Análisis ambiental

Michael Malcolm

June 18, 2011

In the previous unit, we considered how a consumer allocates a fixed income over consumption of various goods. In this unit, we extend our basic constrained consumer choice framework to other kinds of problems faced by consumers. In this section, we study the consumer’s choice to allocate his time between labor (which produces income) and leisure time.

1

Setup and Constraint Set

A consumer derives utility both from consumption of goods and from leisure time. We denote number of hours of leisure time asN and denote consumption asY, which just represents the monetary income earned by the consumer to be used for purchases of consumption goods.

Naturally, there is a tradeoff between the two. The more leisure time N that the consumer takes, the less moneyY he will have available for consumption spending. When the consumer’s wage isw per hour, the ”budget set” is sketched in figure 1.

One option isN= 24 hours of leisure time and no consumption spending. Another option isN = 0 hours of leisure – then at a wage ofwper hour, the consumer’s income for consumption spending isY = 24w. The slope of the budget line is−w. Each hour of additional leisure time requires that the consumer give upwof consumption spending. The ”cost” of an hour of leisure is precisely the opportunity cost of the wage that could have been earned over that hour.

We assume that the consumer’s preferences over leisure and consumption obey diminishing MRS, so that a typical indifference curve is convex as shown in figure 2. A consumer with very little leisure time would be willing to give up a lot of consumption to have more leisure. A consumer who already has a lot of leisure time is not willing to give up as much consumption spending in order to get more.

2

Numerical Example

Consider a consumer whose utility function over consumption spending Y and leisure time N is given by

U(Y, N) =Y N, which makes sense in that bothY andN raise the consumer’s utility.

As for the constraint set, we assume that the consumer can work as many hours as he wants each day for a wage ofw. If the consumer takes N hours of leisure, then he works 24−N hours and so the amount of money that he has for consumption spending would be:

Y =w(24−N) The constraint set could equivalently be written as 24 =N+Y_w. So then, the problem stated formally is:

Figure 1: Budget constraint over leisure and consumption

Notice that, rearranged to be set equal to 0, the constraint can be rewritten as

Y −24w+wN = 0 The Lagrangian is:

L=Y N+λ(Y −24w+wN) The first order conditions are:

∂L ∂Y =N+λ= 0 ∂L ∂N =Y +λw= 0 ∂L ∂λ =Y −24w+wN = 0

Solving the first two first order conditions forλ:

N+λ= 0⇒λ=−N Y +λw= 0⇒λ=−Y

w

Equating the expressions forλ:

−N =−Y

w

N = Y

w

Substituting this back into the constraint set:

Y =w(24−N) Y =w 24−Y w Y = 24w−Y Y = 12w

Substitute this back into our expression forN:

N = Y

w N = 12w

w ⇒N = 12

This consumer always takes N = 12 hours of leisure, and conversely works 12 hours, for consumption spending of 12w. In other words, the consumer’s labor supply is perfectly inelastic at 12 hours. He always works 12 hours and takes 12 hours of leisure, regardless of the wage.

Figure 3: Deriving the labor supply curve

3

Labor Supply Curves

In the example above, the consumer always took 12 hours of leisure and worked for 12 hours, regardless of the wage. This may not always be true, though. Consider the consumer whose preferences are shown in figure 3. When the wage isw= 4, the consumer’s optimal bundle is to takeN = 16 hours of leisure and to work 8 hours. However, when the wage rises tow= 8, the consumer’s optimal bundle involves taking only

N = 14 hours of leisure and to work 10 hours.

This indifference diagram gives us two points on the labor supply curve. When w = 4, the consumer works for 8 hours, but whenw= 8, the consumer works for 10 hours. Figure 3 sketches the corresponding labor supply curve.

4

Income and Substitution Effects from Wage Changes

Suppose that the wage rises. There are two effects.

• Substitution Effect: The relative price of leisure rises. As the wage rises, the opportunity cost of an additional hour of leisure time rises since more income is sacrificed for each hour of leisure. Since leisure is more expensive, the substitution effect will cause the consumer to reduce leisure hours and correspondingly increase work hours.

• Income Effect: The consumer’s real income rises. Since leisure is assumed to be a normal good, one of the things that the consumer buys is more leisure time. Since purchasing power is higher, the income effect will cause the consumer toincrease leisure hoursand correspondinglyreduce work hours.

Informally, when wage rises, someone might say: ”It’s more expensive to sit at home and watch TV now. It’s more worthwhile to work, so I’m going to work more.” This is the substitution effect. At the same time, the person might say: ”I’m wealthier now and don’t need to work as much to make the amount of money I want to make, so I’m going to work less.” This is the income effect. The point is that the two are pulling the

consumer in opposite directions. When wage rises, the substitution effect is pushing the consumer to work more while the income effect is pushing the consumer to work less.

Regarding the assumption that leisure is a normal good: consider a consumer who receives a raw income transfer, rather than a wage increase; in other words, an income shift with no corresponding substitution effect. If you get $1 billion cash, then you might buy a new house or a new car, but presumably one of the things you would do is to work less – in effect, you are using your income to ”buy” more leisure time for yourself. This is exactly what it means for leisure to be a normal good. A pure income increase spurs consumers to take more leisure time.

Because the income and substitution effects work in opposite directions, a wage increase could potentially cause consumers to work more or work less. The net effect is uncertain.

We can show the income and substitution effects graphically. In figure 4, the wage rises fromw0 tow1,

causing the consumer’s budget line to pivot out fromB0 to B1. Comparing the optimal bundles on these

two budget lines, the consumer’s optimal number of leisure hours rises fromN0 toN1.

To decompose this total effect into income and substitution effects, we follow the same procedure as before. Taking the new budget line B1 and shifting it until it is tangent to the original indifference curve

gives us the hypothetical budget lineBH.

Now, the twist fromB0 toBH represents a change in the price ratio (here, the wage rate) but with no

change in purchasing power. This piece is the substitution effect. The increase in the price of leisure (wage rate) causes the consumer to reduce his leisure time fromN0toNH as a result of the substitution effect.

The shift fromBH toB1is a pure increase in purchasing power with no change in the relative price ratio.

The higher purchasing power causes the consumer to increase his leisure time fromNH toN1as a result of

the income effect.

Notice that the increase in leisure time resulting from the income effect is greater than the reduction in leisure time resulting from the substitution effect. Thus, the total effect upon combining the two is an increase in leisure time resulting from the higher wage. Figure 4 summarizes the changes.

5

Backwards-Bending Labor Supply Curves

Economic theory gives an ambiguous prediction about the effect of wage increases on labor supply. Depending on whether the substitution effect or the income effect is stronger, either an increase or a decrease in hours worked is possible.

Evidence seems to suggest that the substitution effect dominates for small wage increases – small increases in pay for working overtime hours often induce consumers to work more hours. For modest increases in wage, the income effect is fairly small, so the substitution effect dominates and labor supply rises. However, for large wage increases, the induced change in purchasing power becomes more substantial and so the income effect takes over and labor supply falls. If your wage rises by 10%, you might well choose to work more hours. On the other hand, if your wage increases to $1 million per hour, it seems doubtful that you would continue to work 40 hours a week for the rest of your life.

This gives rise to a backwards bending labor supply curve. That is, small wage increases cause labor supply to rise, but beyond a certain point, further wage increases cause labor supply to fall.

This situation is illustrated in figure 5. The wage increase fromw0tow1 causes the consumer to reduce

his leisure time from 18 hours to 14 hours, increasing his work hours from 6 hours to 10 hours. Labor supply is upward sloping over this interval. The substitution effect is stronger over this interval.

However, the wage increase fromw1tow2causes the consumer to increase his leisure time from 14 hours

to 15 hours, reducing his work from 10 hours to 9 hours. Labor supply is downward sloping over this interval. The income effect is stronger over this interval.

Figure 4: Income and substitution effects from a wage increase

Figure 6: Budget line with outside incomeY0 and income tax rateτ

6

Labor Supply and Public Policy

Reducing the income tax rate is equivalent to raising the hourly wage. A common assertion is that lowering income tax rates will cause people to work more because they find it more worthwhile to work. But, this is only half the story since this is basically an argument about the substitution effect. The income effect works in the opposite direction – lower tax rates give people more purchasing power, some of which is used to purchase more leisure, which drives work hours down. Thus, the effect on labor supply of lower income taxes is, in fact, uncertain.

A lump-sum transfer is even worse. There is no substitution effect since a lump-sum transfer does not change the relative trade-off between working and taking leisure time. However, the income effect causes leisure to rise and work hours to fall. Thus, the effect is an unambiguous decline in labor supply. However, notice that a lump-sum tax (that is, a tax that is not tied to work hours and income earned), would have the opposite effect and would unambiguouslyraise labor supply.

7

Non-Labor Income

Consider now a case where, in addition to income earned from working, the consumer had some outside income ofY0, regardless of whether he works or how much he works. Also, the consumer pays a tax rate of

τ on any labor income that he earns. The constraint set is shown in figure 6.

There are two changes. First, even when taking 24 hours of leisure, the consumer still has income of

Y0. In effect, the non-labor income shifts the whole budget line up by Y0. Furthermore, because of the

income tax, the government takes a portion τ of the consumer’s income, so each hour of work yields extra consumption of (1−τ)w. This is the relevant opportunity cost of an hour of leisure.

Mathematically, we can express the new budget constraint as follows:

Y =Y0+w(1−τ)(24−N)

The consumer’s income isY0 plus his labor income. Since he works 24−N hours and earns (1−τ)wfor

Let us consider a consumer with the same utility function over consumption and leisure as in the earlier example, namelyU(Y, N) =Y N.

So then, the problem stated formally is:

maxY N s.t. Y =Y0+w(1−τ)(24−N)

Rearranged to be set equal to 0, the constraint can be rewritten as

Y −Y0−24w(1−τ) +w(1−τ)N= 0

The Lagrangian is:

L=Y N+λ(Y −Y0−24w(1−τ) +w(1−τ)N)

The first order conditions are:

∂L ∂Y =N+λ= 0 ∂L ∂N =Y +λw(1−τ) = 0 ∂L ∂λ =Y −Y0−24w(1−τ) +w(1−τ)N = 0

Solving the first two first order conditions forλ:

N+λ= 0⇒λ=−N Y +λw(1−τ) = 0⇒λ=− Y

w(1−τ)

Equating the expressions forλ:

−N =− Y

w(1−τ)

N = Y

w(1−τ) Substituting this back into the constraint set:

Y =Y0+ 24w(1−τ)−w(1−τ)N Y =Y0+ 24w(1−τ)−w(1−τ) _Y w(1−τ) Y =Y0+ 24w(1−τ)−Y Y =Y0 2 + 12w(1−τ)

Substitute this back into our expression forN:

N= Y

w(1−τ)

N= Y0

2w(1−τ)+ 12

For comparative statics, notice in this case that as wrises,N falls and so work hours rise. That is, the substitution effect is stronger than the income effect. Similarly, asτ rises,N rises and work hours fall. This makes sense since an increase in the tax rate is equivalent to a lower wage. Finally, asY0rises,N rises and

work hours fall. But again, this makes sense because this is a pure income effect. With more non-labor income, a consumer will buy more leisure time and so his labor supply will fall.