5.1 Labor Supply
For most households, labor is the single most important source of income.
Budget Constraint and Indifference Curves
For analyzing supply, a budget constraint is usually drawn between leisure and consumption. If an individual can work as many hours as he or she wants at a wage rate of w1, then the individual’s budget constraint between leisure and consumption is a straight line whose slope is –w1. At a given point, leisure is the number of hours between the origin and that point on the horizontal axis, and the number of hours is the space between that point on the horizontal axis and the horizontal intercept.
The horizontal intercept of the budget constraint line between leisure and consumption is the time endowment (T). This is the upper limit of time that can be dedicated to labor or leisure by an individual in a specified period.
The equation for the budget constraint is consumption (c) equal to wage per hour (w) times the number of hours worked (T-n), where n is the number of hours at leisure.
(5.1) c=w*(T-n) can rewrite as: c+w*n=w*T
The amount of money an individual would have if he worked every available hour is his value of the time endowment, this is w*T.
To characterize tastes, we use a set of convex indifference curves between leisure and consumption. The optimal amount of labor to supply is defined by a tangency between the budget constrain and an indifference curve. (Image pg. 124)
Comparative Statics and the Consumption-Leisure Model
Depending on the slope of the indifference curves, a decrease in wages can either increase or decrease the hours of labor supplied. The substitution effect and income effect work in opposite directions, and the result depends on which effect dominates.
If the substitution effect dominates, labor supply decreases when wage decreases. This is consistent with the statement “Now that my wage rate has fallen, it’s really not worth it for me to work as much as I used to.”
If the income effect dominates, labor supply increases when wage decreases. This is consistent with the statement, “Now that my wage rate has fallen, I have to work more to maintain my standard of living.”
[Images page 125.]
Labor Supply Curve
The labor supply curve is a schedule showing the relationship between the quantity of labor supplied and the wage rate, all else equal. The supply of labor curve is found by determining the utility-maximizing number of hours associated with each wage rate (this can be found by looking at various budget constraint possibilities and their associated indifference curves).
The supply curve will slope upward if the substitution effect dominates the income effect. If the income effect dominates that substitution effect, then the supply curve will slope downward. It is possible that the substitution effect will dominate at low wage rates and the income effect will dominate at high wage rates. This results in a labor supply that bends back on itself (starts downward sloping, then becomes upward sloping.)
The Work Incentives of AFDC
The Aid to Families with Dependent Children (AFDC) program provided cash to families with dependent children and an absent or incarcerated parent. The program reduced the grant by one dollar for each dollar of earnings – it was argued this might reduce incentives to work. The budget constraint line for a person on this program is kinked. It has a normal slope down to the level of AFDC, and then becomes horizontal. On that horizontal section, when a person increases work by one hour, they increase their wage, but also decreases the AFDC payment, so the total income is still the same. Based on this result, we can see that a rational person may work for wages higher than the AFDC payment, but would never work any hours if they were on the horizontal section of the line (wage is less than or equal to the AFDC payment). This is because on this line, they can increase leisure “for free,” or in other terms, their wages are being taxed 100 percent along this area.
The producer surplus is the amount of income an individual receives in excess of what she would require in order to supply a given number of units of a factor. The area above the supply curve and below the wage rate is the producer surplus.
The concept of consumer surplus can be used to help understand unemployment insurance. A person who initially works 2000 hours at $20 per hour does not need $40,000 to compensate him for his job loss. Instead, he should be compensated equal to the area of his producer surplus. Although unemployment forces the person to consume more leisure than he would prefer, he does place some value on that leisure, and this should be taken into account in determining the net cost to him of being unemployed.
The Supply of Labor to Occupations
The total number of hours supplied to a given occupation is the total numbers of hours in the economy times the proportion of hours deveoted to this profession.
Market Supply Curve of Labor
The market supply curve of labor is a schedule showing the aggregate quantity of labor that all individuals in the market are willing to supply at each wage rate, all else equal. This can be found by summing horizontally he supply curves of labor for all individuals.
People are concerned both about the wages they make as well as the nonmonetary aspect of their jobs (clean, safe jobs are preferred to dangerous, dirty jobs). The wage premium paid to compensate workers for taking jobs with undesirable characteristics is the compensating differential.
For a low-skilled individual, the available set of combinations of safety and wage rates lies below that for a high-skilled individual. However, within their constraints, both prefer a higher-paying and safer job. [Image page 136]
5.2 Capital Supply
Real capital are those things made to aid future production, such as drill presses, factory buildings, office desks, and computers. These are purchased using money borrowed from households’ savings.
Financial capital is the money that is lent to firms to purchase or rent real capital (this comes from people’s savings).
The life-cycle model is a model that says people’s consumption and saving decisions during the given year are the outcome of a decision-making process that takes into account their lifetime economic circumstances.
Intertemporal Budget Constraint
The intertemporal budget constraint is the budget constraint in the life-cycle model, showing the trade-off between consumption levels at different periods. It has current consumption on the horizontal axis, and future consumption on the vertical axis, so that the budget constraint shows the trade-off between current and future consumption. It has a slope of –(1+i) and must pass through the endowment point.
The endowment point is the feasible consumption bundle if the individual makes no trades with the market. In the life-cycle model, it is the bundle an individual can consume if he or she neither saves nor borrows. This is the point (I0,I1), where I0 is his current income and I1 is his future income.
If the person decides to save some of his money, he increases future consumption by (1+i)*S (where i is the interest rate, and S is the number of dollars he decides to save).
If a person decides to borrow money to increase future consumption, he can increase current consumption by B, and reduce future consumption by (1+i)*B.
The present value of the endowment Is the maximum level of present consumption that can be obtained, given the endowment. (This is the horizontal intercept of the intertemportal budget constraint). It is equal to I0+I1/(1+i) where I0 is his current income and I1 is his future income.
Intertemporal Indifference Map
Indifference curves showing the preferences for future as opposed to present consumption have a marginal rate of substitution that is called the marginal rate of time preference. For an ‘impatient’ individual, the marginal rate of time preference exceeds one when future consumption equals present consumption; that is, the negative of the slope is greater than one around the 45 degree line.
Equilibrium in the Life-Cycle Model
The equilibrium is the point where the intertemporal budget constraint is tangent to the indifference curve. If this point is to the left of the endowment point, the person will be a saver. If this point is to the right of the endowment point, this person will be a borrower.
Comparative Statics with the Life-Cycle Model
Saving and Interest Rates
A change in the interest rate can either lower or increase saving. The effect depends on whether the substitution effect or income effect dominates.
1. The Substitution Effect: When the interest rate decreases, the opportunity cost of present consumption decreases due to the reduction in the amount of future consumption that is sacrificed for each dollar of present consumption. This tends to increase present consumption, and therefore reduce saving.
2. Income Effect: If you are a saver, when the interest rate decreases, you become poorer because the people to whom you lend pay you less money. Because present consumption is a normal good, this decreases in your income tends to lower present consumption, and therefore increase saving.
Supply of Saving
The market supply curve of saving is the aggregate amount of saving that all individuals are willing to supply at each interest rate, all else equal. It is found by using a horizontal summation of the individuals supply curves.
The Taxation of Interest Income
Some argued that lowering the tax rates on interest would induce a great surge in saving, others argue that there won’t be much effect. When interest income is taxed at rate t, the effective interest rate is (1-t)/i. Hence, the opportunity cost of reducing current consumption by a dollar is [1+(1-t)i] of future consumption. This is minus the slope of the after-tax budget constraint to the left of the endowment point (on the saving section of the line). To the right of the endowment point (on the borrowing section of the line), the slope is still –(1+i). So the budget constraint is a kinked line. Therefore, for individuals who were borrowers before the tax was imposed, the system has no effect at all. If people were savers before the tax, we cannot predict whether the change will cause them to save more or less (see table above). This analysis tells us that the effect of interest taxes on the supply of saving is ambiguous.
5.3 More on Present Value
The present value is the amount of money you would be willing to pay today for the right to receive a given amount of money at a specified date in the future.
This depends on the discount rate. The discount rate is the interest rate used in the computation of present value.
In general, when the annual interest rate is I, the present value of a promise to pay $M in T years is:
If there is an income stream, rather than a one-time payment, then we add the present value of each year’s payment: PV=M0+M1/(1+i)+M2/(1+i)2+…+MT/(1+i)T
Present Value in Action
Michael Wittkowski’s Lottery Price
Michael Wittkowski won $40 million in the Illinois State Lottery. The prize was to be distributed in 20 equal installments, with $2 million paid each year. Using the equation for present value, we can see that with an interest rate of about 12 percent, this prize was only worth about $16 million.
Truth in lending laws require mortgage providers to tell borrowers the sum of all the payments that will have to be made over the life of the loan. However, these calculations do not take into account present value. Without this information, it may be difficult to tell the true cost of a 15 or 30-year loan, as well as the difference between the two.
A perpetuity is a stream of income that lasts forever. To find the value of a perpetuity that pays $M annually, we need to determine how much money would have to be insvested now at an annual rate of i percent to obtain $M each year. The present value of a perpetuity is: PV=M/i
Though few payments are really perpetuities, it can be a good approximation for a stream of income that comes in over a finite number of years (such as 20).
5.4 Human Capital
Human capital are the investments that individuals make in education, training, and health care that rise their productive capacity.
Human Capital as the Only Asset
The human capital production function is the relationship between investments in human capital and future gains in income.
When the individual does not have access to financial capital markets, optimal human capital investment is determined at the tangency of an indifference curve and the human capital production function.
Human and Physical Capital
When the individual has opportunities to borrow and lend at the market interest rate, the optimal amount of human capital investment is defined by a tangency between the human capital production function and a straight line whose slope is –(1+i). This occurs regardless of preferences between present and future consumption.
The separation theorem states that the existence of markets allows a person to separate his production decision from his consumption decision.
If the individual can borrow and lend at the going interest rate, he should make whatever human capital investment maximizes the present value of his endowment. Once the present value is maximized, particular preferences determine how much to consume in the present and how much in the future.