Applications of Consumer Theory

1. IntroductionWe will apply the theory of the consumer to a number of examples and focus on how the consumer responds to the incentives they face. Many of these situations have public policy implications. Which tax system is better, the so-called flat tax or the system we have now that is mildly progressive? Should we tax consumption or income? How has social security affected savings and labor supply behavior? Do people save less because of social security? Do people retire earlier than they otherwise would have because of social security? How does Medicare affect the demand for health care? Does the minimum wage cause unemployment? Will consumers always choose the product with the lowest price? The answer to these questions relies on our theory of the consumer.

Our strategy is the following. In the last chapter we studied an abstract model involving two goods, x1 and x2, and we assumed the consumer had utility for the two goods and confronted a budget constraint involving the two goods. We will redefine the variables in the applications that follow, interpret one of the goods as x1 and the other as x2, and then use what we learned in the previous chapter about the abstract "x1 – x2 model." Everything in the last chapter will be relevant. For example, if the consumer receives more income, we can map out an Engel curve for the two goods, regardless of what the two goods are.

2. Differential TaxationSuppose there are two goods, "food" and "everything else." We can match up this new situation perfectly with the abstract "x1 – x2 model." Let F be the quantity of food consumed, let E be everything else, let I be income, and suppose the prices of food and everything else are Pf and Pe. Replace x1 in the abstract model with F, replace x2 with E, and so on. Utility becomes U(F, E) and the budget constraint is I = PfF + PeE, where we have also replaced the relevant prices. Put F on the horizontal axis in place of x1 and put E on the vertical axis in place of x2. It follows that the graph of the budget line is the same as in the last chapter. The intercepts are found in exactly the same way. The horizontal intercept is I/p1 = I/Pf and the vertical intercept is I/p2 = I/PE. These intercepts are depicted in the figure below. The budget line is simply the line connecting the two intercepts. Finally, the indifference curves have the "usual" shape and one is depicted in the figure. The consumer chooses a consumption bundle of food and everything else to maximize his utility subject to his budget and point A is the bundle that satisfies this decision problem. No other bundle is affordable and provides greater utility.

Now suppose the government imposes a tax on everything else while food is exempt.1 How will exempting food from taxation affect the consumer's behavior? The tax raises the price of the taxed items, everything but food, relative to food. Only the vertical intercept is affected by the tax; it will decrease because the price of "everything else" in the denominator has gone up by the tax. The budget line will swivel inward as depicted on the right.

Next, how will the consumer alter his consumption bundle? There are several possibilities. Suppose the consumer chooses point C after the tax. Then consumption of food is unaffected by the tax while consumption of everything else falls as the consumer moves from

1 The state of WA has typically not taxed food, or income, but taxes durable goods under the sales tax. Income and food expenditures tend to be fairly stable when state GDP fluctuates. However, the demand for durables is very unstable. In a recession, demand for durables like cars, refrigerators, and LCD television sets, drops off, while food expenditures and income remain stable. This can cause a much larger drop in tax revenues than if we taxed income or food.

1

point A before the tax to point C after the tax. Suppose instead the consumer chooses point B after the tax. As the consumer moves from point A to B, consumption of everything else is unaffected by the tax while food consumption falls. This is somewhat paradoxical; the tax doesn't affect the items that are taxed, only the item that is not taxed. A third possibility is that the consumer chooses a point somewhere in between points B and C on the budget line, in which case his consumption of both food and everything else falls with the imposition of the tax. In this case, F and E are complements; the price of F has gone up and both E and F fall in response.

There are two other cases. The consumer could choose a point on the budget line to the northwest of B. In that case, food consumption falls a lot while the consumption of everything else actually increases in response to the tax on everything else. This would violate the law of demand; the price of everything else has risen by the tax and the consumption of everything else has also increased. Finally, the consumer could choose a point to the southeast of C. In that case, food consumption increases while the consumption of everything else falls a lot in response to the tax. In this case, F and E are substitutes.

Also, notice that regardless of where the consumer chooses to be on the new budget line, he will achieve a lower level of utility after the tax. So the consumer will be worse off because of the tax. This might explain why many people do not want to pay taxes. Of course, we have ignored the use to which the tax dollars are put. In a more general model, the consumer may care about government spending according to U(F, E, G), where G is government spending on roads, bridges, highways, schools, health care, nuclear submarines, and so on. If the consumer gains enough utility from G, they may be willing to pay his "fair share" of taxes.

Example: Tax on gasoline. Many governments impose taxes on gasoline especially in Europe. Typically, such taxes are earmarked for building new roads, bridges, and highways. In addition, some people have advocated imposing a large tax on gasoline for environmental purposes. A tax on gasoline would raise the price of gasoline relative to other goods and this would cause people to drive less, carpool, walk, ride their bike to work, and so on, and thus conserve gasoline and help the environment. There is evidence that the heavy taxation of gasoline in Europe has caused Europeans to conserve on gasoline and use mass transit.

Example: National sales tax. Suppose food and everything else is taxed at the same rate, 1+t, where t is the percent tax rate, e.g., t = 5%. This would be akin to a sales tax on all purchases and is equivalent in its impact on the budget constraint to an income tax. To see this, consider the following analysis. The budget constraint changes from I = pfF + peE without the tax to I = (1+t)pfF + (1+t)peE with the tax. Factor to get I = (1+t)(pfF + peE), and finally, divide both sides by 1+t, I/(1+t) = pfF + peE. Notice that 1/(1+t) < 1. Define 1 - T = 1/(1+t). In that case the

2

budget constraint becomes (1 - T)I = pfF + peE under an income tax, and T is the income tax rate. So a national sales tax imposed on all purchases is "like" an income tax. Such a tax is used by many governments in Europe, known as the VAT or value added tax.2

3. Are Children an "Inferior" Good?We can ask how parents spend their income on the number of children and the education of each child. Suppose parents care about two "goods," the number of children they have (N) and the education they provide each child (E). Utility is given by U(N, E) and we can put N on the horizontal axis replacing x1 and E on the vertical axis replacing x2. The budget constraint is I = PNN + PEE, where PN is the "price" of producing one child and PE is the "price" of educating the child per unit of education.

In the diagram on the left are several indifference curves and budget lines and points A, B, C, and D represent tangency points. Suppose the parents start off poor and choose point A as the best they can do. What happens if the parents receive more income? The budget line shifts out in a parallel manner as income increases and they choose a new point, B. Clearly, both the number of children and the amount of education per child increases. However, as their income continues to increase, eventually, they choose to produce fewer children but to educate them more. As society evolves from B to D, children per family decreases while education per family increases and it appears that parents are substituting more educated children for the quantity of children. The Engel curve for education is upward sloping; education is a normal good. However, the Engel curve for the "quantity of children" is first upward sloping, indicating a normal good, but then eventually turns downward at high levels of income the "quantity of children" is an "inferior" good.

Alternatively, we can compare two economies that are alike in every respect except one has more income than the other. A poor society might be at point A, while a rich society might be at point D. This behavior tends to negate Malthus' theory of population. Recall that Malthus argued that an increase in prosperity would cause people to have more children who would eventually compete in the labor market and drive wages down because of diminishing marginal productivity. As it turns out, his theory was wrong. Eventually, as economic growth progressed, parents in the US, Japan, and Europe began having fewer children per family but tended to educate them more. This is the so-called “demographic transition” from high population growth to low population growth that most advanced countries went though.

2 For example, suppose t = 5%. Then, 1-T = 1/1.05 = 0.95238. Solve for T: T = 1 - 0.95238 = .04762. So a sales tax rate of 5% is equivalent to an income tax rate of 4.762%.

3

5. Saving BehaviorNext, we will consider an extended example involving consumption over time. People consume over time within the context of their life cycle. How is this behavior affected by their labor income, interest rates, or public policies like social security?

Consider someone who lives for two periods, now and in the future. Label periods #1 for the present and #2 for the future. Let w = lifetime labor earnings, c1 is consumption now, and c2 is consumption in the future. Assume the individual has utility for current and future consumption, U = U(c1, c2). We can reinterpret x1 from our earlier model as c1 in this new model and x2 as c2. Everything we know about the x1-x2 set-up applies to this new model. So, for example, we can draw the following indifference curve picture. We will assume that the consumer recognizes that tradeoffs exist between present and future consumption and that more is preferred to less so the consumer always wants to move in the direction of the arrow.

There is also a budget constraint. Actually, since there are two periods we start off with two budget constraints, one for each period. We will assume that the consumer only works in the first period of life and then when the future arrives is retired. Let w be lifetime labor earnings, r be the interest rate, and s be savings. The first period budget constraint is

w = c1 + s.This says that the consumer can spend her income on consumption now (c1) or she can save it for the future (s). In the second period she receives her savings principal plus interest and then spends it on future consumption,

s + rs = c2,where r is the interest rate. After a few steps of algebra we can derive the lifetime budget constraint,3

w = c1 + (1/(1+r)) c2.Lifetime income is on the left and the present value of her consumption is on the right hand side.

We can match this model up with the x1 - x2 model in the following way,w = 1 c1 + (1/(1+r)) c2I = p1 x1 + p2 x2,

where the variables match up as : w = I, 1 = p1, c1 = x1, 1/(1+r) = p2, and c2 = x2. So 1/(1+r) could be considered the "price of future consumption." An increase in the interest rate lowers the price of future consumption because your saving income is worth more.

3 The second constraint can be written as (1+r)s = c2, or s = c2/(1+r). Solve the first constraint for s, s = w - c1, and substitute into the second constraint for s to get,

w - c1 = c2/(1+r),and take c1 to the other side to obtain the lifetime constraint.

4

We can apply everything we know about the x1-x2 model to this new situation. So the graph of the budget equation is a lifetime budget line and is depicted below.

The two intercepts are given by I/p1 = w/1 = w and I/p2 = w/(1/(1+r)) = w(1+r).

An increase in w is exactly the same as an increase in I in the x1 – x2 model. Both intercepts increase when w increases so an increase in w causes the budget line to shift out in a parallel way as depicted above on the left. What happens if r changes? Suppose r increases. Then the budget line will swivel outward pivoting about the horizontal axis as depicted above on the right.

If we put the indifference map in the same diagram as the budget line we can obtain the consumer's optimal choice of consumption today and in the future. We can also obtain savings from the diagram by subtracting c1 from w on the horizontal axis. To see this suppose w = $100 and c1 = $75. Then s = w - c1 = $25, as in the diagram on the left below. The distance between w and c1 is savings.

Savings will also respond to changes in labor earnings or the interest rate. In the diagram on the right above, an increase in the interest rate swivels the budget line out and the consumer moves from point A in the diagram to point B. Saving increases as a result. The graph depicts a positive relationship between savings and the interest rate, so corr(r, s) > 0. In a sense, the

5

consumer is supplying savings to the capital market. A higher interest rate induces a greater quantity of supply.

This model is a special case of the so-called life cycle model of savings. More generally, a person enters the work force at about 18 or 22, depending on if they attend college or not, works for 40-45 years, then retires. Early in her economic life the consumer may be a net borrower essentially borrowing to go to school, to buy a car or house, and so on. In her middle years she pays off the house, car, and student loans, and begins to save for her children's college

education and her own retirement. After retirement she draws down on her accumulated wealth to finance her retirement consumption. To fix ideas, let w be income from labor earnings and let c be consumption. Suppose the person enters the labor market at the age of 22, retires at the age of 65, and dies at the age of 75. Data strongly suggests that the average white-collar worker's income peaks in his or her mid 50's. This is depicted in the diagram on the left. Data also suggests that consumption is mildly increasing over the life cycle, as depicted in the right hand diagram. In a sense the w path in the figure represents the stream of annual labor earnings w22, w23, ....., w67 for a worker who starts at age 22 and retires at age 67, and c represents the stream of consumption in a similar manner, c22, c23, ..., c80, for someone who lives until 80.

Combining the two diagrams above, we obtain the diagram below. The difference between income (w) and consumption (c) at any age tells us whether the individual is saving or dissaving on net. Net saving is equal to the difference between the two, w – c. When the consumer is in the early part of her life cycle between points A and B below, her consumption is greater than her income, c > w, so the consumer is borrowing to consume. Between B and C in the life cycle, w > c so the person is a net saver. Finally between C and D the person has retired and is dissaving by consuming her accumulated wealth.

People have to form expectations about their future labor earnings, interest rates, and government policies that affect them in the future such as tax policy so they can formulate their consumption, saving, and human capital investment plans now. Perceived changes in these variables, e.g., future wages, may cause a change in behavior today. Suppose you get lucky and receive an unexpected promotion and a big raise. Then most likely this will shift up the entire

6

consumption path now and in the future as in the diagram below. You may even decide to save less as a result. Why? Because you expect your income to be higher in the future so you can maintain a higher level of consumption in the future without having to save more.

Alternatively, suppose you are about to graduate and you pick up the newspaper and read that people in your intended profession are getting paid much better in mid career than previously. In the left panel of the figure below the wage path has shifted up in mid career and is fully anticipated. This means that you can change your consumption today to take advantage of the higher salary you will expect to receive later in life, as depicted in the right hand panel above. This also works on the downside. If you expect taxes to go up in the future, you may respond by consuming less today and saving more to pay the future taxes.

5. Application: Social Security5.1 Pension basicsOne policy that can have a dramatic impact on economic behavior is social security. Most countries have such a program and many of them have the same features, workers pay into the program, while retirees receive benefits.

Pensions generally come in two forms, fully funded and pay-as-you-go or currently funded. If you belong to a fully funded pension, you make contributions to the pension fund and it is invested on your behalf, or the firm you work for makes contributions to it. Over time, an account accumulates and you have a property right to the account. How much you receive in retirement benefits will depend on how much is in your account and how the benefit is defined. And if you die prematurely, your beneficiary will receive whatever is in the account.

Under a pay-as-you-go or currently funded pension system, there really is no specific account with your name on it. In fact, the contributions that go into the fund one month are almost immediately paid out to retirees the next month. Hence, the name "currently funded" or "pay-as-you-go." Most social security programs are currently funded.4

4 Only a small trust fund exists to smooth payments over the business cycle in a currently funded system. Why? Because the business cycle tends to affect tax payments that flow into the trust fund and the retirement benefits that flow out of the fund. When taxes are greater than benefits in any given period, the trust fund increases. When benefits are greater than taxes, the fund decreases. In a recession more people are laid off and do not pay the social

7

There are two basic differences between the two types of pensions. First, under a fully funded system, the contributions to the pension are invested in stocks, bonds, government securities, and so on. This is not true in a currently funded system. Second, you do not have a property right to a pension in a currently funded system. If you die, your beneficiaries would not necessarily receive your pension money.

There are two ways in which benefits can be supported, the number of workers who pay social security taxes and the productivity of each worker. In 1950 there were 16 workers per retiree in the US. In 2000 it was about 2.6. And, as we saw earlier in the course, the growth in labor productivity has slowed down in the last thirty years. So it is now harder to support such a system in the United States than ever before. This is also happening in Europe as Greece, Italy, Denmark, Sweden, and other countries grapple with cutting their generous state pensions.

The second main problem is that benefits are too generous for current retirees in most systems. To illustrate this, consider the case of the very first recipient in the US program, Ida Fuller. She received check #1 in 1940. She contributed $22 that was matched by her employer. She lived another 35 years dying at the age of 100. In her lifetime she received $20,884.52 from social security, about a 17.6% annual return on her contributions. If she had invested her total contributions of $44 in the stock market at its historical rate of return of about 10% for 35 years she would only have received $1457! (= 44exp(0.10x35)) Clearly, she received far more than she could have obtained by investing herself. This indicates that for the first generation or two of recipients social security was a very good deal. However, this is no longer true. Now the rate of return is much lower, only 1 – 2% for many workers and may actually be negative for others. So the system in many countries is financially unstable, making a crisis inevitable.

5.2 ReformHow should the system be reformed? Chile faced the same problem back in the 1980s. They reformed their pay-as-you-go system by using a fiscal budget surplus to finance a large trust fund that is invested in a variety of investments including private stocks and bonds and government securities. Now people pay into a large fund and an account accumulates in their name. It is very similar to a private fully funded pension except that everyone must contribute. The problem with the system in the U.S. is that the trust fund is completely invested in low yielding government securities. Second, the US, and many other countries, has been running a fiscal deficit, not a surplus, for most of the last 30 years. So we cannot follow the example of Chile.

Another solution: get Americans to save more. Economists Laibson and Choi of Harvard and Madrian and Metrick of the University of Pennsylvania have suggested a number of ways in research done in 2004 and 2005. For example, tax refunds and rebates could be paid into taxpayers' savings and retirement accounts. Then the taxpayer may be more aware of saving and hence save more. Another solution is to raise the retirement age to 70 to reflect the greater longevity of Americans.

Others like Peter Diamond have argued that we can raise the payroll tax rate to about 16%, increase the retirement age to 70 with early retirement at 65 (instead of 62), and tax some benefits of wealthier recipients to make the system solvent. Computer simulation models that indicate that this is a possible solution.

security tax (payroll tax). So taxes fall. Second, firms lay off workers in a recession and they tend to lay off more older workers than younger workers to cut cost since older workers usually earn higher salaries than younger workers. Some older workers choose to retire rather than look for a job. So the incidence of retirement goes up during a recession. Social security benefits increase when the incidence of retirement goes up. Since the trust fund is equal to the accumulated taxes minus the accumulated benefits, the trust fund would tend to fall in a recession and increase in an expansionary period. In either case, there is no reason to maintain a large trust fund.

8

One potential solution: privatization. Create an account for each taxpayer for some of their payroll taxes and allow them some discretion in how it is invested. For example, the U.S. stock market earns about 10 – 11% in the long run. This would raise the average rate of return on social security contributions. George W. Bush strongly supported such a plan in 2000 and 2004. However, if the market crashes, your fund could be wiped out. It should be noted that the stock market's return was about 0% during 2000-2009. And many people do not have the knowledge to invest wisely. What if people invest badly? What do we do as a society? Do we bail them out?

5.3 Impact of program on the economyHow has the current program affected the economy? It is believed that the existence of social security lowers private saving. If the government promises to pay a pension, there is less need to save for one's own retirement. Second, social security also provides people with a strong incentive to retire in order to receive the benefit. If everyone saves less, total capital accumulation is reduced and we saw in chapter 2 this could reduce the growth rate. Social security may also cause some to retire early. If people retire before they otherwise would have to receive social security, then the total stock of labor is reduced as well. If the two main inputs, capital and labor, are lower because of social security, then less output can be produced. This may be one price we pay for social security.

There are other motives for saving than saving for retirement. These include leaving a bequest to one's children, saving for a special purpose, e.g., down payment on a house, children's education, or precautionary saving, e.g., saving for a rainy day, so to speak. These motives may respond differently to various changes in the economic environment. For example, an increase in college tuition may not affect a life cycle saver but may affect someone saving for their children's education. Losing a job may not affect saving for a child's bequest but will affect precautionary savers.

6. An Application of Incentives: Supply Side Economic Policy Usually, the economy either experiences inflation or unemployment, but not both at the same time, as mentioned. However, in the mid to late 1970's we experienced both high inflation and rising unemployment simultaneously in the US, as did many countries, and this had never happened before.5 The government was unsure how to proceed. If it stimulated the economy, that could cause more inflation. If it tried to fight inflation, that could cause more unemployment.

Historically, in the post World War Two data inflation and GDP tended to move in the same direction, so corr(inflation, GDP) > 0. How can we explain this data? In the 1960's and 1970's economists used the Keynesian model of aggregate demand and aggregate supply to explain the data. Aggregate supply (AS) captures the total amount of output or income (Y) the economy can produce in the short run given its resources. Thus, AS = Y. Aggregate demand (AD) captures the total amount of spending in the economy on the part of consumers (C), investment by business firms (I), and government (G) and income is equal to spending, thus AD = C + I + G. In a short run equilibrium, AS = AD or Y = C + I + G. This is depicted below where the AS and AD curves intersect. RGDP on the horizontal axis is real GDP (Y) and is a measure of the output the economy can produce and inflation on the vertical axis is the consumer price index, a measure of prices.

5 See the data on unemployment in the Appendix to this chapter.

9

Notice that if the AD curve shifts out and back it generates data that produces corr(inflation, GDP) > 0 and corr(inflation, unemployment) < 0.6 If a recession begins, firms reduce their investment spending and start laying off workers. Consumers become worried about their jobs and begin cutting back their spending. As C and I fall, AD shifts back as in the diagram on the left below. As AD shifts back, inflation falls and GDP falls. Alternatively, when the economy is growing, consumers and firms are confident, they spend more, C and I go up, and the AD curve shifts to the right. As the AD curve shifts out, both the CPI and GDP go up. So inflation and GDP are moving in the same direction.

The point is that prior to 1975 we were able to explain the data generated by our economy by shifting the AD curve around under this theory of aggregate demand and supply. This gives us a signal in deciding on government policy. In a recession the recommendation was to increase G to offset falling C and I, cut taxes to get C and I to increase, and lower interest rates to get consumers buying more houses and cars. During an expansion where the economy was growing too rapidly, the government could decrease G because C and I might be increasing too fast, raise taxes to reduce C and I a bit, and raise interest rates to cool the economy off.

However, in the mid 1970's both problems arose at the same time as Aggregate Supply shifted back.7 What policy should we pursue? Suppose Aggregate Supply shifts back and the 6 When GDP increases, unemployment falls and when GDP falls, unemployment increases.7 In 1974 – ‘75 OPEC raised the price of oil, quadrupling it in a short period of time. This caused both inflation to increase and unemployment to worsen. How can we explain this? We can shift the AS curve back. Why would this occur? Energy is an important cost to an individual firm. An increase in energy costs would raise the firm's costs and cause the firm to cut back by laying off workers, reducing investment, and generally contracting its operations. If most firms are doing this, then the AS curve would shift back causing an increase in inflation and lower RGDP matched by higher unemployment. The question then becomes what can the government do? Gerald Ford was President at the time and did nothing. He lost the election campaign in 1976 and was replaced by Jimmy Carter.

10

economy goes from point a to point b below so that we're experiencing rising prices and falling GDP. If we fight the recession and rising unemployment as in the diagram on the left, inflation will get worse as we move from b to c. If we fight inflation instead, as in the right hand diagram, GDP will fall and unemployment will get worse in moving from b to c.

A new theory evolved called supply side economics to explain this phenomenon and suggest policies that may alleviate it. It was thought that if we cut taxes, people would work more, save more, and invest more, supply would increase as a result, the economy would grow faster, and this would alleviate both problems at once as the AS curve shifts out. Because supply supposedly increased as a result of the government's tax cut, this new theory became known as "supply side economics."

Notice something about these diagrams. If Aggregate Supply were to shift out for some reason. The theory predicts that inflation would go down and RGDP would go up! This would be the best of all possible worlds! And this is why "Supply Side Economics" became so popular so quickly; it seemed like the answer to both problems of rising inflation coupled with rising unemployment. The question is how can we get aggregate supply to increase?

Arthur Laffer put forth the following version of Supply Side Economics. There are three parts to the theory:

1. Cut tax rates and people will respond to that incentive by working, saving, and investing more; 2. If a lot of people do this, total labor and capital increase, and the economy’s output will increase; 3. This will create more income, and income is the main tax base so the tax base will go up and tax revenue will go up. Consider the following diagram. If the tax rate is zero, we won't raise any tax revenue.

And if the tax rate is too high, no one will report their income to the government and we would also obtain zero tax revenue. The curve connecting the two points on the horizontal axis just mentioned is known as the "Laffer Curve." A high tax rate will generate a certain amount of tax revenue. However, according to the theory, if we cut tax rates from the high tax rate to the low tax rate, this will eventually increase the income tax base and we will collect the same amount of tax revenue.

OPEC raised the price of oil once again in 1979 and the Carter Administration tried to fight unemployment by raising G and lowering T. Unfortunately, inflation got worse, as the theory suggested it would.

11

Tax revenue is R = tB, where R is revenue, t is the tax rate, and B is the tax base, for the US the base is income. In step 1 above, t falls, more economic activity is generated, and eventually B goes up so you get the same revenue R. However, If government spending is fixed, a deficit is created when the tax rate t goes down. We have to sell bonds to cover that deficit. Eventually, in step 3 we have to pay off the bonds plus interest.

What actually happened? The Reagan Administration passed the tax cut, the largest of all time to that date and the deficits started. Instead of getting smaller, they got bigger and the stock of government debt went from about $1 trillion in 1980 to about $4 trillion by 1992. The problem was with step #1. The tax cuts didn't cause people to work a lot more. And savings actually fell during the late 1980's. So a cut in the tax rate led to less tax revenue, not more. A more appropriate diagram would be the following, where there is a discontinuity at the top. As the tax rate increases from zero, more tax revenue is collected until we hit point A where taxpayers revolt by reporting zero taxable income. At that point there is a discontinuity in the graph. We have never hit a point like A in general. So the moral of the story is that a cut in the tax rate will generally cause the government to lose tax revenue. The supply side theory was wrong.8 To balance the budget would then require cuts in spending. Finally, the current (2013) $16 trillion government debt must be paid off. How? By running budgetary surpluses in the future.9

8 David Stockman, Reagan’s Director of the Office of Management and Budget, admitted the theory was wrong and word leaked out, which was very embarrassing for the Reagan policy team.9 During the 1996 Presidential campaign Bob Dole reintroduced the supply side tax cut idea in the form of a 15% tax cut across the board in order to boost his ratings. Indeed, many believe this is why he chose Jack Kemp as his running mate. Ironically, Dole was very critical of supply side tax cuts in the 1980's. (So, too, was George H.W. Bush in the 1980 campaign. He was running against Ronald Reagan in the Republican primaries that year and was having trouble getting support; most voters tended to support Reagan. To distance his political positions from Reagan's, Bush strongly criticized supply side tax cuts calling them "voodoo economics," a phrase coined by economist Paul McCracken. Of course, after the primaries, Reagan was the Republican party's nominee and he chose George Bush as his running mate after Bush converted to the supply side theory, among other issues.) In any case, Dole was unable to convince voters he would really cut their taxes and lost the 1996 election. And in 2000, George W. Bush advocated supply side tax cuts to stimulate the economy. Needless to say, the large tax cut in 2001

12

The idea has arisen more recently as well. It was used by the Bush Administration to justify the massive tax cut in 2001. Second, the New York Times recently reported (8/10/03) on a new group that calls itself the Club For Growth headed by supply-sider Stephen Moore whose sole goal is to reduce taxes. It is targeting congressmen and senators who do not follow their views on tax policy for defeat in the next election. In fairness to this group, they also strongly advocate reduced government spending as well. Their primary task is to lower taxes. And, such tax cuts were put forth by Republicans to increase the growth rate in their budget negotiations with the Obama Administration.

Source: OECD, 2009We have depicted the ratio of total taxes to GDP in the chart. As one can plainly see,

taxes in the US are not high, but low relative to the other industrialized countries of the world. The US ranks in the bottom third. It is not immediately obvious why Americans believe they are overtaxed.

7. A Model of Labor Supply and IncentivesWhy did supply side economic policy fail? Why didn't people respond to the incentive they faced by working more, saving more, and investing more, when their taxes were cut? As we saw in section 4, saving need not increase when the interest rate rises. A tax cut on interest income is "like" an increase in the interest rate. So a tax cut may not bring forth much extra saving and investment. In this section we will continue the study of incentives and economic behavior by studying a model of labor supply. We will also show why the supply side tax cut on labor income may not have worked as well.

Consider the following set-up: the consumer-worker cares about general consumption (C) and leisure (L). Her tastes are represented by a utility function of the form, u = U(L, C). So we can immediately fit this new model into our general x1 - x2 model by letting x1 = L, and x2 = C. It also seems reasonable to assume that the consumer-worker is willing to accept tradeoffs between consumption and leisure and that if she were given more of one or both "goods" to consume she would be better off. So our two basic assumptions that there are tradeoffs and that more is preferred to less would also seem to hold in this new set-up.

Her budget constraint is wH = PC, where w = wage per hour, H = hours of work, P = price of consumption. The consumer-worker's labor earnings is WH and her total expenditure is

did not stimulate the economy much. As we saw in chapter two fewer jobs were created in his first administration in the early 2000s than in any administration the previous 70 years.

13

PC. The budget constraint says simply that she spends all of her labor income on consumption. To simplify we will set P = 1. Our indifference curves are in L-C space but the budget constraint is written in terms of H and C. We need to rewrite the budget constraint so it is in the variables L and C. Then we can graph it.10 If there are only 24 hours in a day, the constraint is

w24 = wL + C.The left hand side is sometimes called the consumer's "full income." It is the amount of income the consumer would have if she worked 24 hours per day. Hence the label "full."

We can match this model up with our earlier x1 - x2 model in the following way, I = w24, p1 = w, x1 = L, p2 = P = 1, and x2 = C. From this we know the two intercepts, (24, 0) and (0, w24/P) = (0, w24). This follows because I/p1 = 24w/w = 24 and I/p2 = 24w/P = 24w. The indifference map is depicted on the far left. The budget constraint is depicted in the center above. and the two are combined on the far right. The consumer – worker's optimal behavior, (L*, C*) is also depicted on th far right. This is the best she can do. To get the optimal labor supply notice that since there are only 24 hours in a day, 24 = H + L or H = 24 - L. So H* = 24 - L*.

It follows that an increase in the wage swivels the budget line out, as depicted below, while a decrease in the wage swivels the budget line in. Why? When the wage changes only the vertical intercept of the budget line is affected. Also, the MRS is MUL/MUC since L = x1 and C = x2 in this set-up and the MRT is w/P = w. The two are equal at a tangency point. A change in the wage alters the MRT and thus alters the slope of the budget line. In the diagram on the left below, an increase in the wage causes the worker to increase her labor supply as she moves from point A to B. So corr(H, w) > 0 and the labor supply curve is upward sloping. However, this is not guaranteed. It is possible that an increase in the wage might not cause labor supply to increase, as depicted on the right below. In that case corr(H, W) < 0 and the labor supply curve is downward sloping.

10 Suppose there are only 24 hours in a day. Consider the following, wH = PC and H = 24 – L, so w(24 – L) = PC, or 24w = wL + PC. We are assuming there are only two things you can do with your time, work (H) and leisure (L). The last equation is the consumer's budget constraint, namely, w24 = wL + PC. We will assume P = 1.

14

A cut in the tax rate imposed on labor is "like" an increase in the wage. So it is possible that people could respond to a supply side tax cut on their labor earnings by reducing their labor supply or not changing it much at all. There is no evidence that people increased their labor supply in the 1980's in response to the massive tax cuts that took place.

Notice that the diagram depicts a change in the wage and labor supply. We could graph points A and B in wage - labor supply space, as depicted on the right below, and obtain a supply curve for labor. In the left hand diagram below, it appears that the worker increases labor supply in response to an increase in the wage. So the labor supply curve corresponding to that diagram is upward sloping. This is depicted in the diagram on the right.

In the diagram below, a tax cut is depicted on the left. It swivels the budget line out, since it is like an increase in w, and the consumer moves from A to B. However, labor supply is the same after the tax cut as before. Therefore, the tax cut failed to induce any extra labor supply. This is reflected in a labor supply curve that is perfectly vertical or perfectly inelastic.

The best empirical work that has been done on this issue (See the Handbook of Labor Economics) by people like Professors Hausman, Altonji, Heckman, McCurdy, Mroz, and others,

15

suggests strongly that labor supply of men and women is very inelastic; labor supply curves are very steeply sloped. This has a dramatic effect on the supply side tax cut story. The steeper the labor supply curve the lower the wage elasticity of supply and the smaller the labor supply response will be to a wage tax cut. The evidence tends to support the diagram below on the right, not the left. The elasticity of labor supply with respect to the wage can be defined as

Ew = (H/H) ÷ (w/w).

Researchers find that Ew = 0.1 approximately. This means that a 10% increase in the wage only leads to a 1% increase in labor supply. So a tax cut that increases the take home wage by 10% will hardly have any effect on labor supply at all. This is why the supply side tax cut didn't work. For the theory to work supply would have to be like the diagram on the left below. Instead, it is more like the one on the right.

Why is the actual labor supply curve so inelastic? Because of tastes; people apparently have a taste for working a certain number of hours each week and it is difficult to induce them to work more on a sustained basis. Indeed, over the last century the average workweek has declined from about 59 hours on average to about 41 hours.

8. More ExamplesA. Example: charity. Many individuals donate to charity. There is a large and

growing area of research on this subject. Consider a donor with utility U(C, x) where C is charity and x is a general consumption good. Let d be the individual donor's donation and suppose there are two donors, one and two. Then C is the sum of all donations to the charity, C = d1 + d2. Using this we have for donor one, U(d1 + d2, x1). The donor's constraint is I1 = d1 + x1. He chooses the donation and general consumption to maximize his utility subject to the constraint. We obtain a "donation" function for person one, for example, of the form, d1 = D(I1, d2). For the second donor we have instead, d2 = D(I2, d1). Generally speaking, if there are more donors, then each individual donor's donation depends on all the rest of the donors' donations. For donor one, d1 = D(I1, d2 + d3 + ... + d10,127) and similarly for the other donors, d2 = D(I2, d1 + d3 + ....+ d10,127), and so on.

Notice that one's donation depends on two's donation and two's donation depends on one's donation. This means that the action of one agent affects another agent. This is a classic example of a beneficial externality and it's formally outside a market and so a non-pecuniary externality. When two raises her donation she makes one better off and vice versa. Since each donor ignores the effect of his donation on the other donors, too little is donated to charity. This is the main justification for subsidizing charitable donations.

Empirical evidence strongly suggests that donating to charity is a normal good. If we allow a tax write off for charitable donations, this reduces the cost of making a donation and increases donations.

16

B. Example: Bequests. Parents care about their own consumption and the bequest they leave to their children. Utility is given by u = U(C, B), where C = parental consumption, B = bequest given to the parent's children. The income constraint is I = PC + qB, q = cost of making a bequest, e.g., lawyer's fees in drawing up a will and so on. The parent chooses (C, B) so that MUC/P = MUB/q. Evidence indicates that about 30-35% of all parents choose to leave a bequest to their children. People also make bequests to charity, medical foundations, and religious organizations.

Our theory can instruct us as to how people will respond when they receive more income, for example. From 1995 to 2000 the economy grew at a rapid pace and income increased as a result. It was reported on ABC News (June 2, 2000) that donations to charity were up as a result of the improved economy. This is exactly what our theory would predict since charity is a normal good. What happens if q increases? The relative price of the bequest increases and the budget line swivels in. The parent can respond in different ways. Most likely the bequest will fall. Why might the bequest increase in price? Estate taxation. Many countries impose the so-called death tax, or tax on inheritances.

9. Examples of Substitution: Compensating Grandma for Inflation and Price Clubs.Recently, a controversy has been stirred up over the Consumer Price Index (CPI) and its use as a measure of the standard of living. At the heart of the debate is the so-called substitution effect. The retired elderly live on a fixed income. Inflation erodes the purchasing power of that income. Therefore, many feel that we should compensate the elderly for unexpected inflation by increasing their social security benefit. The question then becomes: How much compensation should we give them? There are several possibilities. The main one seems to be that we should compensate them so they attain the same market basket of goods they were buying before the inflation. Recently, it has been argued, however, that this over compensates them. If true, this means that the elderly are actually better off with inflation than not!

First, suppose all prices increase by 10%. Then we know from our earlier analysis that the budget line will shift in in a parallel way much like a decrease in income. This represents a loss in purchasing power. In the left hand diagram below, a "smooth" inflation, where all prices go up by the same amount, can be offset by an increase in the elderly person's social security income. Inflation moves the budget line in and the inflation adjustment that increases her social security benefit moves her budget line back out.

17

However, most actual inflation is not nice and "smooth." Some prices increase a lot, e.g., pharmaceutical drug prices, some go up a little, and some prices even fall, e.g., computer equipment. When one creates a price index and averages out over all of the prices in the market basket, the "average" price may have increased and we tend to label this "inflation." The point is that if relative prices change during an "uneven" inflation, the slope of the elderly individual's budget line will change as in the right hand diagram.

Here's the problem. Suppose the price of denture cream increases substantially while the price of Ensure goes down, but only by a small amount. Then the average price will increase and inflation will occur. The budget line will shift and swivel in from I1 to I2, as in the diagram below. Suppose we increase the retiree's social security benefit (income) to compensate them for this inflation. Further suppose that we give them enough additional income so the retiree can buy her original consumption point A. The new budget line with compensation is I3. Notice that it goes through the original consumption bundle, point A, so A is affordable. However, will the retiree choose point A again with budget line I3? No, since this will not maximize her utility. She can do better by choosing bundle B. Why? Because relative prices changed as the inflation was occurring. We know that the relative price of denture cream has gone up and consumers generally substitute away from more expensive goods towards cheaper ones; this is a substitution effect. So we would expect the retiree to substitute away from denture cream toward Ensure and actually move to point B.

Now compare her utility at point A with her utility at point B. Clearly, UA < UB and she is actually better off after the inflation with compensation than she was before the inflation took place. This is due to the way she was compensated. She was compensated to maintain her consumption bundle but not her utility level. The general point is that whenever there are substitution possibilities and the inflation is uneven, we will overcompensate Grandma for inflation if we give her enough income so she can buy her original market basket of goods. This is because relative prices have changed in an uneven inflation and consumers generally substitute away from more expensive items.

18

What if there are no substitution possibilities? Consider the L-shaped indifference curves below. They depict a consumer who is unwilling to accept any tradeoffs. The two goods are perfect complements. This is similar to the Leontief technology studied in chapter two. After the inflation the retiree is on the I2 budget line, as before. After compensation the retiree is on the I3 constraint. However, since this person is unwilling to accept any substitution in her market basket of consumption, point A = point B. Now it doesn't matter how she is compensated and overcompensation is not a problem. However, from what we know of consumer theory, most consumers are willing to accept substitution possibilities, in which case, the problem of over compensating people living on a fixed income still exists.

A second area where substitution possibilities create an interesting scenario for consumers involves a situation where there are two markets that are spatially separated and different prices are being charged for the same good. Consider the following situation. Suppose you can buy an expensive stereo locally and pay $3000, or you can travel to another location and pay $2500. Is the full cost of the trip, e.g., dollar cost of gas plus opportunity cost of your time, worth paying $500 less for the stereo in the cheaper location?

Compare two locations and assume that everything about them is the same except that the relative price of a stereo is lower in one. In the figure below, the left-hand diagram depicts the budget line for a consumer living in the location where the stereo is expensive if the consumer buys the stereo locally. The middle diagram depicts the budget line for a consumer living in the cheap location who also buys locally. The budget line is flatter in slope because the stereo is

19

cheaper. In the far right diagram we depict the budget line of a consumer who travels from the expensive location to the cheap location to buy the stereo. This consumer's budget line will differ from that on the far left for two reasons. First, it will have the same slope as the middle diagram because the stereo is cheaper. Second, the consumer on the far left must travel to the location where the stereo is cheaper. The fixed travel cost is like a decrease in income. It must be paid regardless of how much is purchased. So the fixed travel cost shifts the budget line in but doesn't change the slope by itself. What we need to do is compare utility on the far left situation with utility in the far right situation and determine if the person will travel for a cheaper price.

Consider a situation where you live in Spokane and discover that stereos are cheaper in San Francisco. Chances are the travel cost is too high and so you would buy locally in Spokane,

as in the left pane of the diagram. However, if you live in Spokane and discover a stereo shop selling stereos more cheaply in Post Falls, then you might drive the short distance to Post Falls, pay the travel cost, and buy the stereo in Post Falls as in the right diagram. (Remember the shallow budget line represents buying out of town and paying the travel cost.)

Example: It was reported on the CBS News (April, 2000) that many older Americans are traveling to Canada to buy prescription drugs because the price is so much lower. Some Americans living near the border have also traveled to Mexico for less expensive prescription drugs, as reported by the Washington Post in 2005. Clearly, they are willing to pay the cost of a bus ticket and a motel room for a night in order to buy pharmaceutical drugs in another country.

Example: "Price clubs" charge a membership fee to join the club but then charge lower prices for some of the items they sell. "Buying out of town" can be reinterpreted as buying something at the price club while "buying locally" can be interpreted as not joining the price

20

club. The fixed travel cost can then be reinterpreted as the membership fee to join the club and some of the goods, e.g., the stereo, can be bought at the price club at a lower price. Is joining the price club worth it? No, in the left diagram above, but yes in the right diagram. So it depends on how high the membership fee is for joining the price club and how much lower the prices are once you join the club.

10. Substitution and Income EffectsWhy would a consumer-worker respond to a wage increase by working more? Presumably, at the margin it pays them to work more, i.e., they have a greater incentive to work more. However, some workers might work less. Why? Because a higher wage also means they have more income. And they might want to take more leisure time as a result if leisure is a normal good. Such an individual is sacrificing the additional consumption he could have had for more leisure.

In general, there are two effects when any price changes: a substitution effect and an income effect. When a single price changes, relative prices change and consumers will generally respond by substituting toward commodities that have become cheaper and away form commodities that are more expensive after the price change. However, the price change also affects the consumer's real purchasing power. Consider someone who really loves music and buys a lot of music from iTunes. Suppose there is a dramatic increase in the price of a song. How does this affect the individual? They might buy fewer songs from iTunes and shift their consumption to commodities that are now less expensive after the price of a song went up, e.g., movies, videos, and broccoli. They might also have less money to buy other things as well because of the price increase. And this might also affect their spending decisions on the other goods as well.

Next, consider someone who is working. The wage rate the individual earns is the price of his leisure; the wage is the opportunity cost of taking leisure time. A higher wage means the person's leisure time is more expensive; a lower wage means it is cheaper. Suppose the wage increases. Will the person work more because the price of leisure has increased? Or will he work less because he is wealthier (because the wage increased)?

Any price change will have two effects:1. consumers economize through substitution2. real purchasing power is affected.Example: Periodically, the major airlines get into a fare war where prices for commercial

flights drop dramatically. First, people substitute toward air travel by flying more often. Second, their real purchasing power has increased so they can afford more of other goods as well. The lower price reflects the efficiency aspect of competition and the relative scarcity of air travel in comparison with other goods.

Example: Suppose the government imposes a tax on goods except those bought on the internet. The price of goods bought in retail stores increases because of the tax. However, this does not really reflect relative scarcity. Instead, it reflects an artificial increase in the price of goods not purchased on the internet due to the tax. Now when the consumer substitutes away toward buying on the internet it is socially inefficient because the high price does not reflect the relative scarcity of sales in retail stores, but government interference. The tax system usually distorts economic behavior because it alters relative prices.

Example: Suppose the government simply imposes a tax of $100 on every man, woman, and child in the country and suppose you have to pay the tax regardless of how much you work, what you buy, how much property you own, and so on. This is known as a head tax; if you have

21

a head, you must pay the tax. This tax causes a pure income effect; it doesn't affect relative prices at all since the tax is not tied to the purchase of a particular commodity or income. Therefore, it doesn't affect the efficiency of the economy. On the other hand, suppose there is a sales tax imposed on cars only. Then the tax artificially changes relative prices of cars versus everything else and will reduce efficiency in the economy as a result.

We wish to analyze these two effects. Assume a consumer likes movies in the theater and movies streamed online hey can rent. First, consider the substitution effect. People don't really care about their income or relative prices; what they care about is the level of satisfaction or utility they can achieve. They want more income not for the sake of having more income per se, but because of the additional goods and services the extra income can buy and the utility it produces.

Consider the following diagram. A level of utility is depicted as Ua. It is useful to think of the level of utility as the level of real purchasing power the consumer can achieve. Also depicted are two budget lines that can achieve the same level of utility at points a and b. We should stress this. Both budget lines can achieve the same level of utility or purchasing power, yet they involve different levels of income and different relative prices. However, the two budget lines are equivalent in the sense that they allow the consumer to achieve the same level of utility or what we call real purchasing power. Of course, the actual consumption bundle the consumer chooses under the two constraints will differ, a ≠ b. But this won't matter in achieving utility Ua as long as the consumer is willing to accept a tradeoff between the two goods. Notice that the budget line through b is steeper than the one through a. This means that the price ratio Ptheater/Prental is higher at b than at a. A move around an indifference curve (from a to b, for example) is a pure substitution effect. It holds utility or real purchasing power constant and only reflects a change in relative prices.

Next, consider the income effect. When a price changes this also affects the money available for purchasing everything you buy. It alters your utility level, or real purchasing power. The best way to think of the income effect of a change in prices is that it involves a parallel shift of the budget constraint so the consumer moves form one level of purchasing power, or utility, to

22

another. If the budget constraint shifts out, the consumer can attain a higher level of utility. If it shifts in, the consumer attains a lower level. Thus, the pure income effect is “like” a change in income in moving the consumer from one indifference curve to another. The key is that the slope of the budget line doesn't change under the income effect. You just move from one indifference curve to another, or to put it another way, from one level of real purchasing power to another as in the diagram below.

Now let's combine the two effects. Suppose Ptheater increases. We would observe the consumer's budget line swiveling in and the consumer would be observed going from bundle a to bundle b in her consumption pattern depicted below. We can decompose the move from a to b into two parts: the income effect and the substitution effect. To obtain the decomposition, shift the new budget line through b back out to the original indifference curve to pick up point c. Now imagine the following thought experiment. Suppose that Ptheater increases and we compensate the consumer by giving her more income so she can still achieve the same level of utility or real purchasing power as she could at point a. So as Ptheater goes up, we also increase her income. This would swivel the budget line around the indifference curve and move her from point a to c. Her real income or real purchasing power is the same because the extra income compensates her for the price of movies in the theater going up. The move from a to c captures the pure substitution effect because it holds utility or real purchasing power constant. Finally, suppose we take the extra income away. This causes a pure income effect. As we take the extra income away, her budget line shifts back in a parallel fashion and she moves from c to b.

Let's examine her response to a price increase. The price of a movie in the theater goes up and we actually observe her changing her consumption bundle from a to b in both diagrams. We can decompose this into two effects, the substitution effect (a to c) and the income effect (c to b). When the relative price of a movie goes up, she substitutes away from movies in the theater toward rentals and other goods in general (a to c). So the quantity of movies in the theater falls and rentals increase. However, her real purchasing power or utility falls because of the price increase as well (c to b). If movies are a normal good, the loss in real purchasing power also causes her to reduce the quantity of movies she goes to. Notice that as she moves from c to b she also reduces the quantity of rentals since rentals are also a normal good. The substitution and income effects work in the same direction for a normal good. However, the response of rentals is ambiguous. She buys more rentals under the substitution effect but fewer rentals under the income effect. As depicted, the substitution effect dominates so on net she buys more rentals

23

(a to b). However, it is possible that the income effect could dominate. In that case, we would observe her buying fewer rentals in response to the increase in the price of a movie in the theater.

The opposite response will occur if the price of a movie falls. She will substitute toward movies and away from CDs due to the substitution effect and will buy more movies and CDs if they are both normal goods because her real purchasing power has gone up. The two effects work in the same direction for movies when a movie is a normal good. She will go to more movies because the price is lower and because her real income has gone up if a movie is a normal good. The two effects work in the opposite direction for CDs. She buys fewer CDs because of the substitution effect but more CDs because of the real purchasing power or income effect when CDs are a normal good. The observed response of CDs will depend on which effect is larger.

When a good is inferior some interesting possibilities emerge because of the income effect. Suppose the price of hamburger goes up and hamburger is an inferior good. People will substitute away from hamburger because of the substitution effect but will buy more hamburger because of the income effect. If the income effect dominates, it is possible that we could observe people buying more hamburger in response to an increase in the price of hamburger. This is a violation of the law of demand and is somewhat rare. Therefore, if we observe a violation of the law of demand it must be because the good in question is an inferior good. Of course, if the substitution effect dominates the income effect, we will observe people buying less hamburger when its price increases and the law of demand will still hold even though the good is inferior.

Example: A large grocery store chain back east did an extensive survey in the early 1970's and discovered that when the price of hamburger increased, their sales of hamburger would increase. Demand for hamburger appeared to slope upward. As soon as they discovered this, they raised the price of hamburger a small amount and sales actually increased slightly!

Whether there is a large or small substitution effect depends on whether there are goods available that are highly substitutable for the good in question, or not. Goods like cigarettes and alcohol have few substitutes and so there are smaller substitution effects for those goods. Other

24

goods like "food eaten in restaurants" have greater substitution effects because they have close substitutes, "food eaten at home." So a 10% increase in the price of cigarettes will cause a small substitution effect while a 10% increase in the price of food eaten in restaurants will cause a much larger substitution effect.

Application: Suppose we consider the labor supply model once again. Let's work through the intuition without any graphs. When the wage increases, some people report that they would be willing to work more hours, while others report they would actually work fewer hours. What is going on here? Why are the responses different? Recall that the wage is the "price of leisure." When the wage increases, the individual has a strong incentive to substitute toward more hours worked and away from hours taken in leisure. This is the substitution effect. However, an increase in the wage also means the worker is wealthier; her income and hence real purchasing power is higher. If leisure is a normal good (so that hours worked is an inferior good), then the income effect works in the direction of reducing hours worked and increasing leisure. So the two effects work in opposite directions when leisure is a normal good. For workers who report they would work more when the wage rises, the substitution effect dominates. For workers who would choose to work less, the income effect must dominate.

Suppose the two effects are almost offsetting. In that case, labor supply will hardly respond at all to the higher wage rate and the labor supply curve will tend to be very steep or inelastic. Empirically, this turns out to be the case and is another way of stating why the supply side tax cuts failed to stimulate more labor supply. A tax cut is "like" an increase in the wage that induces offsetting income and substitution effects.

11. Uncertainty: A First Look at Game Theory.Many private agents face situations of uncertainty where the outcome is unknown. For example, you must pick a major not knowing exactly what kind of job you might get. You must invest your money not knowing whether your asset will increase in value or not, and so on. These situations can be modeled using a tool known as Game Theory.

We will usually consider games where there are two people each playing strategically. Each individual chooses an action, and when taken together, there is a payoff to the actions that are chosen. Each agent tries to choose an action that maximizes his payoff. And, each player takes into account how the other player will play the game. This is where strategic behavior comes into play. In this chapter we will consider a special case of this where one player is passive, Nature, who chooses probabilities.

Consider the following example. Suppose there are two players in the game, you and "nature." Suppose there is a possibility that it might rain today. You can choose to carry an umbrella to school, or not carry an umbrella. "Nature" chooses whether it will rain or not. "Nature" is considered a passive player since it is not behaving strategically. Your payoffs or utility from each set of actions are listed in what is called a payoff matrix, as depicted below. For example, if it rains and you choose not to carry an umbrella, your payoff is only 10 units of utility. If you carry the umbrella it is 100 instead.

25

The first step is to calculate the so-called expected payoff from each action that you can take. This is the average utility or payoff you receive from each action. To calculate the expected payoff from carrying the umbrella, multiply the probability that it will rain by the payoff and add across choices of Nature. Let p be the probability of rain and 1 - p be the probability of no rain. So, if there is a 20% chance of rain, p = 0.20. It follows there is an 80% chance it won't rain. The probabilities must add up to 100%. The expected utility from carrying the umbrella is

EP(umbrella) = p100 + (1 - p)30 = p100 + 30 - p30 = 30 + 70p.So once the probability of p is known, we can calculate the expected payoff from carrying the umbrella. For example, if it is raining as you leave home, p = 1 = 100% and EP(umbrella) = 100. That is your utility from carrying the umbrella. And, the expected payoff from not carrying the umbrella is,

EP(no umbrella) = p10 + (1 - p)70 = 70 - 60p.So if p = 1 = 100%, then EP(no umbrella) = 10. The second step is to compare the expected payoff from the different actions and choose the action with the largest expected utility or payoff. So, for example, if p = 1, EP(umbrella) > EP(no umbrella) so the optimal decision is to carry the umbrella.

Consider another example. You have the opportunity of going on a blind date or not. Nature chooses whether the date will be a good one or not. Suppose the payoffs are as listed in the matrix below. (This assumes that you can find out perhaps later whether the date would have been a good one or not and achieve your payoff then.) Let p = probability that it is a good date.

The expected payoffs from "go" and "don't go" areEP(go) = p200 + (1 - p)10 = 10 + 190pEP(don't go) = p10 + (1 - p)50 = 50 - 40p

Suppose p = 1/3. What should you do?EP(go) = 10 + 190p = 10 + 190(1/3) = 73.334EP(don't go) = 50 - 40p = 63.33

So you should go on the date. What if p = 1/25? (You probably shouldn't go if p = 1/25.)This is how we can model situations of uncertainty. It is very general. So, for example,

we can model a situation where the consumer might be uncertain about his future income and derive a demand curve under uncertainty. As it turns out, the law of demand will still hold up in such a situation.

We can apply this methodology to the health insurance issue. Many young people are healthy and have good jobs but don’t feel a need to buy health insurance. The question is why. The Affordable Care Act forces them to, now however, but before the Act many chose not to buy coverage on their own but rather chose to go without insurance. We can model their decision as one involving the uncertainty of getting seriously ill. Consider the action of buying or not buying insurance. Nature chooses whether the individual gets seriously sick or not. Let W = income, P =

26

premium for the insurance, A = payment if sick, Z = loss if sick. Assume W – Z < 0, A – P – Z > - Z. Note that for a serious illness like cancer Z might be very large relative to income W.

NatureCatastrophic illness No catastrophic illness

Buy Insurance W – P + A - Z W - PBuyer

Don’t buy W - Z W

Calculate the expected payoffs:EP(Buy) = q(W – P +A – Z) + (1 – q)(W – P) = W – P + q(A – Z);EP(Don’t buy) = q(W – Z) + (1 – q)W = W – qZ.

The rule is, Buy the insurance ifW – P + q(A – Z) ≥ W – qZ;

or, buy ifqA ≥ P.

If the expected gain when the insurance is needed (qA) is at least as great as the premium (P), then buy the insurance. This is more likely to hold if the probability of a serious illness (q) is high enough, the payment under insurance (A) is large enough, and if the premium (P) is low enough. Young people with good jobs that don’t have insurance may choose not to buy insurance because they believe the probability of getting cancer, for example, is zero.

12. ConclusionWe applied our theory of the economic behavior of the consumer to various areas including differential taxation, whether children are an "inferior" good or not, savings behavior, labor supply, home ownership, and bequeathing. We also studied two important policy areas, social security and supply side tax cuts. Bequests and the distribution of wealth is also an important research topic receiving much attention lately. In addition, we applied the theory to compensating agents for inflation, price clubs, and the fixed costs of travelling to buy a good at a lower price. The key to understanding these last phenomena is the difference between substitution and income effects. Finally, we also studied the important special case of uncertainty and introduced several important concepts from game theory.

27

AppendixUnemployment data since 1960. Original source: BLS.

Notice that the unemployment rate was falling in the 1960s to about 3.4%. It increased to about 6% in 1971, fell to about 4.7% as the economy recovered from the recession of 1970 – ‘71, but then increased dramatically after the oil embargo of 1974 when OPEC raised the price of oil. After a period of recovery where it fell to about 5.87%, it increased once again when OPEC raised the price of oil again in 1979. So unemployment throughout the 1970s was much higher than in the 1960s and became a cause for concern.

Important ConceptsSavingsLife cycle modelSocial security

fully funded programpay-as-you-go program

Supply side tax cutsThe Laffer curveLabor supplyCompensating GrandmaPrice clubsIncome and substitution effectsUncertaintyPayoff matrix

28

Review Questions1. What does utility depend on over time? How would you define the marginal utility of

future consumption? What is the MRS of current for future consumption?2. How does the budget line shift when lifetime labor earnings increases? When the interest

rate falls?3. How does savings respond when the interest rate increases?4. How was the social security program originally set up? How was it changed in 1983?5. What is the logic behind supply side tax cuts? Did the massive tax cut in 1981-83 work?

Why or why not?6. If we give a consumer enough income after a price change has occurred so she can

consume her initial consumption bundle are we over or under compensating her and why?

7. Does a price club necessarily save you money?8. What is the substitution effect? What is a pure income effect? Is it possible for a demand

curve to slope upwards? Is it likely?9. How can we model a situation involving an uncertain outcome?10. What are the motives for saving?

Practice Questions 1. An increase in the interest rate necessarily increases savings.

a. True b. False

2. As an individual gets older and more experienced they earn more income. The smart consumer understands this even when he/she is young and plans with this in mind. Trace out what you think the income consumption curve would be like in the diagram provided.

3. In the life cycle theory of savings a. a person saves mostly in her early years.b. a person saves mostly in her middle years.c. a person saves mostly in her late years.d. a person saves throughout her life cycle.

4. If MU1 = u/c1 = 5, MU2 = u/c2 = 8, p1/p2 = 1/(1+r), and r = 1, what should the consumer do?

a. Consume more c1 and less c2.b. Consume less c1 and more c2.c. Consume more of both goods.d. Consume less of both goods.

5. Consider the effect of the interest rate (r) on consumption today (c1). Recall that 1/(1+r) is the "price" of future consumption, c2. Suppose the interest rate increases and consumption today is a normal good. How does consumption today respond?

a. c1 will increase because both the income and substitution effects work in the same direction.

b. c1 will increase if the income effect dominates the substitution effect.c. c1 will increase if the substitution effect dominates the income effect.d. c1 will decrease if the income effect dominates the substitution effect.

29

6. Consider a situation of uncertainty where terrorists are considering an attack on another country. Will the US intervene or not? Nature chooses whether the US intervenes. The terrorist must decide whether to attack or not. The payoffs are depicted in the table. Assume: W < T, W > V, V > 0. Let q = probability of intervention.

NatureUS intervenes US doesn’t intervene

Attack W - T WTerrorist

Don’t attack V V

Find the expected payoff from each action.EP(Attack) =

EP(Don’t attack) =

7. From the last question, find the decision rule. When should the terrorist attack?

30

Answers1. b.2. The individual who anticipates that his/her income will increase later in life can use capital markets to consume more now (buy a house) than they otherwise would have. And they can consume more in the future as well. So the income consumption curve is probably upward sloping.

3. b.4. a.5. b.6. EP(Attack) = q(W – T) + (1-q)W = W – qT

EP(Don’t attack) = qV + (1 – q)V = V7. Attack if W – V ≥ qT

Don’t attack if W – V < qT.

(Note W – V = marginal benefit from attacking versus not attacking, and qT = expected cost when the US attacks.)

31