Homework 1 / Solution
C
slope = Katie’s
wage
32 H
Katie
C
slope = Aaron’s
wage
32 H
Aaron
- (4 points) Consider the indifference curves illustrated below. Compare Katie and Aaron’s preferences for work (i.e. who dislikes work more). Explain. What conclusion can be drawn about Katie and Aaron’s wages if they both choose to work the same number of hours? For instance, if they both decide to work 32 hours per week, who earns a higher wage? Explain.
Because the slope of an indifference curve is the marginal rate of substitution and calculated as it represents a worker’s willingness to substitute consumption for leisure. A relatively steep indifference curve (such as the one drawn for Katie) suggests that the additional utility gained from one more hour of leisure is large relative to the additional utility gained from another dollar of consumption. The opposite is true for a flat indifference curve like Aaron’s. Given this interpretation, Katie’s gain from an additional dollar of leisure is larger (relative to her utility gain from consumption) than Aaron’s. As such, Katie has a stronger preference for leisure, or equivalently, a strong preference not to work.
The slope of the budget line is equal to the worker’s wage. Furthermore, it is at the tangency between the indifference curve and budget line that the worker chooses the optimal consumption and leisure bundle. Therefore, if Katie and Aaron are both working the same number of hours, it is at that point that their wage is equal to the slope of the indifference curve. Since Katie’s indifference curve is steeper than Aaron’s, she must be earning a higher wage (see figure).
- The utility function of a worker is represented by U(C,L) = C x L. The marginal utility of leisure is then given by MUL = C and the marginal utility of consumption is given by MUC = L. Suppose that this person currently has a weekly income of $600 and enjoys 70 hours of leisure per week. How many additional dollars of income would it take to make a non-worker give up 10 hours of leisure time?
C U=42000
700
600
60 70 L
Initially, this worker enjoys utility in the amount of U(C,L) = C∙L = 600∙70 = 42,000. If he or she is willing to enjoy only 60 hours of leisure, in order to be equally happy, their income (consumption spending) must be:
U(C,L) = 42,000 = C∙60
C =
Graphically, this simply means the worker must remain on the same indifference curve as before. Initially, the consumption/leisure bundle is given by point M. When 10 hours of leisure are forfeited, there must be a $100 increase in weekly income.
- Suppose Jeff experiences an increase in wages from below his reservation wage to above his reservation wage. What can you conclude about the income and/or substitution effect(s)? Do they both exist? If so, which one will dominate?
The substitution effect states that there is a positive relationship between a worker’s wage and the number of hours he decides to work. When the wage rate increases, there is an increase in the opportunity cost of consuming one more hour of leisure. When the opportunity cost of a good (leisure in this case) increases, people demand less of that good. Therefore, in increase in the wage rate reduces the demand for leisure, which is equivalent to an increase in the time spent in the labor market. The income effect suggests that there exists a negative relationship between the wage rate and number of hours worked. When the wage rate increases, workers have more income to spend on leisure activities. Assuming leisure is a normal good, more of it will be demanded when income increases. As a result, fewer hours will be worked.
When the wage rate was initially below Jeff’s reservation wage, he would have opted out of the labor market. Now, however, an increase in the wage above his reservation value will induce him to work some positive number of hours. This result is consistent with the substitution effect, as staying home has now become more expensive. The income effect suggests he would now choose to work fewer hours than before. This is not possible, however, since working fewer hours than zero is not an option. Therefore, only the substitution effect exists when a worker was initially out of the labor force.
C
P
Q
R
V
L1 L2 T L
C
P
Q
R
V
L2L1 T L
- What happens to hours of work when the wage rate falls? Decompose the change in hours of work into income and substitution effects.
There are two possible final outcomes of a decrease in the wage rate. Both are shown at the right. In both cases, a decrease in the wage rate decreases the opportunity cost of enjoying one more hour of leisure, thus increasing the demand for leisure. This is called the substitution effect. Simultaneously, a decrease in the wage rate affords the worker fewer opportunities in terms of leisure activities because less money (income) is earned. As a result, less leisure is demanded, which is the income effect. The income and substitution effects produce competing results, and whichever dominates will determine whether more or less hours are worked. The top graph illustrates a dominant substitution effect, and the lower graph depicts a situation in which the income effect dominates.
To decompose the change in hours worked into income and substitution effects, the income effect can be isolated by drawing a hypothetical budget line tangent to the new indifference curve and parallel to the original budget line (since the slope of the budget line is the wage rate, producing a pure income effect, holding the wage rate constant, will segregate the income effect). In the graphs provided, the income effect is represented with the shift from P to Q, and the substitution effect is given by the move from Q to R.
- The way the worker’s compensation system works now, employees permanently injured on the job receive a payment of $X each year whether they work or not. Suppose the government were to implement a new program in which those who did not work at all got $0.5X but those who did work got $0.5X plus workers’ compensation of 50 cents for every hour worked (of course, this subsidy would be in addition to the wages paid by their employers). What would be the change in work incentives associated with this change in the way workers’ compensation payments were calculated?
C
1
C
B
X BL1
A
BL2
.5X
T L
If workers now only receive ½X in the form of non-labor income and experience a $0.50 per hour wage increase, the original budget line BL1 will become steeper (BL2) and reflect an endowment that is half of its original value. There are three types of workers to consider:
A: Worker A was a non-worker prior to this change in the worker’s compensation program. Non-workers never experience income effects, so this worker might only consider entering the labor force as a result of the substitution effect. Specifically, the wage offer available (w +0.5) may now exceed the worker’s reservation wage, inducing him to choose H* > 0.
B: Worker B experiences a decrease in total income, as seen by the downward shift of the budget constraint, and an increase in his wage rate. The decrease in non-labor income will induce him to increase H* because he now has less money with which to purchase leisure (and leisure is a normal good) by the income effect. The higher wage rate will also encourage him to increase hours of work since, by the substitution effect, the opportunity cost of consuming leisure has increased, thus making him demand less of it.
C: Worker C similarly enjoys an increase in the wage rate, but total income also increases since the budget constraint has shifted away from the origin. Thus, the substitution effect suggests less leisure will be consumed since it is now more costly. The income effect suggests that the worker will demand more leisure. Since total income has increased, normal goods (including leisure) will be in greater demand. The final effect on H* is ambiguous.
- Suppose Margaret derives utility from consumption and leisure according to the utility function U = 10CL. Accordingly, her marginal utility functions are MUL = 10C and MUC = 10L.
- State and interpret the utility-maximizing rule (condition) that workers use to determine how many hours to work and leisure, as well has how much money to spend on consumption.
Rule:
Interpretation: The above condition can be rearranged as for easier interpretation. The left-hand side represents the additional utility the individual gains from spending an additional dollar on leisure, whereas the right-hand side represents the additional utility gained from spending an additional dollar on consumption. When equal, the worker is maximizing utility.
- Suppose that Margaret initially earns an hourly wage of $12.50 and that she spends $2000 on consumption goods and services. Use the utility-maximization rule from part (a) to determine how many hours she devotes to leisure. How would this answer change if her hourly wage decreased to $12.00 and she spent $1800 on consumption?
The condition from part (a) can be used in accordance with Margaret’s utility function information to solve for the optimal number of leisure hours. Specifically, Using this formula, wage and consumption information can be substituted in to solve for L*.
w = $12.50, C = $2000: (note that this means H1* = T – L1* = 240)
w = $12.00, C = $1800: (note that this means H1* = T – L1* = 250)
- Use your answers from part (b) to determine Margaret’s elasticity of labor supply when the wage rate changes from $12.50 to $12.00 per hour. Verbally interpret this elasticity and characterize it as elastic or inelastic.
The elasticity of labor supply is defined as and is calculated as An elasticity of -1.0417 means that a 1% increase in wages
will cause Margaret to decrease the number of hours she works by 1.0417%. Finally, because |σ| > 1, this is considered elastic labor supply, which suggests Margaret is responsive in terms of hours worked to wage changes, though only slightly in this example.
- Use your answers from part (b) to graph Margaret’s labor supply decision. Draw the initial and new budget constraints, as well as the appropriate indifference curves.
C
BL1
P
BL2 R U1
U2
V
150 160 T L
When Margaret experiences a wage decrease from $12.50 per hour to $12.00 per hour, her budget line rotates down, as illustrated. Part (b) indicates that when the wage rate decreases, Margaret works more hours and consumes less leisure. Additionally, the elasticity in part (c) reinforces that there is a negative relationship between the wage rate and the number of hours Margaret works. Therefore, the substitution income dominates. Margaret initially chooses the leisure-consumption bundle labeled P, and her final choice is bundle R. To isolate the income and substitution effects, a hypothetical budget line has been drawn (in red) which is parallel to the original budget constraint and tangent to the new indifference curve. Therefore, P to Q is the income effect (when w ↓, income ↓, and since leisure is a normal good, L* ↓), and Q to R is the substitution effect (when w↓, the opportunity cost of leisure ↓, so L* ↑). Note here that the substitution effect is illustrated to have a very small impact on the worker’s labor supply choices.