ECON 115 - Labor Economics Drake University, Spring 2024 William M. Boal

Version A

I. Multiple choice

(1)c. (2)b. (3)d. (4)c. (5)b. (6)a. (7)b. (8)c. (9)a. (10)b. (11)c.

II. Problems

(1) [Elasticity of labor demand: 8 pts]

1. decrease.
2. 2 percent. Set -0.2 = percent change in jobs / 10 percent, and solve.
3. increase.
4. 8 percent. Percent change in workers' income = percent change in jobs + percent change in wage.

(2) [Budget constraint: 6 pts]

• Endowment point is at L=60 hours and C=\$200.
• Initial slope = - wage = -\$10 per hour. If Alex works 20 hours, will have \$400 in total income for consumption. But then the tax starts, so the kink point is at 40 hours of leisure and \$400 in consumption.
• Further income is taxed at 50%, so Alex's after-tax wage becomes \$5. So to the left of the kink point, the slope of the budget constraint is -\$5 per hour.
• If Alex works all 60 hours, then the first 20 hours are paid at \$10 per hour and the last 40 hours are paid at \$5 per hour. Together with \$200 nonlabor income, Alex would be able to spend \$600 on consumption, the intercept on the consumption axis.

(3) [Individual labor supply--optimal choice: 12 pts]

1. MRS = MUL/MUC = (C-50)/L.
2. The reservation wage equals the MRS at the endowment bundle. Inserting nonlabor income for C and total available time for L gives reservation wage = (150-50)/60 = 5/3 = \$1.67.
3. Budget constraint is spending = income, or C = 150 + (60-L) 10 = 750 - 10 L.
4. Tangency condition is MRS = wage, or (C-50)/L = 10. Solve this equation jointly with the budget constraint found in part (c), to get L*= 35 hours, C*= \$400.
5. h* = total available time - L* = 25 hours.

(4) [Individual labor supply--income and substitution effects: 22 pts] Note that the "hypothetical budget line" is drawn tangent to the new indifference curve, but parallel to budget line #1.

1. 50 hours.
2. \$200. The graph shows that if Beth works zero hours, she can still enjoy \$200 in consumption.
3. \$10 per hour. The wage is the negative of the slope of the budget line.
4. \$20 per hour. The wage is the negative of the slope of the budget line.
5. Income effect: work less. Reason: An increase in the wage is like an increase in nonlabor income in that the budget line is now farther from the origin. So the income effect is to "purchase" more leisure and more consumption. More precisely, the income effect is a movement between indifference curves and parallel budget lines, here from tangency with budget line #1 to tangency with the hypothetical budget line.
6. -10 hours.
7. Substitution effect: work more. Reason: An increase in the wage is an increase in the "price" of leisure compared to the price of consumption. So the substitution effect is to "purchase" less leisure and more consumption. More precisely, the substitution effect is a movement along a single indifference curve from tangency with hypothetical budget line to tangency with budget line #2.
8. +15 hours.
9. Total effect: work more. Reason: total effect = substitution effect + income effect.
10. +5 hours.
11. (w,h) = (\$10,30 hours), (\$20,35 hours). This labor supply curve slopes up.

(5) [Household specialization: 10 pts]

1. Household's joint production-possibility curve has intercepts at \$200 of market goods and 300 units of household services, because these are the points where both parties are working in the market or both are working in the household.
To find the rest of the curve, begin where both parties are working in the household, producing zero market goods and 300 units of household services. If market goods are desired, who should work in the market? Party B has a comparative advantage in market work. Sending Party B into the market creates a line segment with slope = -1, because giving up 10 units of household services gains \$10 in market goods for the household. This continues until Party B is working full-time (10 hours) in the market, producing \$100 in market goods but no household services. At that point, the household enjoys \$100 in market goods (produced by Party B) and 200 units of household services (produced by Party A). This kink point should be circled. The remainder of the PP curve is a line segment joining the kink point to the vertical intercept.
When the production possibility curve is plotted on the graph with the indifference curves, it is obvious that this household reaches its highest attainable indifference curve at the kink point, where ...
2. Party A works zero hours in the labor market, and
3. Party B works 10 hours in the labor market (i.e, full time).

(6) [SR labor demand: 9 pts]

1. Value of marginal product = output price × MPE = 30 (K/E)1/2.
2. The firm maximizes profit by choosing E so that wage = VMPE, or 20 = 30 (K/E)1/2. Solve to get E* = 36.
3. Using production function, q* = 2 (16×36)1/2 = 48 units of output.
4. Profit = total revenue - total cost
= total revenue - cost of labor - cost of capital
= \$1440 - \$720 - \$160
= \$560.

(7) [LR labor demand--scale and substitution effects: 16 pts] Note that the "hypothetical isocost line" is drawn tangent to the old isoquant curve, but parallel to isocost line #2.

1. Slope of any isocost line = -w/r, where r = price of capital. The graph shows slope of isocost line #1 is -1. We are given that r = \$10, so wage must = \$10 on isocost line #1.
2. Similarly, the slope of isocost line #2 is -3, so wage must = \$30 on isocost line #2.
3. Substitution effect: use less labor. Reason: Substitution effect is movement along isoquant from tangency with isocost line #1 to tangency with hypothetical isocost line with the new slope.
4. -10 units of labor (from 70 units of labor to 60 units).
5. Scale effect: use less labor. Reason: Scale effect is movement between isoquants and parallel iscost lines, from hypothetical iscost line to isocost line #2.
6. -30 units of labor (from 60 units of labor to 30 units).
7. Total effect: use less labor. Reason: total effect = substitution effect + scale effect.
8. -40 units of labor.

III. Critical thinking

(1) The slope of the budget line equals the negative of the wage because each additional hour of leisure requires forgoing the earnings from an hour of work. If the hourly wage is constant, then the budget line must be a straight line. If the effective wage changes depending on the number of hours worked, then the budget line would not be straight. Examples where the budget line has kinks include a tax on income above some threshold, time-and-a-half pay above 40 hours per week, or a cash grant that must be repaid as a fraction of earnings. (Full credit requires a graph of a typical budget line.)

(2) Isoquants trace out combinations of labor and capital inputs that yield the same amount of output. Typically, both inputs are productive--that is, they both have positive marginal products. Thus for example if labor input were increased and capital input were held constant, then output would increase. The only way that output could remain constant when labor input were increased would be for capital input to be decreased. So isoquants typically must slope down.

Version B

I. Multiple choice

(1)d. (2)a. (3)e. (4)b. (5)c. (6)b. (7)d. (8)a. (9)b. (10)d. (11)b.

II. Problems

(1) [Elasticity of labor demand: 8 pts]

1. decrease.
2. 3 percent. Set -0.3 = percent change in jobs / 10 percent, and solve.
3. increase.
4. 7 percent. Percent change in workers' income = percent change in jobs + percent change in wage.

(2) [Budget constraint: 6 pts]

• If Alex does not work, but receives a cash grant, then leisure = 60 hours and consumption = \$200. So this is one kink point.
• If Alex then works, the grant is reduced by 50 cents for every dollar of earnings, so the "after-tax" wage is \$5. Thus the budget constraint here has a slope of -5.
• Once Alex has worked 40 hours (that is, L=20), then the entire cash grant will be repaid, and Alex's wage returns to \$10. So next kink point is at 20 hours of leisure and \$400 of consumption.
• Subsequent work does not involve repaying the cash grant, so the slope of the budget constraint is -10.
• Intercept on consumption axis is at 60 × \$10 = \$600.

(3) [Individual labor supply--optimal choice: 12 pts]

1. MRS = MUL/MUC = C/(L-10).
2. The reservation wage equals the MRS at the endowment bundle. Inserting nonlabor income for C and total available time for L gives reservation wage = 200/(60-10) = \$4.
3. Budget constraint is spending = income, or C = 200 + (60-L) 20 = 1400 - 20 L.
4. Tangency condition is MRS = wage, or C/(L-10) = 10. Solve this equation jointly with the budget constraint found in part (x), to get L*= 40 hours, C*= \$600.
5. h* = total available time - L* = 20 hours.

(4) [Individual labor supply--income and substitution effects: 22 pts] Note that the "hypothetical budget line" is drawn tangent to the new indifference curve, but parallel to budget line #1.

1. 50 hours.
2. \$100. The graph shows that if Beth works zero hours, she can still enjoy \$200 in consumption.
3. \$20 per hour. The wage is the negative of the slope of the budget line.
4. \$10 per hour. The wage is the negative of the slope of the budget line.
5. Income effect: work more. Reason: A decrease in the wage is like a decrease in nonlabor income in that the budget line is now closer to the origin. So the income effect is to "purchase" less leisure and less consumption. More precisely, the income effect is a movement between indifference curves and parallel budget lines, here from tangency with budget line #1 to tangency with the hypothetical budget line.
6. 15 hours.
7. Substitution effect: work less. Reason: A decrease in the wage is a decrease in the "price" of leisure compared to the price of consumption. So the substitution effect is to "purchase" more leisure and lessconsumption. More precisely, the substitution effect is a movement along a single indifference curve from tangency with hypothetical budget line to tangency with budget line #2.
8. -10 hours.
9. Total effect: work more. Reason: total effect = substitution effect + income effect.
10. +5 hours.
11. (w,h) = (\$20,25 hours), (\$10,30 hours). This labor supply curve bends backwards.

(5) [Household specialization: 10 pts]

1. Household's joint production-possibility curve has intercepts at \$300 of market goods and 400 units of household services, because these are the points where both parties are working in the market or both are working in the household.
To find the rest of the curve, begin where both parties are working in the household, producing zero market goods and 400 units of household services. If market goods are desired, who should work in the market? Party A has a comparative advantage in market work. Sending Party A into the market creates a line segment with slope = -1, because giving up 20 units of household services gains \$20 in market goods for the household. This continues until Party A is working full-time in the market, producing \$200 in market goods but no household services. At that point, the household enjoys \$200 in market goods (produced by Party A) and 200 units of household services (produced by Party B). This kink point should be circled. The remainder of the PP curve is a line segment joining the king point to the vertical intercept.
When the production possibility curve is plotted on the graph with the indifference curves, it is obvious that this household reaches its highest attainable indifference curve at a tangency with the upper segment, where ...
2. Party A works 10 hours in the labor market (i.e, full time), and
3. Party B works 5 hours in the labor market (producing \$50 in market goods and 10 units of household services).

(6) [SR labor demand: 9 pts]

1. Value of marginal product = output price × MPE = 10 (K/E)1/2.
2. The firm maximizes profit by choosing E so that wage = VMPE, or 10 = 10 (K/E)1/2. Solve to get E* = 25.
3. Using production function, q* = 4 (25×25)1/2 = 100 units of output.
4. Profit = total revenue - total cost
= total revenue - cost of labor - cost of capital
= \$500 - \$125 - \$250
= \$125.

(7) [LR labor demand--scale and substitution effects: 16 pts] Note that the "hypothetical isocost line" is drawn tangent to the old isoquant curve, but parallel to isocost line #2.

1. Slope of any isocost line = -w/r, where r = price of capital. The slope of isocost line #1 is -4. We are given that r = \$10, so wage must = \$40 on isocost line #1.
2. Similarly, the slope of isocost line #2 is -1, so wage must = \$10 on isocost line #2.
3. Substitution effect: use more labor. Reason: Substitution effect is movement along isoquant from tangency with isocost line #1 to tangency with hypothetical isocost line with the new slope.
4. +20 units of labor (from 20 units of labor to 40 units).
5. Scale effect: use more labor. Reason: Scale effect is movement between isoquants and parallel iscost lines, from hypothetical iscost line to isocost line #2.
6. +30 units of labor (from 40 units of labor to 70 units).
7. Total effect: use more labor. Reason: total effect = substitution effect + scale effect.
8. +50 units of labor.

III. Critical thinking

Same as Version A.