S. Klepper, Economics 73-100, Fall 2011

# Solutions to Exam II

1. If labor is the only variable input and the amount of labor needed to produce each level of output declines by x%, then the total variable cost of each level of output will decline by x%. Then the average variable cost and marginal cost of each level of output will also decline by x%.  Therefore, both marginal and average variable cost at every level of output will be at least 20% lower at the second plant than the first plant.  If average variable cost is at least 20% lower at every level of output, then the minimum value of the average variable cost across all output levels must be at least 20% lower in the second plant than the first plant.  Consequently, the minimum price at which producers would be willing to produce a positive level of output would be at least 20% lower in the second plant than the first plant.  The total fixed cost and hence average fixed cost at every level of output would be the same in the two plants.  At every level of output, average total cost equals average variable cost plus average fixed cost. With only average variable cost lower in the second plant than the first, the average total cost of the second plant at every level of output would not be 20% lower than the first plant if 20% less labor was needed to produce each level of output in the second plant than the first plant. Therefore, the average total cost of the second plant might be less than 20% lower than the average total cost of the first plant at each level of output.

Suppose labor is 20% more productive in the second plant than the first plant.  Then the marginal cost schedules of the two plants in rounds 1-3 (multiply each by 10 for rounds 4-6) would be:

 Output MC plant 1 MC plant 2 1 6 4.8 2 4 3.2 3 2 1.6 4 4 3.2 5 6 4.8 6 8 6.4 7 9 7.2 8 11 8.8 9 13 10.4

Then the optimal choices in rounds 1-6 for plants 1 and 2 would be:

 Round Price Q in plant 1 Q in plant 2 MC last unit, plant 1 MC last unit, plant 2 1 3.50 0 4 - 3.2 2 5.15 4 5 4 4.8 3 8.40 6 7 8 7.2 4 30 0 0 - - 5 44 4 4 40 32 6 61 5 6 60 48

The new breakeven price would be \$3.20 in rounds 1-3 and \$32 in rounds 4-6.  Consequently, in the second plant no output would be produced in round 4 (if labor was only 20% more productive in the second plant) but 4 units of output would be produced in round 1. Therefore, if labor was only 20% more productive in the second plant, in one round it would be profitable to produce a positive level of output in the second plant but not the first.  It would not, however, be profitable to produce a positive level of output in the second plant in every round.  When it would be profitable to produce a positive level of output in a plant, the output chosen would be where the marginal cost of the last unit, as reflected above, is less than the price, but the marginal cost of the next unit (not produced) is greater than the price. Given the lower marginal cost at every output in the second plant, it would always be profitable to produce at least as large an output in the second plant as the first plant.

If a positive level of output was produced in the first plant (and hence the second plant), then at least as great an output would be produced in the second plant.  Then the profits earned from the output produced in the second plant would exceed the profit earned from the production of the output in the first plant. To see this, note that if the same level of output were produced in the two plants, then it would be less expensive to produce that output in the second plant and hence the profits earned in the second plant would be greater than the profits earned in the first plant. The only reason a greater level of output would be produced in the second plant than the first plant is if even greater profits could be earned in the second plant by doing so. Therefore, the profits earned in the second plant would exceed the profits earned in the first plant if a positive level of output were produced in both plants.  Note that in round 2 the marginal cost of the last unit of output produced would actually be greater in the second plant than the first plant, which results from the greater level of output produced in the second plant than the first.  Note that in this round, 4 units of output using 16 units of labor would be produced in the first plant whereas 5 units of output using 17.6 units of labor would be produced in the second plant.  Consequently, it is possible that more labor might be employed in the second plant than the first plant.

Based on this discussion, the answers to the individual questions, with points allotted in brackets, are:

 1. False

 2. True

 3. True

 4. False—not if the labor saving was only 20%.

 5. True

 6. True

 7. False

 8. False

 9. False

2. A theater cannot be maximizing its revenues if the quantity of tickets demanded at the price it charges is greater than the capacity of the theater; if it were then the theater could raise its price by some amount and still sell out the theater, thereby increasing its revenues.  In general, setting a price at which the quantity demanded equals the capacity of the theater will not maximize its profits.  To see this, suppose that at this price the price elasticity of demand were less than one.  Then by raising its price, and in the process lowering the quantity demanded below the capacity of the theater, the theater would increase its revenues and hence its profits.  More generally, if the price elasticity of demand were less than one then the movie theater could always increase its profits by increasing its price.

If a theater set a price such that the quantity demanded were less than the capacity of the theater and the price elasticity of demand were greater than one, it also could not be maximizing its profits.  To see this, note that the theater could lower its price by some amount without the quantity demanded exceeding the capacity of the theater, and the revenues and profits of the theater would rise since the price elasticity of demand were greater than one.  Last, the income elasticity of demand will have no bearing on the price that maximizes a theater’s profits since it does not indicate anything about how the quantity demanded is affected by the price charged.

Based on this discussion, the answers the individual questions, with the points allotted to them in brackets, are:

 10. False

 11. True

 12. True

 13. False

 14. True

3. The annual growth in real GDP due to the growth in labor is computed as f * % annual growth in labor, where f is the fraction of GDP paid to labor, and the annual growth in real GDP due to the growth in capital is computed as (1 –f) * % annual growth in capital.  Therefore, the growth in real GDP in both countries due to the growth in labor and capital was:

Chile: fC * 8% + (1-fC) * 9%, where fC is the fraction of GDP paid to labor in Chile

Brazil: fB * 8% +(1-fB) * 12%, where fB is the fraction of GDP paid to labor in Brazil

Regardless of fC and fB, the annual growth in real GDP due to the growth of labor and capital in both countries must have exceeded 8%.  Furthermore, since fC was greater than fB, the annual growth in real GDP due to the growth in labor must have been greater in Chile than Brazil.  Since both countries experienced the same annual growth in real GDP of 9%, a greater fraction of this growth can be attributed to the growth of labor in Chile than Brazil.

For Chile and Brazil, the residual annual growth in real GDP attributable to technical change was:

Chile: 9% - fC * 8% - (1-fC) * 9%.

Brazil: 9% - fB * 8% - (1-fB) * 12%.

Independent of the precise value of fC, the residual annual growth in real GDP attributable to technical change must be between 0% and 1% for Chile, so some part of its growth is due to technical change.  For Brazil, independent of the precise value of fB, the residual annual growth in real GDP attributable to technical change must be less than 1% and could even be negative (which would occur if fB was less than 3/4).

In the four Asian Miracle countries, the annual growth in real GDP attributable to technical change was estimated to range from 0.52% to 2.11% and averaged 0.705%.  In both Chile and Brazil, the residual annual growth in real GDP attributable to technical change could have been less than 0.705% depending on the values of fC and fB.  The average high income Western country was estimated to have a residual annual growth in real GDP attributable to technical change of over 1%, so regardless of fC and fB, both Brazil and Chile experienced a lower residual annual growth in real GDP attributable to technical change than the average high-income Western country. Since fC is greater than fB and both countries experienced the same annual growth in real GDP and labor but Brazil experienced a higher annual growth in capital, it must be the case that the residual annual growth in real GDP due to technical change over the past 15 years was greater in Chile than Brazil.

Based on this discussion, the answers the individual questions, with the points allotted to them in brackets, are:

 15. True

 16. True

 17. False

 18. True

 19. True

 20. False