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5. Applications of Consumer Theory In this chapter you will learn:

Taxes can change relative prices and alter incentives; Incentives, the Laffer curve, and supply side tax cuts. Do they work?; Compensating Grandma for inflation; Price clubs and transactions costs; Substitution and income effects; Uncertainty.

5.1 Introduction We will apply the theory of the consumer from the last chapter to a number of different areas including food consumption, savings behavior, labor supply, uncertainty, bequests, charity, and housing, among others. We will also discuss several public policies that are typically used in most advanced economies like taxation, social security, and supply side tax cuts because they illustrate the importance of incentives. In the last chapter we studied an abstract model involving two goods, x1 and x2, In the applications in this chapter we will study models with two goods, and interpret one of the goods as x1 and the other as x2, and then use what we learned in the last chapter about the abstract “x1 – x2” model. Everything in the last chapter will carry over to this chapter. For example, if the consumer receives more income under our theory of the labor - leisure choice, we can map out an Engel curve for leisure. If leisure is a normal good, an increase in income will cause consumers to take more leisure time and work less as consumers “spend” more of their time on leisure.

5.2 Taxation Taxes alter relative prices and change incentives as a result. Suppose there are two goods, "food" and "everything else." We can match up this new situation perfectly with the abstract “x1 – x2” model. Let x1 = F be the quantity of food consumed, let x2 = E be everything else, let I be income, and suppose the prices of food and everything else are Pf and Pe. The utility function is u(F, E) and the budget constraint is I = Pf

. F + Pe . E. Put F on the horizontal axis in place of x1

and put E on the vertical axis in place of x2. The horizontal intercept of the budget line is I/p1 = I/Pf and the vertical intercept is I/p2 = I/Pe. The budget line connects the two intercepts. Finally, the indifference curves have the "usual" shape. The consumer chooses (F, E) to maximize utility subject to the budget and point A is the bundle that satisfies this decision problem.

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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, and is “like” an increase in the price of “everything else.” Only the vertical intercept is affected by the tax; it will decrease from I/Pe to I/(Pe + t) because of the tax t. The budget line will swivel inward as depicted on the right. So the slope falls because of the tax.

How will the consumer respond? 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 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 A to B, consumption of everything else is unaffected by the tax on everything else, while food consumption falls. This is somewhat paradoxical; the tax doesn't affect the good that is 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 consumption of both food and everything else falls with the tax on everything else. In this case, F and E are complements. Fourth, 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 and seems unlikely. 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, the two goods are substitutes.

Which case is most likely? We can probably rule out the cases where consumption of the taxed good increases and where the taxed good doesn’t respond at all. Most likely the response will be between point B and the horizontal intercept. So either the two goods are complements (between B and C), food consumption doesn’t respond (point C), or substitutes (southeast of C).

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 so 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 their "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. This would reduce pollution and help the environment, in addition to raising revenue for maintenance of our infrastructure. There is significant evidence that the heavy taxation of gasoline in Europe has caused Europeans to conserve on gasoline and use mass transit instead.

Example: National sales tax. Suppose both food and everything else are 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

1 Food is not taxed in many countries and states in the US, including the state of WA. Instead, WA relies heavily on the sales tax and much of the sales tax revenue is collected from the sale of durable goods like refrigerators, LCD tvs, and cars. Unfortunately, in a recession the demand for durable goods declines dramatically causing a budget crisis.

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purchases and is equivalent to an income tax. To see this, note that the budget constraint changes from I = pfF + peE without the tax to I = (1+t)pfF + (1+t)peE with the tax. Divide both sides by 1+t, I/(1+t) = pfF + peE, and notice that 1/(1+t) < 1. Define 1 - T = 1/(1+t). In that case the 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. A national sales tax, also known as the VAT or value added tax, is used by many governments in Europe.2

5.3 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 where the family optimizes. 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 and they choose a new point, B. Clearly, both the number of children and the amount of education per child increase. However, as their income continues to increase, eventually, they choose to produce fewer children but to educate them more. The Engel curve for education is upward sloping since education is a normal good. However, the Engel curve for the "quantity of children" bends backward. The quantity of children is a normal good for low levels of income but an inferior good for higher levels.

Alternatively, we can compare two economies, one poor, and the other wealthy. A poor

society might be at point B, while a rich society might be at point D. 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 did not include a tradeoff between quantity of children and education, mainly because education was not as important in 1800. With economic growth parents in the US, Japan, and Europe began having fewer children but tended to educate them more. This is the so-called “demographic transition” from high population growth to low population growth.

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%.

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5.4 Saving Behavior Next, 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 the 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 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.

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 retirement 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 (s) plus interest (r) on her savings and then spends it on future consumption,

s + rs = c2. 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. We

can match this model up with the x1 - x2 model in the following way, w = 1 c1 + (1/(1+r)) c2

I = 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. We can apply everything we know about the x1-x2 model to this new situation. So the graph of the lifetime budget equation is a line, where the two intercepts are given by I/p1 = w/1 = w and I/p2 = w/(1/(1+r)) = w(1+r).

3 The second constraint can be written as (1+r)s = c2, or s = c2/(1+r). Substitute this into the first constraint for s to get the lifetime budget constraint.

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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 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 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.4

Savings will also respond to changes in labor earnings or the interest rate. In the diagram on

the right, an increase in the interest rate swivels the budget line out and the consumer moves

4 If she worked on both periods, w1 = c1 + s and w2 + (1+r)s = c2, or after combining, w1 + w2/(1+r) = c1 + c2/(1+r). With three periods her lifetime constraint is, w1 + w2/(1+r) + w3/(1+r)2 = c1 + c2/(1+r) + c3/(1+r)2. If she only works in the first period but lives for three periods, we have instead, w1 = c1 + c2/(1+r) + c3/(1+r)2. She must save enough in the first period to finance second and third period consumption.

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from point A to point B. Current consumption falls and saving increases as a result. The graph depicts a positive relationship between savings and the interest rate, so corr(r, s) > 0; as r goes up, s goes up (A to B), as r falls, s falls (B to A). The consumer is supplying savings to the capital market. A higher interest rate induces a greater quantity of supply.

This model is a special two-period case of the so-called life cycle model of savings. More generally, a person enters the work force at 22, 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.

Suppose the consumer’s consumption path as she ages is 𝑐!!, 𝑐!",… , 𝑐!" , “consumption at age 22,” “consumption at age 23,” and so on, her labor earnings path is (𝑤!!,𝑤!",… .𝑤!"), “labor earnings at age 22,” “labor earnings at age 23,” and so on, and her utility is 𝑈 𝑐!!, 𝑐!",… , 𝑐!" , where she retires at the age of 65, and lives to 75. To fix ideas, let the w graph below represent her labor earnings path and let c represent her consumption path as she ages. Data strongly suggests that the average white-collar worker's labor earnings peaks in their 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. The w path in the figure represents the stream of annual labor earnings for a worker who starts at age 22 and retires at age 65, and c represents the stream of consumption in a similar manner, c22, c23, ..., c75.

In general, if she earns wt at age t, consumes ct at age t, and has savings plus interest income

from the previous period of (1+r)st-1, then her constraint for age t is wt + (1+r)st-1 = ct + st so her current savings is given by 𝑠! = 𝑤! + 1+ 𝑟 𝑠!!! − 𝑐!. Essentially, the consumer would like to consume along the c-path as she grows older over time since it maximizes her lifetime utility subject to her lifetime budget constraint, but she earns the w-path. She would never need to save or dis-save if the w-path matched the c-path perfectly, period by period. However, if the two paths do not match, then she will need to save and/or dis-save. When she does this she is converting the w-path into the c-path and essentially “smoothing” her labor earnings to match her desired consumption. When income is lower than desired consumption, she borrows. When income is greater than desired consumption, she saves.

Combine the two diagrams. Early in the life cycle w < c between points A and B so she borrows. In her middle years between points B and C labor earnings is greater than consumption so she repays what she borrowed earlier and saves for retirement so she is a net saver. Finally between C and D she has retired and is dissaving by consuming her accumulated wealth.

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People have to form expectations about their future labor earnings, interest rates, and

government policies so they can formulate their consumption and savings plan now. Expected changes in future wages, interest rates, and public policies, e.g., future taxes, may cause a change in behavior today. Suppose you get lucky and receive an unexpected promotion and a big raise that comes as a surprise. Then most likely this will shift up the entire consumption path now and in the future as in the diagram. 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 now.

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 at the start when you are 22. This means that you can change your consumption today at age 22 to take advantage of the higher salary you will expect to receive later in life, as depicted in the right hand panel. 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.

5.5 An Application of Incentives: Supply Side Economic Policy 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

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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), so AD = C + I + G. In a short run equilibrium, AS = AD or Y = C + I + G. This is depicted below as point a.

Notice that if the AD curve shifts out both inflation and GDP increase as the economy moves from a to b in the left hand diagram, and if it shifts back going from b to a, both inflation and GDP fall so shifting the AD curve generates the post war data we actually observed.5 When the economy is growing, consumers and firms are confident, they spend more, C and I both go up, and the AD curve shifts out. As the AD curve shifts out, both inflation and GDP go up. In a recession consumers and firms are cutting back, C and I fall, so AD shifts back causing inflation and GDP to fall. So prior to about 1969 we were able to explain the data through the tools of AD – AS where the AD curve shifted. This gives us a signal in choosing government policy. In a recession the recommendation was to increase G to offset falling C and I and thus increase AD, cut taxes to get C and I to increase AD, and lower interest rates to get consumers buying more houses and cars and businesses investing more to increase AD. During an inflationary period, the government could decrease G, raise taxes, and increase interest rates to cool the economy off by shifting AD back a bit.

However, in the early to mid 1970's both problems got worse at the same time, which we had never experienced before.6 The theoretical explanation is that Aggregate Supply shifted back moving the economy from a to c. What policy should we pursue? All we really understood back then was aggregate demand management since that was our experience since World War Two. Suppose Aggregate Supply shifts back and the economy goes from point a to point b below so that we're experiencing greater inflation, falling GDP, and rising unemployment. If we fight the recession and 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 d. What should we do?

5 When GDP increases, unemployment falls and when GDP falls, unemployment increases. The covariance between inflation and unemployment for the period 1950 – 1969 is cov = – 0.00227 < 0. So in this era, when inflation is up, unemployment is down. However, for the period 1969 – 1975, cov = 0.001 > 0. So when inflation is up, unemployment is also up! Now you can see why policymakers were so perplexed. (Data source: FRED.) 6 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 a 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.

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A new theory evolved called supply side economics to explain this phenomenon and suggest policies that may alleviate both problems at the same time. This is why the new theory became so popular so quickly; it seemed to solve both problems. Arthur Laffer put forth the following version of the new theory.7 There are three parts to it:

1. Cut tax rates and people will work more, save more, and invest more; 2. If a lot of people do this, total labor and capital increase, and GDP 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 also go up. Not only would the policy solve both problems at the same time, it would also be cost

free; we wouldn’t lose any tax revenue after the tax cut!8

Consider the following diagram on the left below. 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.

Tax revenue is R = tB, where R is tax revenue, t is the tax rate, and B is the tax base. For the US the base is income. In step #1, t is reduced as the “tax cut,” more economic activity is generated, and eventually B goes up in step #3 so you get the same revenue R. Points a and b are on the same horizontal level of revenue in the graph. However, If government spending is fixed, a deficit is created in step #1 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. Any deficit must be financed either by selling bonds or printing money. One doesn’t want to print money since that

7 Arthur Laffer was at an expensive luncheon with the up and coming young Republicans in 1974 including Dick Cheney and Donald Rumsfeld, both advisors in the Ford Administration. Laffer drew his famous “Laffer curve” on an expensive napkin and signed it for Rumsfeld. It is now in the Smithsonian. The theory actually dates to the 14th century, at least. A clear statement of it is by Ibn Khaldun, in his book The Muqaddimah, “It should be known that at the beginning of the dynasty, taxation yields a large revenue from small assessments. At the end of the dynasty, taxation yields a small revenue from large assessments.” It is not clear why the “assessment” (tax rate) increases as the dynasty proceeds. See http://blog.acton.org/archives/65499-obamacare-laffer-curve-napkin.html 8 Supply Side Economics only involves the financing of government spending and not government spending. There were other conservatives who also wanted to cut government spending. The problem is that the three largest areas of federal government spending are social security, health care, and defense. They comprised about 61% of the budget circa 1980 and about 12.5% of GDP. Today these three areas comprise about 70% of the budget or 16% of GDP. The biggest increase is in health care spending, which is mostly Medicare, which increased from 2% of GDP to about 5.5%.

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may cause inflation. So the government issues more bonds and these bonds must be paid off. So revenue actually has to increase enough to pay off these bonds.

What actually happened? The Reagan Administration managed to get the tax cut passed, the

largest of all time to that date, and we immediately began running massive deficits. 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, only a little. And savings actually fell during the late 1980's. People did respond to the incentive of the lower tax rate, but the magnitude of response was very small. So a cut in the tax rate led to less tax revenue, not more. The supply side theory was wrong.9 A more appropriate diagram would be the one on the right. In fact, there is no evidence that cutting tax rates increases revenue. The moral of the story is that a cut in the tax rate will generally cause the government to lose tax revenue. So if one supports tax cuts, one should also support spending cuts. Finally, the government debt of $17 trillion (2015) must be paid off eventually. How? By running budgetary surpluses in the future.

Senator Bob Dole resurrected the supply side theory in the 1996 presidential campaign. It was resurrected once again by the Bush Administration to justify the massive tax cut in 2001, and once again in 2008, when the Great Recession hit full force. It did little to stimulate the economy in 2001, or in 2008, nor did it bring forth a massive increase in savings and labor supply. Instead the deficits exploded after 2008 to over $1 trillion per year!!!

The supply side theory has also been used in a number of conservative states including Kansas, North Carolina, and Mississippi, among others. It has been a resounding failure in all of them. In Kansas they cut tax rates dramatically and they immediately started running massive deficits. Since the state must balance its budget, this requires it to impose massive spending cuts, tuition increases, and the complete elimination of some government services.10

We have depicted the ratio of total taxes at all levels of government to GDP for the countries in the OECD in the chart. This is the collection of the most advanced countries in the world economy. The data covers the period right before the Great Recession began. As one can plainly

9 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. See

http://www.theatlantic.com/magazine/archive/1981/12/the-education-of-david-stockman/305760/ 10 See http://www.cnbc.com/id/102659501, http://money.cnn.com/2015/01/11/pf/taxes/kansas-tax-cuts/ and http://www.nytimes.com/2015/02/12/us/politics/education-is-newest-target-of-kansas-budget-cuts.html?_r=0

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see, taxes in the US are not high, but low relative to the other industrialized countries that we compete with. The US ranks in the bottom third.

(Source: OECD, 2009. The data covers 2005 and 2007.)

5.6 A Model of Labor Supply and Incentives Why did supply side economic policy fail? As we saw in section 5.4, 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. The same is true of labor supply.

Consider the following set-up: the consumer-worker cares about leisure (L) and general consumption (C). Her tastes are represented by a utility function of the form, 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 about utility also hold in this new set-up.

Her budget constraint is wH = PC, where w = wage per hour, H = hours of work, and P = price of consumption. The consumer-worker's labor earnings is WH and her total expenditure is 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. If there are only 24 hours in a day, then the time constraint is 24 = L + H, or H = 24 – L. Substitute for H in the budget constraint, w(24 – L) = C, or 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).11 The indifference map is depicted on the far left. The budget constraint is depicted in the center, and the two are combined on the far right. The consumer – worker's optimal choice, (L*, C*), is depicted on the far right. 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*, i.e., subtract the distance L* from distance 24 in the right hand diagram to get hours worked. 11 This follows because I/p1 = 24w/w = 24 and I/p2 = 24w/P = 24w.

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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. In that case corr(H, W) < 0 and the labor supply curve is downward sloping.

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.12

Notice that the diagram above 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.

12 In fact, labor force participation among men (the per cent of men working) has been falling since 1950 from 86.2 to about 69.2 in 2014. See https://research.stlouisfed.org/fred2/series/LNS11300001

C

L

C

L

C

L

Tastes The Budget Optimizing Behavior

24

w24

L*2424 - L* = H*

C*

U = U(L, C) 24w = wL + PC

A

C

L24

A

B

An increase in the wageincreases labor supply

C

L24

A B

An increase in the wage reduces labor supply

13

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 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, 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 = (ΔH/H) ÷ (Δw/w).

Researchers find that Ew = 0.1, approximately, which is very small. 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

C

L24

A

B

H* = 8H* = 10

w

H8 10

supply

A

B

C

L24

A

B

H* = 8

w

H

8

supply

A

B

H* = 8before tax cut:

after tax cut:

w

H

taxcut

w

H

taxcut

S

∆H is large

S

∆H is small

Supply Side Tax Cut Works Supply Side Tax Cut Fails

14

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.

5.7 More Examples 5.7.1 Charity. Many individuals donate to charity. 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, donors one and two. 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, U(d1 + d2, x2) and I2 = d2 + x2. And her donation function is, 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 donor 2, 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. 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.

5.7.2 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” on inheritances.

B

C

B

CI/P

I/q

B

C

15

5.8 Price changes, compensation, and price clubs. 5.8.1 Compensating Grandma for Inflation Recently, controversy has arisen over use of the Consumer Price Index (CPI) as a measure of the standard of living. The retirees live on a fixed income and 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 toward the origin and the slope won’t change so the shift is a parallel one. This is the same as a decrease in income. This represents a loss in purchasing power. In the left hand diagram, a "smooth" inflation, where all prices go up by the same amount, can be perfectly offset by an increase in the elderly person's social security income. Inflation moves the budget line in and the increase in her social security benefit moves her budget line back out.

However, most actual inflation is not nice and "smooth” where all prices go up in the same manner. It is uneven. Some prices increase a lot, e.g., energy prices, pharmaceutical drug prices, some go up a little, e.g., meals in a restaurant, and some prices even fall, e.g., computer equipment. A price index averages across all the different prices. If the "average" price has increased, we tend to label this "inflation." The point is that if relative prices change during an "uneven" inflation, this will cause the person to economize by substituting away from the more expensive goods and toward less expensive goods. If energy costs, for example, have risen a lot, but the costs of going to a baseball game have only risen a little, then people, including the elderly, will economize by using less energy and going to more baseball games.

Now, how should we compensate the individual for an uneven inflation? Should we

compensate them so they can use as much energy and go to the same number of baseball games as before the uneven inflation occurred? This would compensate them for inflation AND a change in relative prices. Normally, we don’t compensate agents for relative prices changes. So how should we compensate Grandma?

Assume Grandma consumes denture cream and Ensure and she starts at point A on budget line I1 below. Suppose the price of denture cream increases substantially while the price of Ensure goes down 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. Suppose we increase the retiree's

Ensure

Denture cream

A

Ensure

A

Denture cream

“Smooth” inflation “Uneven” inflation

16

social security benefit (income) so the retiree can buy her original consumption bundle at 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, the retiree will not choose point A again with budget line I3 since relative prices have changed. Staying at A will not maximize her utility. She can do better by choosing bundle B on budget line I3. So we would expect the retiree to substitute away from denture cream toward Ensure and 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.

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. 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.

Ensure

Denture CreamI1I2

I1

I2

I3

A

B

I3

UA

UB

Ensure

Denture CreamI1I2

I1

I2

I3

A = B

I3

UA = UB

17

5.8.2 The law of one price? 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. If there were absolutely no transactions costs, there could only be one price in a marketplace. Why? Suppose there are two prices being charged for the same commodity. One can earn profits by buying at the low price and finding a buyer who is willing to pay the higher price. However, this activity would drive the two prices together until there was only one price being charged. This is sometimes called the law of one price.

However, if there are transactions costs in engaging in market trades this may not happen. Consider the simplest case of a transactions cost, travel to another market. 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 sold at 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, 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 in the middle diagram because the stereo is 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 right 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 with utility on the far right 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 below. 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 small 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.)

Travel to the InexpensiveLocation and Buy the StereoOut-of-Town

Buy Stereo Locallyin the InexpensiveLocation

Buy Stereo Locallyin the ExpensiveLocation

everythingelse

everythingelse

everythingelse

stereo stereo stereo

18

Application: Buying drugs in Canada. 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. Another recent story (NBC, 2015) documented this is still occurring.

Application: "Price clubs." Price clubs like Costco and Sam’s Club charge a membership fee to join the club but then charge lower prices for many 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 club and buying the good elsewhere. 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.

5.9 Substitution and Income Effects Why would someone respond to a wage increase by working more? Why would someone else work less? Presumably, they both have a greater incentive to work more when the wage increases. Why is the demand for some goods, e.g., cruise ship vacations, very price elastic, while the demand for others, e.g., Mariner’s tickets to a serious fan, is very price inelastic?

In general, there are two effects when any price changes: a substitution effect, where consumers shift toward cheaper goods, and an income effect because any price change will also change their real income, i.e., their level of utility. 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. Well, maybe not movies. 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.

Any price change will have two effects: 1. consumers economize through substitution 2. real purchasing power or the standard of living 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 since it’s now cheaper. Second, they are better off after the price falls so their real income or purchasing

EverythingElse

Stereo

EverythingElse

Stereo

Buy Out-of-Town

Buy Locally

19

power has increased. 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 purchased in a store 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 artificially.

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 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. Therefore, it doesn't affect the efficiency of the economy.

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 real 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. 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 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 of a price change 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.

20

Now let's combine the two effects. Suppose Ptheater increases. The consumer's budget line swivels in and the consumer would be observed going from bundle a to bundle b depicted below. We can decompose the move from a to b into two parts: the substitution effect and the income 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 point c. Her real purchasing power is the same because the extra income compensates her for the price increase. 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.

The opposite response will occur if the price of a movie falls. She will substitute toward movies and away from rentals due to the substitution effect and will buy more movies and rentals if they are both normal goods because her real purchasing power has gone up. The two effects work in the same direction when a good is a normal good.

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 (since it’s inferior). If the two effects offset perfectly, then we have the case of a perfectly inelastic demand curve. So goods that are in inelastic demand must be inferior goods.

Application: Inelastic labor supply 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 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

21

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 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.

5.10 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.

Example: Carrying an umbrella. 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 the probability of 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.

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 payoff from carrying the umbrella is

EP(umbrella) = p100 + (1 - p)30 = p100 + 30 - p30 = 30 + 70p. So once the probability 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. And, the expected payoff from not carrying the umbrella is,

EP(no umbrella) = p10 + (1 - p)70 = 70 - 60p. 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 highest expected utility or payoff. This leads to a decision rule. So, for example, if p = 1, EP(umbrella) > EP(no umbrella) so the optimal decision is to carry the umbrella. Decision rule: carry the umbrella if 30+ 70𝑝   ≥ 70− 60𝑝, or after a few steps of algebra, carry the umbrella if 𝒑   ≥ 𝟒/𝟏𝟑.

Example: Health Insurance. (Think of the duck saying, “Aflac.”) 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. Is this a rational decision? The Affordable Care Act

Nature

You

umbrella 100

10 70

30

noumbrella

Rain No Rain

22

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 and misses work, or not. Let W = income, P = premium for the insurance, A = payment from insurance 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.

Calculate the expected payoffs: Let q = probability of illness, 1 – q = probability of no illness. EP(Buy) = q(W – P +A – Z) + (1 – q)(W – P) = W – P + q(A – Z) = W –P + qA - qZ; EP(Don’t buy) = q(W – Z) + (1 – q)W = W – qZ. The decision rule is: Buy the insurance if W – P + q(A – Z) ≥ W – qZ; or, after a few steps

of algebra, buy if qA ≥ 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 (q) is zero, or that the loss due to illness (Z) is really small. Important Concepts

Savings Life cycle model Supply side tax cuts The Laffer curve Labor supply Compensating Grandma Price clubs Income and substitution effects Uncertainty Decision rule

Review Questions

1. 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. What is the logic behind supply side tax cuts? Did the massive tax cut in 1981-83 work?

Why or why not?

23

5. 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?

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

curve to slope upwards? Is it likely? 8. How can we model a situation involving an uncertain outcome?

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.

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 = Δu/Δc1 = 5, MU2 = Δu/Δc2 = 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.

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.

Nature US intervenes US doesn’t intervene

Attack W - T W Terrorist Don’t attack V V

24

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?

25

Answers

1. 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 = V 7. 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.)

C

C1

2