Economist's Toolkit 3451

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ECP 3451-01 Economics and the Law

The Microeconomist’s Toolkit

Market Equilibrium Models

The simplest mathematical representation of a demand function or a supply function is a linear function in two variables, quantity (Q) and price (P).

Demand Function: QD = a – bP

Supply Function: QS = c + dP

The terms a, b, c, and d are called the parameters of the functions. The coefficient on price in the demand function, b, is always preceded by a negative sign which means that the demand curve is always negatively sloped. The coefficient on price in the supply function, d, is positive which means that the supply function is positively sloped. [Question: If the demand curve or the supply curve is perfectly elastic (i.e., horizontal), what is the value of –b or d? If the demand curve or the supply curve is perfectly inelastic (i.e., vertical), what is the value of –b or d?] Market Equilibrium This is a system of two equations (the demand function and the supply function) in two variables (quantity and price). In principle, then, one should be able to solve for the two variables. How? Equilibrium in the market occurs at the price that makes quantity demanded equal to quantity supplied, or QD = QS. To find the equilibrium price and quantity: 1. Set the demand and supply functions equal to each other:

QD = QS

a – bP = c + dP 2. Solve for the equilibrium price in terms of the parameters:

bP + dP = a – c

P equilibrium = (a-c)/(b+d) 3. Use the equilibrium price to solve either the demand function or the supply function to find the

equilibrium quantity:

Q equilibrium = a – bP equilibrium = c + dP equilibrium.

Factors Affecting Demand and Supply Other than the Price of the Good Demand depends on variables other than the price of the good. These other variables include consumers’ incomes, number of consumers, consumer tastes and preferences, prices of substitutes and complements, and so on. Similarly, supply depends on variables other than the price of the good. These other variables include the costs of resources, taxes, technology, the prices of other goods that can be produced with the same resources, and so on. A more complete market model includes additional terms representing these variables, for example:

ECP 3451-01: Economics and the Law The Microeconomist’s Toolkit

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Demand Function: QD = a – bP + fI

Supply Function: QS = c + dP - gT

where I is consumer income and T is a tax on suppliers. It is not possible to represent more than two variables in a two-dimensional diagram. Therefore, in the two-variable model and in the usual diagrammatic representation, the effect of these other variables is included in the intercept parameter, a or c. It is as if we added a + fI to create a new intercept term, a’, in the demand function and added c - gT to create a new intercept parameter, c’, in the supply function:

Demand Function: QD = a’ – bP

Supply Function: QS = c’ + dP.

If any of the other variables influencing demand changes, the intercept parameter of the demand function, a’, changes and the demand curve shifts. An increase in consumers’ incomes, for example, increases the intercept parameter so that the demand curve shifts up. If any of the other variables affecting supply changes, the intercept parameter of the supply function, c’, changes and the supply curve shifts. An increase in costs (including taxes), for example, increases the intercept parameter so the supply curve shifts up.

The Inverse Demand and Supply Functions The demand and supply functions above show price as the independent variable. However, even though economists usually treat price as the independent variable, they put it on the dependent or y-axis (that is, the vertical axis) in a diagram. Therefore, to diagram the demand and supply functions above, it is best to transform them so that P appears as the dependent variable on the left-hand side:

Demand Function: P = (a/b) – (1/b)QD = α - βQD

Supply Function: P = – (c/d) + (1/d)QS = γ + δ QS These are called the inverse demand and supply functions. When diagrammed, these functions generate the familiar linear demand and supply curves. The term α (=a/b) is the vertical or y-intercept of the demand curve; it shows the value of P when quantity demanded, QD, is zero. The term -β (=-1/b) is the slope of the demand curve. The term γ (=-c/d) is the vertical or y-intercept of the supply curve; it shows the value of P when quantity supplied, QS, is zero. The term δ (=1/d) is the slope of the supply curve. Examples

Suppose the wheat market in a small country in 1975 was described by the following linear approximations of demand and supply where P is the price per bushel and the quantities are in millions of bushels:

Demand Function: QD = 200 – 20P


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Supply Function: QS = -40 + 10P

1. What is the equilibrium price per bushel of wheat and the equilibrium quantity of wheat in millions of bushels?

QD = QS

200 – 20P = -40 + 10P

30P = 240

Pequilibrium = $8 per bushel

Qequilibrium = 200 – 20*8 = 40 million bushels

2. What are the inverse demand and inverse supply functions?

Inverse Demand Function: P = (200/20) – (1/20)QD = 10 – .05QD

Inverse Supply Function: P = (40/10) + (1/10)QS = 4 + .10QS

3. Diagram the inverse demand function and the inverse supply function and identify the equilibrium price and quantity of wheat on the diagram.

Consumer and Producer Surplus

Consumer Surplus The demand function or demand curve shows the maximum price consumers are willing to pay for a small (marginal) increase in the quantity of the good, service, or activity. This willingness to pay reflects their marginal benefit (or marginal utility or marginal value—they all mean the same thing) for the good. The marginal benefit is different at different quantities. In fact, the marginal


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benefit decreases as the quantity increases. This is the principle of diminishing marginal benefit or diminishing marginal utility. The more of the thing you already have, the less value you place on having additional units of it. The difference between the marginal benefit of a good to consumers and the price they pay for it is their (marginal) consumer surplus. It is the net gain to consumers from having a marginal increase in the good instead of the next best alternative good. Diagrammatically, (marginal) consumer surplus at any quantity is the vertical distance between the demand curve and the price. At any quantity of the good that is below the equilibrium quantity, consumers’ willingness to pay for a marginal increase in the quantity of the good is greater than the price they must pay so marginal consumer surplus is positive. At any quantity of the good that is above the equilibrium quantity, consumers’ willingness to pay for a marginal increase is less than the price so marginal (but not total) consumer surplus is negative. The aggregate or total consumer surplus is the difference between consumers’ willingness to pay for all units of the good consumed and their total expenditure on the good. Diagrammatically, aggregate consumer surplus at any quantity is the area below the demand curve, above the price, and ranging horizontally from zero up to the given quantity. At the equilibrium quantity, consumer surplus is a triangle. At any quantity below the equilibrium, consumer surplus is a trapezoid. At any quantity above the equilibrium, consumer surplus is the sum of the (positive) triangle to the left of the equilibrium and the (negative) triangle to the right of the equilibrium. [Note: Recall that the formula for the area of a triangle is A=0.5*base*height. A trapezoid is composed of one or more triangles and a rectangle whose areas can be computed separately and added together.] Producer Surplus The supply function or supply curve shows the minimum price sellers are willing to accept for a marginal increase in the quantity of the good, service, or activity given the quantity they already supply. This minimum price equals the marginal cost of the good. The marginal cost is different at different quantities. In fact, the marginal cost increases as the quantity increases. This is the principle of increasing marginal cost. The more of the thing you already supply, the greater is the additional cost to supply additional units of it. The difference between the marginal cost of a good to producers and the price they receive for it is their (marginal) producer surplus. It is the net gain to producers from providing a marginal increase in the supply of the good instead of using their resources to increase the supply of the next best alternative good. Diagrammatically, (marginal) producer surplus at any quantity is the vertical distance between the price and the supply curve. At any quantity of the good that is below the equilibrium quantity, producers’ marginal cost is less than the price they receive so marginal producer surplus is positive. At any quantity of the good that is above the equilibrium quantity, producers’ marginal cost is greater than the price so marginal (but not total) producer surplus is negative. The aggregate or total producer surplus is the difference between the total cost of supplying a given quantity and the total revenue. Diagrammatically, aggregate producer surplus at any quantity is the area below the price, above the supply curve, and extending horizontally from zero up to the given quantity. At the equilibrium quantity, producer surplus is a triangle. At any quantity below the equilibrium, producer surplus is a trapezoid. At any quantity above the equilibrium, producer surplus is the sum of the (positive) triangle to the left of the equilibrium and the (negative) triangle to the right of the equilibrium.


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Social Surplus The demand curve shows the marginal benefit of the good to consumers. The supply curve shows the marginal cost of supplying the good. The difference between the marginal benefit to consumers and the marginal cost is the net gain to society from having a marginal increase in the good instead of using the resources for the next best alternative. This is the social surplus. The social surplus is the sum of consumer surplus and producer surplus. Social surplus is independent of price. It depends only on consumers’ marginal benefit and producers’ marginal cost. The price determines how much of the social surplus is received by consumers and how much is received by sellers, but the price does not determine the total amount of the social surplus. The social surplus is greatest when consumers’ marginal benefit equals producers’ marginal cost. At any smaller or larger quantity, the social surplus is less. One interpretation of economic efficiency is maximization of social surplus, which explains why marginal benefit equal marginal cost is the criterion for an efficient allocation of resources. Examples 1. At the equilibrium quantity, 40,000 bushels, and the equilibrium price, $8 per bushel, which

area shows the consumer surplus? Answer: Consumer surplus is the difference between demand and total expenditure, or the blue triangle below the demand curve and above the price of $8 and lying between 0 and the quantity 40,000.

2. At the equilibrium quantity, 40,000 bushels, and the equilibrium price, $8 per bushel, which

area shows the producer surplus? Answer: Producer surplus is the difference between supply and total revenue, or the purple triangle above the supply curve and below the price of $8 and lying between 0 and the quantity 40,000.

3. At the equilibrium quantity, 40,000 bushels, and the equilibrium price, $8 per bushel, which

area shows the social surplus? Answer: Social surplus is the difference between demand and supply, or the blue and purple triangle below the demand curve and above the supply curve and lying between 0 and the quantity 40,000.

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Efficiency of a Competitive Equilibrium This is the most basic concept. We show that the allocation of resources in a competitive market equilibrium is economically efficient. The argument is divided into steps: 1. first, we define the term “allocation of resources”; 2. then, we define the concept of economic efficiency; 3. finally, we explain why competitive markets supply goods and services efficiently as long as

there are no increasing returns to scale, no externalities, and no “public” or collective consumption goods.

Define Allocation of Resources Suppose there are m consumers and n goods. Let qij be the quantity of good i consumed by individual j. Then, the allocation of resources can be represented by: q11 q12 … q1n Q = q21 q22 … q2n … … … … qm1 qm2 … qmn 1. Each row of Q represents an individual and each column of Q represents a good, service, or

activity. 2. The allocation of resources, Q, specifies the quantity of each good, service, and activity

consumed by each individual. Thus, q11 is the quantity of the first good consumed by the first individual; q12 is the quantity of the second good consumed by the first individual; and q1n is the quantity of the nth good consumed by the first individual. Similarly, q21 is the quantity of the first good consumed by the second individual, and so on.

3. Note that this is only a partial definition of an allocation of resources. A complete definition

would also specify the quantity of each input supplied by each individual to produce each of the goods, services, and activities.

Define Economic Efficiency (or Pareto Optimality) An allocation of resources, QEff., is efficient if there is no alternative allocation of the resources that makes at least one person better off without also making someone else worse off. Economic efficiency occurs when, for each good or service, the social surplus is maximized. The social surplus is maximized by choosing that quantity of each good, service, or activity at which MSB=MSC. The efficient quantity, QEff., is the quantity such that MSB(QEff.)=MSC(QEff.). Intuitively, suppose the current allocation of resources is Q1, which includes a particular quantity of oranges and a particular quantity of lemons. Suppose at this quantity of oranges, MSBoranges>MSCoranges. This means the value to some consumer of one more orange (its MSB) is greater than the value of the additional resources needed to produce another orange (its MSC). This consumer is willing to give up more than enough lemons (or give up purchasing power over more than enough lemons) to free up the resources needed to produce the additional orange. This consumer is better off with an alternative allocation of resources, Q2, which includes one more orange than Q1 but a smaller number of lemons. Furthermore, reallocating the resources from Q1


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to Q2 doesn’t make any other individual worse off. We do not have to take goods or resources away from anyone else to produce another orange for our orange consumer. Allocation Q1 is not economically efficient because there is an alternative allocation, Q2, that makes our orange consumer better off without making anyone else worse off. Now, consider allocation Q3. In allocation Q3, MSBoranges<MSCoranges. This means our orange consumer is unwilling to give up enough lemons to free up the resources needed to produce one more orange. In fact, our consumer would actually prefer more lemons and fewer oranges. One less orange would free up enough resources to produce more than enough lemons to satisfy our consumer. Our consumer would be better off with fewer oranges and more lemons, and no one else would be worse off. Therefore, Q3 is inefficient; there is an allocation involving fewer oranges and more lemons that makes our orange consumer better off without making anyone else worse off. Any allocation in which MSB>MSC for any good is inefficient because there is another allocation with more of this good that would make at least one consumer better off and no one else worse off. Any allocation with MSB<MSC for any good is also inefficient because there is another allocation with less of this good and more of some other good that would make this consumer better off and no one else worse off. Therefore, the only efficient allocation is one with MSB=MSC for every good. Proof That a Competitive Equilibrium Is Economically Efficient Consumer equilibrium: Each consumer maximizes her/his consumer surplus by consuming all units of each good for which marginal benefit is greater than price and not consuming any units for which marginal benefit is less than price. Therefore, each consumer is in equilibrium when her/his marginal benefit for each good, service, and activity equals the price of the good, service, or activity. For each consumer i and each good j,

MBij(QD)=Pj Producer equilibrium: Each supplier maximizes producer surplus by producing all units of a good for which price is greater than marginal cost and not producing any units for which price is less than marginal cost. Therefore, each producer is in equilibrium when the price of each good, service, or activity produced equals her/his marginal cost for the good, service, or activity. For each good j,

Pj=MCj(QS) Competitive Equilibrium: Equilibrium requires QD=QS=QEquil.. At the competitive equilibrium quantity and price, for each consumer i and each good j,

MBij(QEquil.)=Pj=MCj(QEquil.) Efficiency of Competitive Equilibrium: Marginal social benefit is the sum of marginal benefit to the consumer and marginal external benefit to any economic agents other than the consumer:

MSB(Q)=MB(Q)+MEB(Q) Marginal social cost is the sum of marginal cost to the producer and marginal external cost to any economic agents other than the producer:

MSC(Q)=MC(Q)+MEC(Q)


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If there are no externalities, so that for all Q

MEB(Q)=0 and MEC(Q)=0 then,

MSB(Q)=MB(Q) and MSC(Q)=MC(Q)

Therefore, we can rewrite the equilibrium condition as

MSBij(QEquil.)=Pj=MSCj(QEquil.) But this is the same as the efficiency condition above so that the equilibrium quantity is also the efficient quantity:

(QEquil.)=(QEff.). Three Characteristics of Efficiency An efficient allocation of resources is characterized by all three of the following features: 1. (Pareto Criterion) An allocation of resources is efficient if there is no alternative allocation of

the resources that makes at least one person better off without also making someone else worse off.

2. For each good, service, or activity, MSBij(QEquil.)=MSCj(QEquil.), or equivalently if there are no

externalities, MBij(QEquil.)=MCj(QEquil). 3. Social surplus (consumer surplus plus producer surplus) is maximized. These are not three separate conditions or characteristics. They all hold simultaneously.

Externalities External Benefits (or Positive Externalities): External benefits or positive externalities occur when an individual’s decision about consumption or production provides benefits to other individuals who are not party to the decision. Individuals choose the quantity of each activity that makes their marginal private benefit (MB) equal to their marginal cost. They ignore or may even be unaware of the marginal external benefit (MEB) received by other individuals. By definition:

MSB=MB+MEB But when an activity generates positive externalities:

MEB>0 so that:

MSB>MB. Because the benefit from the activity to the direct consumer or producer (marginal private benefit) is less than the benefit of the activity to society (marginal social benefit), the quantity chosen by the individual (where MB=MC) is less than the efficient quantity (where MSB=MC).


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External Costs (or Negative Externalities): External costs or negative externalities occur when an individual’s decision about consumption or production imposes costs on other individuals who are not party to the decision. Individuals choose the quantity of each activity that makes their marginal private cost (MC) equal to their marginal benefit. They ignore or may even be unaware of the marginal external cost (MEC) imposed on other individuals. By definition:

MSC=MC+MEC But when an activity generates negative externalities:

MEC>0 so that:

MSC>MC. Because the cost of the activity to the direct consumer or producer (marginal private cost) is less than the cost of the activity to society (marginal social cost), the quantity chosen by the individual (where MB=MC) is less than the efficient quantity (where MB=MSC). Internalizing an Externality: Actions taken to correct the inefficiency caused by an externality are called “internalizing the externality”. Actions to internalize externalities may include, among others, subsidies, taxes, regulation, and redefinition of property rights, but also private actions.

Economist's Toolkit 3451

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