Module 11: Comparative Statics: Analyzing and Assessing Changes in Markets

 

“Photovoltaik Dachanlage Hannover” by AleSpa on commons.wikimedia.org is licensed under CC BY-SA

THE POLICY QUESTION
SHOULD THE FEDERAL GOVERNMENT SUBSIDIZE SOLAR POWER INSTALLATIONS?

In 2006, in an effort to spur the use of solar power in the country, the US government created a solar investment tax credit (SITC) allowing US households and businesses to deduct 30 percent of the amount they invested in solar power installations. Since the program began, the growth rate in solar installations has been 76 percent per year.

The program has been seen as very successful in promoting solar energy installations. But what does economics say about the trade-offs of subsidizing these types of installations? Does the economic welfare they provide justify their costs?

This chapter looks at a method for evaluating the effects on price and quantity equilibriums when supply or demand—or both—changes within a market. Subsidies are just one potential cause of such changes. We will also look at the effects of taxes, price floors, and price ceilings. And throughout we will focus on the significance of changes in supply and demand for overall economic welfare.

11.1 CHANGES IN SUPPLY AND DEMAND

Learning Objective 11.1: Describe the causes of shifts in supply and demand and the resulting effects on equilibrium price and quantity.

The competitive market supply-and-demand model is one of the most powerful tools in economics. With it we can predict the impact of economic changes on consumers’ consumption decisions, producers’ supply decisions, and the market itself. We call the analysis of such changes comparative statics: the analysis of how equilibrium prices and quantities change when other exogenous variables—variables that shift demand and supply curves—change. The term static emphasizes the fact that we are comparing two different market equilibriums at discrete points in time—one before and one after the changes occur—as opposed to analyzing the dynamic process of price and quantity changes.

Changes in Demand

Since we know where supply and demand curves come from, we know precisely what can cause them to shift. Let’s start with demand. Recall the consumer choice problem: consumers want to maximize their individual utilities by choosing bundles of goods available to them in their budget sets. How do these bundles change? Price is one of the main mechanisms, but this is accounted for in the curve itself. As price falls, demand increases and vice versa—this is what the slope of the demand curve represents. But how do other factors affect demand?

• A change in income.
• It will shift the budget line and cause changes in quantities demanded. For example, if a tax refund causes consumers to demand more automobiles, the demand curve shifts to the right.
• Changes in preferences.
• They can alter the ultimate quantity choice. For example, a new study that finds negative health effects of drinking coffee could lower the demand for coffee, which would be reflected in a shift of the demand curve to the left.
• Changes in the prices of goods that are substitutes or complements.
• This can also affect the demand for a good. For example, if the price of Coca-Cola increases, the demand for Pepsi Cola might increase, causing a rightward shift in the demand curve.

Figure 11.1 shows the effect on price and quantity equilibriums when demand increases, and figure 11.2 shows the effect when demand decreases.

Figure 11.1 Increase in demand causes equilibrium price and quantity to rise.

Figure 11.2 Decrease in demand causes equilibrium price and quantity to fall

Changes in Supply

Shifts in supply arise from changes in the following:

• The cost of production for the firms.
• For example, autoworkers in the United States represented by a union might negotiate a new labor contract for higher wages. This represents a cost increase on the part of firms and will result in a leftward shift of the supply curve.
• Technology that affects the production function.
• For example, a new robotic technology that aids in the assembly of automobiles in the factory could improve the efficiency of the plant, resulting in a rightward shift of the supply curve.
• The number of sellers in the market.
• For example, perhaps a new company, like Tesla, decides to begin to manufacture cars in the United States, or a foreign company, like FIAT, begins to sell in the US market—either way, the number of sellers has increased, and the number of cars for sale is likely to increase as well, leading to a rightward shift in the supply curve.
• The presence of alternative markets for sellers.
• For example, an economic boom in Canada might cause US auto manufacturers to send more of the cars they produce to Canada instead of selling them in the United States, leading to a lower supply of cars in the US market represented by a leftward shift in the supply curve.

Figure 11.3 shows the effect on price and quantity equilibriums when supply decreases, and figure 11.4 shows the effect when supply increases.

Figure 11.3 Decrease in supply causes equilibrium price to rise and quantity to fall

Figure 11.4 Increase in supply causes equilibrium price to fall and quantity to rise.

Changes in Both Supply and Demand

To see what happens when both supply and demand change, let’s consider an example of each individually and then combined. On the demand side, a tax refund increases consumers’ incomes and results in an increase in demand for automobiles. The rightward shift in the demand curve will lead to a new equilibrium price and quantity. Compared to the original equilibrium price and quantity, the new equilibrium is characterized by higher prices and greater quantity.

On the supply side, we might have a situation of increased productivity resulting from the introduction of new robotic technology in automobile manufacturing. This would shift the supply curve to the right. Compared to the old equilibrium price and quantity, the new equilibrium price is lower and the new equilibrium quantity is greater.

What if both changes occurred at the same time? Both the demand curve and the supply curve would shift to the right. The effect on equilibrium quantity is clear: the new quantity would be greater than the original quantity, as both shifts move the quantity in the same direction. The effect on equilibrium price is ambiguous, since the demand shift puts upward pressure on price and the supply shift puts downward pressure on price. In the end, the actual change is the net of these two effects and will depend on the shape of the two curves and the magnitude of the two shifts. Figure 11.5 illustrates the situation where price rises, and figure 11.6 illustrates the situation where price falls.

Figure 11.5 Increase in demand and supply that causes equilibrium price and quantity to rise

Figure 11.6 Increase in demand and supply that causes equilibrium quantity to rise and price to fall

Similarly, for a simultaneous demand and supply decrease, quantity would unambiguously fall, but the effect on price will depend on the magnitude of the shifts, so without more information, we cannot predict the direction of the change.

What happens when supply and demand changes are in opposite directions? Figure 11.7 illustrates the situation when demand increases and supply falls. In this situation, the change in price is unambiguous—it will rise—but the effect on quantity depends on the relative magnitude of the shifts in the curves. Quantity can rise, fall, or remain unchanged. In figure 11.7, the quantity increases slightly.

Figure 11.7 Increase in demand and decrease in supply cause equilibrium price to rise and have an ambiguous effect on quantity

To summarize, when one curve moves and the other doesn’t, we can make predictions about the direction of the change in both price and quantity. When both curves move, we can make predictions about the direction of the change in one but not of both price and quantity. Table 11.1 summarizes the effects on price and quantity for changes in demand and supply.

Calculating and Comparing the New Market Equilibrium

So far, we have not considered the magnitude of the changes in equilibrium price and equilibrium quantity due to shifts in supply and demand. Without knowing the specific demand and supply functions, it is impossible to determine the precise magnitudes. With specific supply and demand functions, however, we can perform comparative statics analyses by solving for the original and the new equilibrium price and quantity and comparing them.

Let’s return to the demand and supply functions from chapter 10:

$Q_1^D=1,800-20P$

$Q_1^S=50P-1,000$

In chapter 10, we solved these and found the equilibrium price of $40 and the equilibrium quantity of one thousand.

Now suppose two things happen. One, some new information causes demand to increase. The new, higher demand is now described by the demand function:

$Q_2^P=2,400-20P$

Two, a new manufacturing process increases the productivity of firms, resulting in an increase in supply. The new increased supply is described by the supply function:

$Q_2^S=50P-400$

Solving for the new equilibrium, we get

$2,400-20P=50P-400$

$2,800=70P$

$P_2^S=\$40$

$Q_2^{\ast }=1,600$

Comparing these to the original equilibrium price and quantity reveals that quantity increased by six hundred and that price remained at $40. From table 11.1, we expected the quantity to increase, but we could not predict the direction of the change in price. For this specific example, we see that price has not changed. The graphical model is useful in making predictions about the directions of some equilibrium price and quantity changes, but we need a specific model in order to pin down the magnitude of changes in both price and quantity.

11.2 WELFARE ANALYSIS

Learning Objective 11.2: Apply a comparative static analysis to evaluate economic welfare, including the effect of government revenues.

We can apply the principles of comparative static analysis to measuring economic welfare. In chapter 10, we looked at welfare in terms of consumer surplus, producer surplus, and their combination, total surplus. For purposes of evaluating overall economic welfare, the total surplus is what economists care about. So we measure the effect of changes in supply and demand on welfare by comparing the total surplus before and after the change.

Figure 11.8 illustrates the change in welfare from an increase in demand. Increased demand leads to more transactions and to more consumer and producer welfare from all transactions. The blue-shaded area shows the net increase in welfare that results from the increased demand.

Figure 11.8 Increase in total welfare from an increase in demand

Predicting the effect on total welfare is straightforward when one curve shifts but complicated when both curves shift. Any increase in only supply or demand will increase total welfare, and any decrease in only supply or demand will decrease total welfare. It is also easy to see that welfare will increase when both supply and demand increase and will decrease when both supply and demand decrease. But what happens when one increases and the other decreases?

Figure 11.9 shows a situation where demand increases at the same time that supply decreases. The blue triangle represents the original total welfare, the red triangle represents the new total welfare, and the purple triangle represents neither a gain nor a loss in welfare. Whether there is a net gain or loss in welfare depends on the relative sizes of the areas marked “gain in welfare” and “loss in welfare.” As is easy to see, whether this is a net loss or gain depends on the magnitude and position of the shifting curves and can easily go either way.

Figure 11.9 Welfare effects of shifts in both curves

Sometimes the government takes actions that lead to positive government revenue (e.g., imposing a tax) or negative government revenue (e.g., spending money). Economists account for government revenues or expenditures by including them in the total surplus calculations. Government revenues are public resources and spending is with public money, so both should be accounted for the same way consumer and producer surpluses are—they are all part of the societal gains or losses that we consider in total surplus:

$TotalSurplus=ConsumerSurplus+ProducerSurplus+GovernmentRevenue$

Note that government revenue can be positive (tax receipts) or negative (government spending).

11.3 PRICE CEILINGS AND FLOORS

Learning Objective 11.3: Show the market and welfare effects of price ceilings and floors in a comparative statics analysis.

Price ceilings and price floors are artificial constraints that hold prices below and above, respectively, their free-market levels. Price ceilings and floors are created by extra-market forces, usually the government. A classic example of a price ceiling is a rent-control law like those that exist in New York City. Figure 11.10 illustrates the effect of a price ceiling in a market for rental housing. The price ceiling holds prices below the market equilibrium price, and there are more consumers wishing to rent apartments at the ceiling price than there are rental units available. The result is excess demand for rental housing.

Market equilibrium price is sometimes referred to as the market-clearing price. This term references the fact that the market is cleared of all unsatisfied demand and excess supply at the equilibrium price.

Figure 11.10 Welfare effects of a price ceiling

The effect of a rental price ceiling on welfare is clear. Fewer apartments will be rented than at the market equilibrium price, and all the surplus that would have been created by those rentals is not realized, resulting in deadweight loss. Of the total surplus that is created, a greater proportion accrues to the consumers than the producers, but in welfare terms, only the total surplus matters. So price ceilings do two things:

1. They lower total surplus and create deadweight loss.
2. By creating excess demand, they create winners and losers among consumers. In the case of a rental price ceiling, some consumers are lucky enough to find an apartment to rent at the ceiling price, while others cannot.

The consumer surplus shown in figure 11.10 assumes that the market will somehow allocate the apartments to the users whose valuation of them is the highest (those that represent the highest points on the demand curve). But no mechanism exists to assure that this will happen. Instead, there is likely to be a random assignment of apartments among those willing to pay the ceiling price. This means that figure 11.10 likely overstates the resulting consumer surplus and understates the deadweight loss. We will continue with the assumption of efficient allocation of apartments to keep the analysis simple, but it is important to understand the consequences of this assumption.

Figure 11.11 illustrates the situation for a price floor. When the price of a good is not allowed to sink to its market equilibrium level, a situation of excess supply occurs where producers would like to make and sell more goods than customers would like to buy. The price floor creates a deadweight loss in the same way a price ceiling does: it limits the number of goods that are bought and sold and therefore limits the amount of surplus created relative to the potential surplus. In the case of the price floor, more of the surplus that is created is in the form of producer surplus than consumer surplus. An example of a price floor might be the regulated fares for taxis in many cities. The success of Uber and Lyft, so-called ride-sharing services, suggests that there is indeed an excess supply of potential taxis that would be quite happy to give rides for less.

This analysis assumes that firms with the lowest marginal cost supply the good, but there is nothing in the market that ensures this. So the true welfare effects are likely to be lower producer surplus and higher deadweight loss than depicted in figure 11.11.

Figure 11.11 Welfare effects of a price floor

For both price ceilings and price floors, the welfare impact is clear: there is a reduction in total surplus relative to the market equilibrium price and quantity.

11.4 TAXES AND SUBSIDIES

Learning Objective 11.4: Show the market and welfare effects of taxes and subsidies in a comparative statics analysis.

Governments levy taxes to raise revenues in many areas. Governments at all levels—national, state, county, municipality—tax things such as income, hotel rooms, purchases of consumer goods, and so on. They tax both producers of goods and consumers of goods. Governments also subsidize things, such as the production of dairy goods or the purchase of electric cars. In this section, we will explore how taxes and subsidies affect the supply-and-demand model and the impact of supply and demand on welfare. We will discover that it does not matter upon whom you levy a tax or to whom you provide a subsidy, producers or consumers; the burden or benefits will end up being shared among producers and consumers in identical proportions.

Taxes on Sellers

To examine the effects of a tax on a market, let’s perform a comparative static analysis. We will start with a market without a tax and then compare it to the same market with a tax. Let’s consider the market for tomatoes at the farmers’ market in Lawrence, Kansas. The current equilibrium price is $1.00 per tomato, and the equilibrium quantity is five hundred. Now suppose that the city of Lawrence decides it needs to raise revenues to support the improvement of infrastructure at the market and imposes a sales tax on tomatoes to do so. What would be the impact of this tax on the market?

A sales tax on tomatoes can be imposed on either buyers or sellers. Generally, sales taxes are collected by sellers and then remitted to the government. This means that for a $1 tomato that has a 10 percent tax, the seller collects both the $1 and the $0.10 for the government. We call this type of tax an ad valorem tax because it depends on the value of the good itself. Alternatively, the government could have imposed a set amount, like $0.20, on one tomato, regardless of the price of the tomato itself—this is known as a specific tax. For this analysis, we will use a specific tax because it is easier to analyze, but either way, the analysis is similar. So let’s consider a $0.20 tax on each tomato to be paid by sellers. Since the seller collects the tax, we can illustrate its effect on the market through the supply curve.

Consider any single point on the supply curve. This point is the seller’s willingness-to-accept price for a tomato at a certain supply quantity. Suppose this price is $0.50 at a given point. If the government imposes a $0.20 tax on each tomato, the seller’s new willingness-to-accept price will rise to $0.70. This price is the sum of the original willingness-to-accept price and the $0.20 that the seller will collect from the buyer and give to the government. The same logic applies to every point on the supply curve, so the tax has the effect of creating a new supply curve shifted upward by $0.20 from the original supply curve. Figure 11.12 shows the original supply curve ($S_1$) and the shifted supply curve ($S_2$).

Figure 11.12 Effect of a tax on sellers of tomatoes at the Lawrence farmers’ market

The original equilibrium price of a tomato is $1. With the new tax, there is a difference between what consumers pay and what producers receive—the difference being the $0.20 tax per tomato. So equilibrium price is no longer a single value at which quantity supplied is equal to quantity demanded. Now in order for the market to be in equilibrium, the quantity demanded at the price consumers pay, which includes the tax, must equal the quantity supplied at the amount the sellers receive after the government takes out the tax.

Figure 11.12 shows the new equilibrium. For now, don’t worry about where the prices and quantities come from; we will take them as given. In figure 11.12, the new equilibrium quantity is shown as 450, the price paid by consumers is $1.10, and the price received by producers is $0.90. The consumer surplus was $A+B+E$ but is now only A, and the producer surplus used to be $C+D+F$ but is now only D. The revenue generated by the government from the tax on tomatoes is the area $B+C$. The tax creates a deadweight loss from the reduction in sales, and the deadweight loss is $E+F$. Total surplus includes consumer surplus, producer surplus, and government revenue: $A+B+C+D$. What policy makers must decide in general is whether the objectives achieved by the tax are worth the loss in efficiency represented by the deadweight loss, $E+F$.

Taxes on Buyers

Suppose the city of Lawrence decided to collect the $0.20 tax from consumers as they left the farmers’ market. For example, there might be a person with whom you have to check out as you leave, and this person counts your tomatoes and charges you the $0.20 per tomato. How does this scenario change the graphical analysis?

Since the value to the consumer of a tomato has not changed, the willingness to pay remains the same. Thus a consumer that was willing to pay $1 for a tomato is still willing to do so, but $0.20 of that $1 now goes to the city. So in essence, the consumer was willing to pay the farmer $1 for the tomato but is now only willing to pay the farmer $0.80. This is represented by the blue demand curve ($D_2$) in figure 11.13, which is the same as the previous demand curve but lowered by $0.20, or the amount of the tax.

Figure 11.13 Effect of a tax on buyers of tomatoes at the Lawrence farmers’ market

The effect on the new market equilibrium of a tax on buyers is identical to the effect of the tax on sellers. The tax itself creates a wedge between what buyers pay and what sellers receive, and the new equilibrium quantity is the same as before. Consumer surplus, producer surplus, government revenue, and deadweight loss are all the same as before. The lesson here is that it makes no difference on whom the government levies the tax; the tax does not stay where the government puts it. In this example, whether the tax is applied to the consumer or the seller, the consumer pays $0.10 more than without the tax, and the seller receives $0.10 less than without the tax.

Distribution of the Tax Amount

In our tomato tax example, the tax amount is shared equally between the sellers and buyers ($0.10 each), but is this always true? The answer is no. How much of the tax burden falls on buyers and sellers depends on the elasticities of the supply and demand curves. The following figures illustrate this concept. Figure 11.14 shows a relatively elastic demand curve with a relatively inelastic supply curve. The original pre-tax equilibrium price is $P_1$, the post-tax price buyers pay is $P_B$, and the post-tax price sellers receive is $P_S$. In this case, the greater share of the burden of the tax falls on sellers, as can be seen by the fact that $P_1-P_S>P_B-P_1$. We call the division of the burden of a tax on buyers and sellers the tax incidence.

Figure 11.14 Tax incidence with inelastic supply and elastic demand

In figure 11.15, the supply curve is relatively elastic, the demand curve is relatively inelastic, and the majority of the tax incidence falls on the buyers, as can be seen by the fact that $P_B-P_1>P_1-P_S$.

Figure 11.15 Tax incidence with elastic supply and inelastic demand

These tax-sharing effects make sense intuitively if we think about the meaning of elasticity. A more elastic curve means a larger quantity response to a change in price. Only a small part of the tax burden can be given to market participants that are more responsive to price and a larger part can be given to those that are less responsive to price because the quantity demanded and supplied must equal in the end.

Subsidies for Buyers

The effects of subsidies on markets are similar to taxes in that they create a difference between what consumers pay and what suppliers receive. They are also similar in that it does not matter if you subsidize the purchase of a good or the sale of a good; the market equilibrium effects are the same. In the next section, we’ll consider in more detail the market equilibrium effects of a government policy to subsidize buyers of solar power installations. For now, we’ll examine a simpler example to understand the mechanics of such a subsidy.

Let’s go back to the Lawrence farmers’ market. Suppose the city government decides that it is important to support the tomato growers. It will subsidize consumers in the amount of $0.20 for each tomato they purchase. Suddenly, potential buyers who would have bought a tomato if the price was $1 are now willing to buy a tomato that is priced at $1.20 because their out-of-pocket cost is the same. We can see this effect in figure 11.16. In the figure, the demand curve is shifted up by the amount of the subsidy leading to an increased quantity purchased, $Q_2$, and an increased equilibrium price, $P_S$.

Figure 11.16 Effect of a subsidy on tomatoes at the Lawrence farmers’ market paid to buyers

Tomato sellers are certainly helped by this policy. Producer surplus was $C+D$, and it is now $B+C+D+E$, so it has increased by $B+E$. Consumer surplus has also increased. Originally it was $A+B$. After the subsidy, it is $A+B+C+F+G$, so it has increased by $C+F+G$. To see this, remember that consumers are actually spending $P_C$ on each tomato. So both consumer and producer surpluses have increased.

The government spends $0.20 on each tomato sold, and $Q_2$ are sold, so the cost to the government of this policy is shown as the area $B+C+E+F+G+H$.

And how does welfare change with the implementation of the subsidy? Within the area of government expense, $B+E$ is the new producer surplus, and $C+F+G$ is the new consumer surplus. That leaves only the area $H$ as government expenditure not offset by the new surplus. Therefore, $H$ is the area of deadweight loss, and net welfare has decreased by $H$.

Subsidies for Sellers

Suppose instead of giving $0.20 to buyers of tomatoes, the government gives it to the sellers of tomatoes. This subsidy is shown in figure 11.17 where the supply curve is shifted down by the amount of the subsidy. In this case, sellers who would have been willing to accept $0.70 for a tomato will now accept $0.50 for the same tomato because the government will give them the extra $0.20. The effect on the market with respect to consumer surplus, producer surplus, government expenditure, and deadweight loss is identical to the case where the subsidy is paid to the buyers.

Figure 11.17 Effect of a subsidy on tomatoes at the Lawrence farmers’ market paid to sellers

In fact, for both forms of subsidies, the true benefit is not the $0.20 paid by the government but the price paid by consumers relative to the price without the subsidy. So the consumer benefit is $P_1-P_C$. Similarly, the producer benefit is the difference between the new post-subsidy price and the pre-subsidy price, or $P_S-P_1$. The distribution of these benefits depends on the relative elasticities of the two curves, just as we saw with taxes.

11.5 THE POLICY QUESTION
SHOULD THE FEDERAL GOVERNMENT SUBSIDIZE SOLAR POWER INSTALLATIONS?

Learning Objective 11.5: Apply a comparative static analysis to evaluate government subsidies of solar panel installations.

The SITC is a 30 percent federal tax credit to buyers of solar systems for residential and commercial properties. This credit reduces the federal income taxes that a person or company pays dollar for dollar based on the amount of investment in solar property.

We know from section 11.4 that the impact of a subsidy on a market does not depend on the designated receiver of the subsidy. In this case, the subsidy is paid to buyers of solar panels, both residential and commercial. What we need to properly analyze in this market is some notion of elasticity of the demand and supply for solar panels.

In general, studies have found residential energy usage to be fairly inelastic. This is because measures that consumers can take to save energy in response to price increases—like lowering the thermostat in winter and raising it in summer, turning off lights diligently, and using more energy-efficient lightbulbs—can only mitigate usage somewhat.

If this logic carries over to the solar panel market, the effect of the subsidy would look like figure 11.18. In this figure, the consumer surplus increases $C+F+G$, and the producer surplus increases $B+E$. It is clear from the figure that the larger share of the benefits from this policy accrues to consumers, but there is still a fair amount of deadweight loss, $H$. If the goal of the policy is to increase the consumption of solar panels, it has clearly succeeded, shifting consumption from $Q_1$ to $Q_2$.

Figure 11.18 Effect of the solar investment tax credit (SITC) program on the market for solar panels with inelastic demand

However, it is probably not the case that the market for solar panels is the same as the market for residential energy. This is because solar panels represent only one form of energy, and close substitutes such as electricity delivered from the power plant, natural gas, and propane exist and are easily accessible for most homeowners. So it is quite likely that consumers have demands for solar panels that are quite elastic, as small movements in price could alter the comparative calculus between solar and other forms of energy a lot. If this is true, then our market would look a lot more like figure 11.19.

While similar to figure 11.18, figure 11.19 is different in one crucial aspect. Our subsidy is likely to increase consumption more with an elastic demand curve than with an inelastic demand curve. It is also true that the increases in consumer and producer surplus are more equal and that there is still substantial deadweight loss.

Figure 11.19 Effect of the solar investment tax credit (SITC) program on the market for solar panels with elastic demand

Our analysis shows that the SITC is likely to be successful in increasing the consumption of solar panels, but it will also be quite costly and create a lot of deadweight loss. Is it worth it? To answer this question, policy makers have to evaluate the benefit to society of increased reliance on solar power and compare it to the cost of the program in terms of both accounting costs ($B+C+E+F+G+H$) and deadweight loss $\left(H\right)$.

EXPLORING THE POLICY QUESTION

1. What other ways could the government sponsor the installation of solar energy systems, and how would their effects differ from a subsidy to buyers?
2. Do you support the SITC subsidy? Why or why not?

Results of adjustments to demand and supply graphs in a grid for comparison

https://cutt.ly/N7X5eCm

Welfare economics I: Efficiency and optimal allocation

 

Pareto efficiency

This efficiency criterion was developed by Vilfredo Pareto in his book “Manual of Political Economy”, 1906. An allocation of goods is Pareto optimal when there is no possibility of redistribution in a way where at least one individual would be better off while no other individual ends up worse off.

A definition can also be made in two steps:

-a change from situation A to B is a Pareto improvement if at least one individual is better off without making other individuals worse off;

-B is Pareto optimal if there is no possible Pareto improvement.

This can be easily understood using an Edgeworth box. Starting from point C, two Pareto improvements can be made:

-from C to D: individual 1 would increase its utility, since a further indifference curve would be reached, while individual 2 will remain with the same utility;

-from C to E: individual 2 would maintain its utility while individual 2 increases theirs.

Once we are at point either D or E, no further Pareto improvements can be made. Therefore, D and E are Pareto optimal.

Following the same steps for every indifference curve, we can say that every point in which indifference curves from different individuals are tangent is Pareto optimal. The curve that links these infinite Pareto optima is called the contract curve.

Edgeworth box

In 1881, Francis Y. Edgeworth came up with a way of representing, using the same axis, indifference curves and the corresponding contract curve in his book “Mathematical Psychics: an Essay on the Application of Mathematics to the Moral Sciences”. However, the representation given, using as an example the work being done by Friday and wages being paid by Robinson Crusoe, was not the one we commonly know nowadays.

It was Vilfredo Pareto, in his book “Manual of Political Economy”, 1906, who developed Edgeworth’s ideas into a more understandable and simpler diagram, which today we call the Edgeworth box.

This diagram is widely used in welfare economics, game theory or general equilibrium theory, to name a few. It is easy to draw and can be easily explained. In the adjacent image, we can see two examples of an Edgeworth box, and how it is drawn.

The first example is mainly used for welfare economics and distribution matters. As we see, this “box” is formed using two sets of typical indifference maps, which in this case represent the indifference curves of agents A (green) and B (red), who must choose quantities of goods x and y. When the indifference map of agent B is rotated, and put on top of the map of agent A, the box is formed. When indifference curves are tangent to each other, which is the case in this example, a contract curve (blue) can be drawn using these tangency points.

Our second example is mainly used to explain Ricardian trade theory graphically. In this case, we draw the production-possibility frontier for countries 1 (green) and 2 (red). When we rotate the diagram of country 2, we end up with an Edgeworth box, which here will help understand how great the gains of international trade are and therefore helps illustrate how trade is not a sum zero game.

Video – Edgeworth box:

Production possibility frontier

The production possibility frontier (PPF) represents the quantity of output that can be obtained for a certain quantity of inputs using a given technology. Depending on the technology, the PPF will have a certain shape.

As you can see on the adjacent figure, this PPF (blue curve) slopes downwards. This slope, which equals the marginal rate of transformation between X and Y, shows us how, in order to increase the output X, the quantity of Y must decrease. In fact, the marginal rate of transformation measures the tradeoff of producing more X in terms of Y.

This frontier determines the maximum output (of both X and Y) that can be obtained given the technology. Production at point A will produce more quantity of Y and less of X than production at point B. However, both are technically efficient, since they maximize the output. For example, production at point C is technically inefficient because, at any point on the PPF, more combined output is produced using given the technology. Also, point D is unattainable given the technology, being this is the reason why it is outside the PPF.

The PPF can be derived from the contract curve on an Edgeworth box. In this box, we see the quantity of inputs (K, L) being used in the production of each good (X,Y). In fact, we can see how, for each quantity of each product, the quantity of each input can change. The isoquants (green curve for X, red for Y) determine how much a certain input has to increase in order to compensate the decrease in the other input, maintaining the quantity of output produced unaltered. The slope of these curves is given by the marginal rate of technical substitution of each output.

The points where the isoquants of different outputs combination intersect, which are Pareto-optimal, allow us to draw the contract curve, from which the PPF can be derived. Since the technology is given, only one PPF can be derived from the contract curve (as opposed to the case of the utility possibility frontier).

 

Video – Production possibility frontier:

General equilibrium

A market system is in competitive equilibrium when prices are set in such a way that the market clears, or in other words, demand and supply are equalised. At this competitive equilibrium, firms’ profits will necessarily have to be zero, because otherwise there will be new firms that, attracted by the profits, would enter the market increasing supply and pushing prices down. Following the first fundamental theorem of welfare economics, this equilibrium must be Pareto efficient. Both will have a fundamental relation as a mechanism for determining optimal production, consumption and exchange.

Initial approach:

Let’s consider an economy where there are:

Two factors of production: capital (K) and labour (L).

Two goods: good X and good Y.

Two agents: A and B

The economic problem that is faced needs to find the most adequate allocation of factors of production in order to produce goods X and Y and how these goods will be distributed amongst consumers A and B. This configuration will be such that there will be no other feasible configuration that will allow an increase in any individual’s welfare without decreasing the other individual’s welfare.

In order to achieve Pareto optimality, a certain set of assumptions need to be held.

-The production function needs to be continuous, differentiable, and strictly concave. This will result in a convex set of production possibilities, also known as production possibility frontier Its shape shows an increasing opportunity cost as we need to use a higher number of resources in order to produce a larger amount of a certain good.

-Consumers’ preferences need to be monotonic, convex and continuous, showing how individuals’ welfare increases with a greater amount of goods, but with a decreasing marginal utility.

–Perfect and free availability of information.

-There has to be an absence of externalities and public goods so the utility of individuals depends directly and uniquely from their possession of goods X and Y.

Production optimisation

The optimisation problem in production relies in the maximisation of total output production taking into consideration that it is subject to a limited amount of capital and labour. Analytically,

We can start by looking at the production of goods X and Y as two different optimisation problems. The firm will have to decide what quantity of capital and labour allocate to the production of good X, as shown on the left side of the diagram below, but also what quantity of capital and labour assign to the production of good Y, as shown on the right. These curves are the isoquants corresponding to each production process. 

These two diagrams can be plotted together using what is known as the Edgeworth box, which makes it easier to compare quantities of capital and labour used, while also comparing quantities of goods X and Y being produced. Indeed, it’s not only easier to analyse, but also makes more sense, since the total available quantities of capital and labour are given. 

The solution to this problem is related to the marginal rate of technical substitution (MRTS). A higher efficiency will be achieved if the reallocation of a unit of labour or capital from one good to another leads to a higher production of the former. When the marginal rate of technical substitution is equal for both goods, it means that all available inputs are being used, which translates into a purely efficient production process. 

Graphically, if we plot all these points we construct what is known as the contract curve (blue curve in the Edgeworth box). These represent all Pareto efficient distributions, such as F, G or H. I is not Pareto efficient, since going from I to either G or H would result in an increase in the production of one of the goods without giving up the production of the other.  From this curve we can derive the production possibility frontier, which shows the quantities of goods X and Y being produced, as shown in the following diagram. It must be noted that both the contract curve and its derivative, the production possibility frontier, show all the solutions that are Pareto efficient from the firm’s point of view. Only when considering input and output prices will we be able to determine a unique solution (because of the concavity of the production possibility frontier).

Consumption optimisation

Bundles of goods cannot be ranked in a reliable way without knowledge of the distribution of the products, especially if a bundle has different amounts of each good. There may be some bundles that have more products of a good but less of another. The optimisation problem will be to maximise the utility of individuals A and B subject to a limited total amount of goods X and Y. Analytically,

In this case we have to achieve the optimal distribution of two, already produced goods (X and Y) between two individuals (A and B). We can follow the same step by step method used before. Here, we’ll plot indifference curves corresponding to the amounts of goods X and Y consumed by A (on the left), and the amounts of goods consumed by B (on the right).

Again, we use the Edgeworth box to graph the different distributions that can be given between two individuals, A and B, and two goods, X and Y. The further the indifference curve is from the origin, the higher the level of utility enjoyed by the consumer. 

Although all the points in the graphic are feasible, not all are efficient, given the utilities and preferences of consumers. The indifference curves join all the points that give consumers the same level of utility. By connecting all points of tangency between the indifference curves of both individuals, the contract curve is constructed and represents all Pareto efficient allocations. The tangency between indifference curves is the point where both consumers have an equal marginal rate of substitution for goods X and Y, and are therefore not willing to trade between them, as it would result in a lower utility.

Global optimum

Until now we have only considered different parts of the economy, and not the economy as a whole. The optimisation problem faced this time is similar to the previous one, although this time an additional restriction is added, since we are here considering both production and consumption: the production level also needs to be efficient. 

As this optimisation problem is based on the previous one, we have the same marginal rate of substitution equalisation, but also these two must be equal to the marginal rate of transformation, the PPF’s slope, 

These solutions are multiple, since there are various points where the condition holds. However, if we consider output prices (given by the consideration of input prices mentioned before), we are able to consider a unique solution. In the adjacent diagram, if output prices were to be PX and PY, the equilibrium would be point E. However, if output prices were instead P’X and P’Y, the equilibrium would be point E’.

Let’s say that prices are set at PX and PY, and that the equilibrium point is E, as seen in the diagram below. Consumers A and B will consume both goods X and Y in different amounts. These amounts are given by the equilibrium in consumption, point E on the contract curve. We have also equilibrium in the production process, given by point E on the production possibility frontier. We know this is a general equilibrium because the marginal rate of substitution is equal to the marginal rate of transformation; or, in other words, the slopes of the indifference curves are equal to the slopes of the production possibility frontier.

Competitive markets result in an equilibrium position such that it is not possible to make a change in the allocation without making someone else worse-off. In reality there are many Pareto optimums and we cannot state that one is better than the other. Even if one consumer got all of the production and the other one none, we cannot say it is an inefficient distribution if all resources are being used efficiently. This is the reason why some economists believe it is an incomplete criterion. However, there are others, such as Milton Friedman and the advocates of the Chicago School, for whom this proves that the economy will act efficiently without the need of government intervention.

Fundamental theorems

There are two fundamental theorems of welfare economics.

 

-First fundamental theorem of welfare economics (also known as the “Invisible Hand Theorem”):

any competitive equilibrium leads to a Pareto efficient allocation of resources.

The main idea here is that markets lead to social optimum. Thus, no intervention of the government is required, and it should adopt only “laissez faire” policies. However, those who support government intervention say that the assumptions needed in order for this theorem to work, are rarely seen in real life.

It must be noted that a situation where someone holds every good and the rest of the population holds none, is a Pareto efficient distribution. However, this situation can hardly be considered as perfect under any welfare definition. The second theorem allows a more reliable definition of welfare

 

-Second fundamental theorem of welfare economics:

any efficient allocation can be attained by a competitive equilibrium, given the market mechanisms leading to redistribution.

This theorem is important because it allows for a separation of efficiency and distribution matters. Those supporting government intervention will ask for wealth redistribution policies.

https://cutt.ly/B7XTS8D