2. Increase in supply

An increase in supply would lead to a lower price and more quantity sold.

https://bit.ly/3h30qXO

Module 4: Consumer Choice

 The Policy Question: Hybrid Car Purchase Tax Credit—Is it the Best Choice to Reduce Fuel Consumption and Carbon Emissions?

The U.S. government offered a tax credit toward the purchase of hybrid cars with the goal of reducing the amount of carbon emissions U.S. cars produce annually. We are using the tools of microeconomic consumer theory to study this policy and assess the effectiveness of this policy in reducing emissions.

We are now very close to being able to predict how consumers will change their driving and gasoline purchases in response to a government tax credit on hybrid cars. As we will see, this is simply a specific example of the general question we first raised in Module 1 of how to predict consumer behavior when prices or incomes change. For our policy example and in general, we address this question by combining the budget constraint with the concept of preferences and utility maximization. All of consumer theory in economics is based on the premise that each person will try to do his or her best given the money they have and the prices of the goods and services they like. This is what we mean by utility maximization – choosing the affordable bundle of goods and services that returns the highest utility.

Think about a consumer who goes grocery shopping. There are many ways to fill a shopping basket – ending up with many different possible bundles of goods. This module is concerned with how each consumer picks the best affordable. As we will see shortly, consumers think about the income they have, and the relative prices of all the possible goods they could buy, and then choose among all of the possible bundle combinations that their budget can support.

Exploring the Policy Question

What is your prediction about how consumers’ driving and gasoline purchasing behavior will change when their income increases or decreases? When the price of gasoline increases or decreases? What implications will these behavior changes have for the hybrid car tax credit policy?

4.1 The consumer choice problem: maximizing utility

LO1: Define the consumer choice problem.

4.2 Solving the consumer choice problem

LO2: Solve a consumer choice problem with the typical utility function.

4.3 Corner solutions and kinked indifference curves

LO3: Solve a consumer choice problem with utility function for perfect substitutes.

4.4 Policy example: The hybrid car tax credit and consumer choice

4.1 The consumer choice problem: maximizing utility

LO1: Define the consumer choice problem.

What is the consumer’s optimal choice among competing bundles? This question summarizes the consumer choice problem. To resolve this problem, we can combine our understanding of the budget constraint and preferences as represented by utility functions. The budget constraint describes all of the bundles the consumer could possibly choose. The utility function describes the consumer’s preferences and relative level of satisfaction from the consumption of bundles. We put these two pieces together by answering the question: Among all the bundles the consumer could possibly choose, which one returns the highest level of utility?

Conceptually, we are overlaying the indifference curve map on the budget constraint and looking for the point or points of intersection, as in Figure 4.1. Recall that there can be more than one indifference curve for bundles of goods A and B. The goal for solving the consumer choice problem is to get on the highest indifference curve – the curve that is the farthest to the upper right – while also satisfying the budget constraint. The highest indifference curve – the one that represents the highest level of utility or satisfaction – is the one that just touches the budget line at a single point. It is not possible to get on a higher indifference curve given the budget constraint, and though it is possible to get on a lower one, doing so necessarily means a lower level of utility or satisfaction.

Figure 4.1 Indifference curve on the budget constraint

Figure 4.1 summarizes the solution to the consumer choice problem: The consumer should pick the one bundle that returns the highest level of utility while also satisfying the budget constraint. This graph also shows us the two fundamental conditions that represent the solution to the consumer choice problem:

1. The consumer’s optimal choice is on the budget line itself, not inside the budget constraint. This is why we can focus on the line rather than the whole set of affordable bundles.

2. At the optimal choice, the indifference curve just touches the budget line and so at this one point they have exactly the same slope.

This second condition has a significant intuitive interpretation. Recall that the slope of the indifference curve is the marginal rate of substitution(MRS) – the rate at which the consumer is willing to trade off one good for the other and remain just as well off. Recall also that the slope of the budget line is the economic rate of substitution (ERS) – the rate at which the consumer is able to trade one good for the other at current prices. What figure 4.1 shows is that at the optimal choice these two things must equal. But why?

Consider a point, like point 1 in Figure 4.2, where the indifference curve intersects the budget line. Here, the MRS is greater than the ERS is absolute value. The MRS tells us the amount of good B the consumer is willing to give up to get one more unit of good A and remain just as satisfied. In this case, the ERS tells us how much of B a consumer must give up in exchange for one unit of A in the marketplace. Here, the consumer is willing to give up a lot of B to get a little more A and remain just as satisfied, but the ERS indicates that the consumer does not have to give up that much of B to get the same amount of A. So it must be the case that if the consumer trades some of B for A in the market he or she will be better off. In other words, a consumer who moves down along the budget line will be on a higher indifference curve.

Figure 4.2 Budget line with 3 indifference curves. Click on points 1 – 4 to see the utility interpretation of each.

So, the graph of the budget line and indifference curves illustrates the two conditions that define the consumer’s optimal choice:

(1) $P_{A}A+P_{B}B=I$,

and

(2)

$MRS=ERS:\frac{M{U}_A}{M{U}_B}=\frac{P_A}{P_B}$

(note that the negative signs cancel)

With these two conditions, we can solve the consumer choice problem mathematically.

Note that the MRS=ERS condition can be rearranged to: $\frac{M{U}_A}{P_A}$

Economists sometimes call this the ‘equal bang for the buck’ condition because the equation indicates that at the optimal choice the consumer must be getting exactly the same marginal utility per dollar from the consumption of goods A and B. Why? Think of a case where these two goods are not equal. If you’re at a pizza parlor eating slices of pizza and bowls of salad, suppose that:

$\frac{M{U}_{Pizza}}{P_{Pizza}}<\frac{M{U}_{Salad}}{P_{Salad}}$

This equation indicates that marginal utility per dollar of pizza is less than the marginal utility per dollar of salad at your current level of consumption. You cannot optimize your utility. Why not?

For simplicity let’s assume the price of both a slice of pizza and a small salad is exactly $1, so:

$M{U}_{Pizza}<M{U}_{Salad}$.

Now, let’s take $1 away from pizza and spend it instead on salad. You lose the marginal utility of that last slice of pizza, but you gain the marginal utility of one more serving of salad. From the condition above, we know that total utility must have increased since:

$M{U}_{Pizza}<M{U}_{Salad}$.

We also know that marginal utility for a normal preference diminishes when you consume more of a good, and therefore increases when you consume less, . Therefore, the marginal utility of pizza will increase and the marginal utility of salad will decrease, getting you closer to equality. As long as this condition is not met (as long as there is an inequality), a consumer can always make these types of trades to become better off.

4.2 Solving the consumer choice problem

LO2: Solve a consumer choice problem with the typical utility function

Formally, the consumer’s optimal choice problem looks like this:

(4.1)

 

$\begin{array}{c}maxU(A,B)\\ (A,B)\\ subjectto{P}_{A}A+P_{B}B\le I\end{array}$

 

As we discussed in the Section 4.1, the second line of this problem (the ‘subject to’ part) is the budget constraint. Since we have the more-is-better assumption, the consumer will always spend all of his or her budget on goods A and B. So, we can rewrite the consumer choice problem as:

(4.2)

 

$\begin{array}{c}maxU(A,B)\\ (A,B)\\ subjectto{P}_{A}A+P_{B}B=I\end{array}$

Let’s use the Cobb-Douglas utility function and solve this problem analytically. Our problem is now:

(4.3)

 

$\begin{array}{c}max{A}^{\alpha }{B}^{\beta }\\ (A,B)\\ subjectto{P}_{A}A+P_{B}B=I\end{array}$

To solve this problem, we will apply what we know from Section 4.1: At the optimal solution, we have two conditions, MRS=ERS, and we are on the budget line.

MRS=ERS in this case requires that we find the marginal utility of bundles A and B. This requires determining the partial derivative of the utility function for good A and then for good B.

$M{U}_A=\frac{\partial \left(A^{\alpha }{B}^{\beta }\right)}{\partial A}=\partial A^{(a-1)}{B}^{\beta }$

$M{U}_B=\frac{\partial \left(A^{\alpha }{B}^{\beta }\right)}{\partial B}=\beta A^{\alpha }{B}^{(\beta -1)}$

The MRS is the ratio of the two marginal utilities:

$\left(4.4\right)MRS=-\frac{M{U}_A}{M{U}_B}=-\frac{\frac{\partial \left(A^{\alpha }B\right)}{\partial A}}{\frac{\partial \left(A^{\alpha }{B}^{\beta }\right)}{\partial B}}=\frac{\partial A^{(a-1)}{B}^{\beta }}{\beta A^a{B}^{(\beta -1)}}=-\frac{\alpha B}{\beta A}$

The ERS is the ratio of the prices:

 $\left(4.5\right)MRT=-\frac{P_A}{P_B}$

Putting these two equations together gives us:

$\left(4.6\right)MRS=MRT\Rightarrow -\frac{\alpha B}{\beta A}=-\frac{P_A}{P_B}\Rightarrow \frac{\alpha }{\beta }\frac{B}{A}=\frac{P_A}{P_B}$

The second part of the consumer choice problem, the budget constraint (that we are on the budget line or the ‘subject to’ part), is straightforward:

$\left(4.7\right)P_{A}A+P_{B}B=I$

At this point, solving the problem is a matter of simple algebra. We have two equations with two unknowns, good A and good B. We can solve these equations by repeated substitution: solve equation (4.4) for good A or B and substitute the result into equation (4.5).

If we solve (4.4) for bundle B, we get:

$\left(4.8\right)B=\frac{\beta }{\alpha }\frac{P_A}{P_B}A$

Substituting into (4.5) gives us:

$P_{A}A+P_B\left(\frac{\beta }{\alpha }\frac{P_A}{P_B}A\right)=I$

Simplifying this equation gives us:

$\frac{(\alpha +\beta )}{\alpha }{P}_{A}A=I$

Solving for A, we get:

(4.9) $A=\frac{I}{P_A}\frac{\alpha }{(\alpha +\beta )}$

Plugging this equation into (4.6) gives us:

$A=\frac{I}{P_B}\frac{\beta }{(\alpha +\beta )}$

Simplifying this equation gives us the solution for good B:

(4.10) $B=\frac{I}{P_B}\frac{\beta }{(\alpha +\beta )}$

Remember that the prices, the income, and the parameter values for alpha and beta are all just numbers that are given. Therefore, these solutions for goods A and B are simply a specific amount of both – one specific bundle of A and B consumption that is the very best choice a consumer has among all the possible choices.

Equations 4.9 and 4.10 are the demand functions for goods A and B, respectively. Demand functions are mathematical functions that describe the relationship between quantity demanded and prices, income and other things that affect purchase decisions. We can use these demand functions to predict what will happen to the consumption of both goods when prices and incomes change. From the demand functions, it is easy to predict that increases in the price of the good will lead to lower consumption and increases in income will lead to greater consumption. We will return to the examination of these demand functions in the next module.

4.3 Corner solutions and kinked indifference curves

LO3: Solve a consumer choice problem with utility function for perfect complements and perfect substitutes.

So far, we have considered the optimal consumption bundle for a consumer who has ‘well- behaved’ preferences, meaning that he or she has indifference curves that are smooth, curved in, and not touching the vertical or horizontal axes. The optimal choice for these well-behaved preferences is characterized by the MRS=ERS. The solution to the consumer choice problem with these preferences is always an interior solution: a utility maximizing bundle that has a positive amount of both goods.

But we know that there are other relatively common preference types, such as perfect complements and perfect substitutes, that have indifference curves that are shaped differently. The solution to the consumer choice problem for these preferences types cannot be characterized by the MRS=ERS condition.

As we saw in Module 2, perfect complements have indifference curves that are kinked at 90-degree angles, and perfect substitutes have indifference curves that are straight lines that begin and end on the axes. For both perfect complements and perfect substitutes the solution to the consumer choice problem is the one consumption bundle that puts the consumer on the highest indifference curve possible. But in both cases, this does not have the tangency condition of the MRS equaling the ERS.

Since perfect complements have indifference curves that are kinked: they have abrupt changes in slope at a single point. At this kink, the MRS is not defined because there is no slope. The solution to the consumer choice problem with perfect complement preferences is interior – you definitely need some of both good to get any utility – but at the kink so there is no MRS to equate to ERS.

The solutions to consumer choice problems with perfect complement preferences are usually corner solutions: a utility maximizing bundle that consists of only one of the two goods. In other words, a consumption bundle that is located at one corner of the budget constraint. Corner solutions are typical when preferences are perfect substitutes but can occur for many other preference types that we will not study.

Let’s see how to find the utility maximizing bundle for these two preference types starting with perfect complements.

Perfect Complements

In the case of perfect complements with strictly positive prices, the optimal bundle is always the one at the 90-degree kink in the indifference curve, as shown in Figure 4.3. We can use this observation similarly to how we used the fact that MRS = ERS in the previous section. If the optimal bundle is always at the kink and is always on the budget line, we can once again find two equations with two unknowns to solve.

Figure 4.3 Solution to the consumer choice problem for perfect complements

Recall that the utility function for perfect complements looks like this:

$U(A,B)=min[\alpha A,\beta B]$ , where α and  are parameters.

POP UP TEXT: Parameter: a fixed value given outside the model, one that never changes. Variable: a value that can change such as prices, income, etc.

Consuming at the kink implies that $\alpha A=\beta B$. This condition makes intuitive sense: Since the utility function takes on only the value of the smaller of the two choices, spending money on some of good A without the corresponding increase in good B will not raise utility at all but will cost the consumer money. So, this choice cannot possibly be optimal. The only optimal decision is to spend money on goods A and B in the precise proportions that they are consumed. With the condition $\alpha A=\beta B$  and the budget constraint, $P_{A}A+P_{B}B=I$  , we can easily solve through repeated substitution. Note that:

$A=\frac{\beta }{\alpha }B$, substitutes into the budget constraint as follows:

$P_A\frac{\beta }{\alpha }B+P_{B}B=I$

Solving for B:

$B=\frac{I}{P_A\frac{\beta }{\alpha }+P_B}$, or $B=\frac{\alpha I}{\beta P_A+\alpha P_B}$

Solving for A using $A=\frac{\beta }{\alpha }B$ :

$A=\frac{\beta }{\alpha }\left[\frac{\alpha I}{\beta P_A+\alpha P_B}\right]=\frac{\beta I}{\beta P_A+\alpha P_B}$

Again, we see that the quantity demanded of each item decreases with an increase in its price and increases with increases in income. But now we also have the interesting result that quantity demanded of one good decreases as the price of the other good increases. This makes intuitive sense because perfect complements are goods, such as hot dogs and hot dog buns, that are only consumed together.

Perfect Substitutes

With perfect substitutes, the optimal bundle is generally either at one corner of the budget constraint or the other. If you stop and think about it for a moment, the intuition behind this observation becomes clear. If you like Coke and Pepsi equally well and think of them as perfect substitutes for one another, you will logically buy only the one that has the lower price. If Coke costs $1 and Pepsi $1.50, and you like them equally well, why would you ever buy any Pepsi at all? There is one exception to this “all-of-one-or-the-other” rule. When Coke and Pepsi have the same prices, you can get all of one, all of the other, or any combination of both. In this case, there is not just one optimal bundle. In this module, we will concentrate on the case where there is one optimal bundle and so you, the consumer, have to purchase all of one or the other.

Consider Figure 4.4, which illustrates a generic budget line (in black) for goods A and B, and a series of indifference curves (in blue) representing preferences for the perfect substitutes of goods A and B. The slope of the indifference curves (the MRS) is always greater than the slope of the budget line (the ERS). As you can see from Figure 4.4, this means that consumers always does better when they consume more A and less B – consumers move to a higher indifference curve. Specifically, consider bundle 1 in the graph. This bundle contains both A and B. If we compare bundle 1 to the bundle in the corner of the budget constraint labeled ‘optimal bundle’, you can see that the optimal bundle is on a higher indifference curve, which means that this bundle is better. In fact this is true of any other bundle in the budget constraint – all other bundles place consumers on a lower indifference curve.

Figure 4.4: Solution to the consumer choice problem for perfect substitutes

Finding this optimal bundle is relatively easy mathematically. Since we know it has to be at a corner , we just need to check the two corners of the budget constraint and see which one yields the higher utility. In general, utility functions that represent perfect complements look like this:

$U(A,B)=\alpha A+\beta B$

Note that the additive form of the utility function is the key—you can always get to the same utility by taking away some of one good and adding some of the other (even if there is none of the the other good in the bundle). Note also that the MRS is:

$MRS=\frac{\alpha }{\beta }$

Since α and β are parameters, this means the slope of the indifference curve is a constant and so the indifference curves for perfect substitutes are straight lines that intersect the axes.

Solving for the optimal consumption bundle for perfect substitutes starts with checking the corners, which means we ask what utility the consumer gets from spending all of his or her income on just one good. So, if:

$P_{A}A+P_{B}B=I$,

and the consumer decides to consume only A, then the total amount consumed of A is:

$\frac{I}{P_A}$.

Similarly the total amount consumed of B if all of the income is spent on B is:

$\frac{I}{P_B}$.

So, all that is left to do is to check whether:

$U(A,0)=\alpha \frac{I}{P_A}>\beta \frac{I}{P_B}=U(0,B)$ ,

or if the opposite is true or if they are equal.

If the above is true then we know immediately that consuming only A is the optimal choice. Similarly if the opposite is true, we know consuming only B is optimal. Finally if they happen to be equal, any combination of A and B consumption along the budget line is optimal. Figure 4.5 illustrates these three possibilities.

Figure 4.5 Solutions to the consumer choice problem for perfect substitutes. Solutions (a) and (b) show the corner solutions.

4.4 Policy example: The hybrid car tax credit and consumer choice

Suppose you are a policymaker considering proposals to reduce the consumption of fossil fuels and carbon emissions. With the tools from Modules 1 through 4, you can now conduct an economic analysis to analyze the likely outcome of a tax credit for electric and hybrid cars. Begin by considering the consumer choice problem prior to the introduction of the tax credit.

First, combine the indifference curve mapping from Modules 1 and 2 with the budget constraint from Module 3, as shown in Figure 4.5. Remember, these are simply graphical representations of the mathematical equations (equations 4.4 and 4.5).

Figure 4.5 Optimal Miles Driven without Car Tax Credit

We are looking for the point on the budget line that puts the consumer on the highest indifference curve possible. This is of course the tangency point between the the indifference curve and the budget line. This tangency is characterized by the equality of the marginal rate of substitution between miles driven and other goods consumed, and the price ratio of the same two goods, the economic rate of substitution. Figure 4.5 shows that , given our assumption of convex indifference curves, there is a unique point that defines a number of miles driven and an amount of money for other goods consumed that make up the optimal bundle for this ‘typical’ consumer.

Mathematically, we can solve for the tangent point by remembering that it is characterized by two things:

1. The equality of the marginal rate of substitution to the price ratio.

2. The budget constraint holding with equality.

The first equation ensures that the bundle is at a point of tangency between the indifference curve and lines with the same slope as the budget line. But remember that the price ratio describes the slope, not the specific budget line, which is why we need the second equation to make sure we are on the specific line that describes the consumer’s budget.

Specifically, if the utility function that represents the ‘typical’ consumer’s preferences over miles driven (M) and all other consumption (C) is:

$U(M,C)=M^{\alpha }{C}^{\beta }$

Then we can solve for the MRS:

$MRS=\frac{M{U}_M}{M{U}_C}=\frac{\frac{\alpha U}{\alpha M}}{\frac{\alpha U}{\alpha C}}=\frac{\alpha }{\beta }\frac{C}{M}$

As noted above, this is one condition that characterized the optimal consumption bundle, the other is the budget line:

Income = (PriceMileDriven ) x (MilesDriven) + $ for other consumption

We can solve this system of equations by the process of repeated substitution to come up with the description of the precise optimal bundle for this consumer.

Now let’s consider what happens after policymakers pass the tax credit into law. From Module 3, we know that the effect of the tax credit on the budget constraint is to make the cost of miles consumers drive less expensive, which represents a drop in the price of miles driven. This will cause the budget line to expand along the vertical axis, but it will not change the horizontal intercept. The consumer must now choice a new the optimal bundle, and we must compare the new bundle to the old one. Let’s first examine this graphically, in Figure 4.6.

Figure 4.6 Optimal Miles Consumers Drive with Car Tax Credit

In this graph, it is clear that the new bundle represents more of both goods. The consumer will choose to consume more other goods with the extra buying power that the more fuel-efficient car affords as well as to consume more miles driven due to the lower cost of the activity. So the policy turns out to have an unintended consequence: in an effort to decrease fuel consumption, the policy actually lowers the marginal cost of driving, inducing consumers to drive more.

By making some fairly basic assumptions about typical consumer preferences and modeling the consumer choice problem, economic theory suggests that we should expect an increase in miles driven as a result. This application illustrates the power of models. By simplifying reality into a model framework, we can discover something about the world and human behavior that was not obvious.

But this is just a theory, so it is suggestive rather than definitive. It is important to note that the assumptions we made may not be completely accurate and so our prediction may be inaccurate. For example, depending on the precise indifference curve mapping we could actually see decreased miles driven after the hybrid car tax credit

We now need to test the theory by evaluating real-world data. So, what does the data suggest? Studies have shown that:

1. An increase in MPG increases how much consumers drive

2. This effect is in the rage of 10 to 30 percent, meaning that a 10 percent increase in MPG would increase how many miles consumers drive by between 1 and 3 percent.

This data does not mean that the policy is a failure. If you increase MPG by 50 percent and miles driven increase by 10 percent in response, you still have succeeded in decreasing fuel consumption substantially. The point is that you do not get the same decrease you would have expected if you didn’t understand the consumer choice problem and how consumers respond to changes in relative prices. Understanding consumer choice is important because there may be a number of different policy approaches, each with its own costs and benefits, and the goal of policymakers is to achieve the policy objective at the lowest cost.

In this case of fuel consumption, economists generally prefer that policymakers increase the gas tax, the subsidy approach and other indirect approaches like the corporate fuel economy standard known as CAFE [http://www.nhtsa.gov/cars/rules/cafe/overview.htm (inactive link as of 05/24/2021)]. The tax on fuel raises the cost of consumption and, as we have now seen, which would decrease fuel consumption. A number of studies have shown that an increase in the gas tax is a more cost-effective way to decrease gasoline usage. [http://www.cbo.gov/ftpdocs/51xx/doc5159/03-09-CAFEbrief.pdf (inactive link as of 05/24/2021)]

With our analysis of the policy based on both theory and evidence, we are now in a position to answer the original question posed at the beginning of Section 1:

Suppose that a hybrid car tax credit were wildly successful and succeeded in doubling the average fuel economy of all cars on U,S, roads . . . What do you think would happen to the fuel consumption of all U.S. motorists? Should the government expect the fuel consumption and carbon emissions of cars driven in the United States to decrease by half in response?

The answer is clearly ‘no.’ Even the most conservative estimates suggest that a 100 percent increase in average fuel efficiency will result in an increase of miles driven by 10 percent meaning that the decrease in fuel consumption and carbon emissions will fall but by less than half.

Exploring the Policy Questions

• Suppose the hybrid car tax credit were accompanied by a tax increase on gasoline. How would the analysis of the policy change? Could you make a prediction about the change in carbon emissions now?

• How would the analysis change if consumers’ preferences change and they begin to lower their gas consumption by carpooling, walking, and biking more instead of driving? How could you show this on Figure 4.6?
• The phenomenon of increased energy efficiency leading to increased consumption is known among economists as the rebound effect, and it is common in other practical contexts besides vehicles and driving. What other examples of rebound effect would you expect to see in the real world?

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