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Chicago Problem Sets

Chicago. Henry Simons’ classic problem set, 1933.

Simons

 

According to Martin Bronfenbrenner, the following problem set devised by Henry Simons for Chicago undergraduates in 1933 was a pedagogical Meisterstück (ok, he just said “one of the most famous problems in economic pedagogy”). It is likely that Paul Samuelson, who considered Simons his best teacher at Chicago, cut his teeth on this problem set as well.

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Economics 65-165

M. Bronfenbrenner

A General Problem in Competitive Price

This problem was originally devised by the late Professor Henry C. Simons for Chicago undergraduate classes in 1933. It has lived on to become one of the most famous problems in economic pedagogy. Give yourself plenty of time to work with it. It is not only long but abounds in pitfalls.

There are 1000 firms in a highly competitive industry which produces a standardized product. Each firm owns and operates one plant, which is of the most efficient size. All firms have identical costs, as follows:

Output per week

Total Cost

Output per week

Total Cost

Fixed

Variable

Fixed

Variable

1

 $100 $10 13 $100 $101
2 100 19 14 100

113

3

 100 27 15 100 126
4 100 34 16 100

140

5

 100 40 17 100 155
6 100 45 18 100

171

7

 100 50 19 100 188
8 100 56 20 100

206

9

 100 63 21 100 225
10 100 71 22 100

245

11

 100 80 23 100 266
12 100 90 24 100

288

The demand curve for the industry is given by: pq = $255,000. Your first task is to make out a demand schedule, and incorporate it in your solution as Appendix 1.

Part i

Draw the supply curve (the sum of the marginal cost curves) and the demand curve of the industry on the same graph (Fig. 1). Read off the equilibrium price and quantity. Prove that your answer is correct by comparing quantities supplied and demanded at prices $1.00 higher and $1.00 lower.

Draw the cost and demand curves of the individual firm on the same graph (Fig. 2). Accompany both graphs (Fig. 1-2) with textual explanation of their construction and of any differences between them.

 

Part ii

Congress unexpectedly imposes a tax of $4.00 per unit on the manufacture of this commodity. The tax becomes effective immediately and remains in effect indefinitely. Assume:

a. No changes in the economic system other than those attributable to the tax.
b. No change due to the tax has any effect on the prices of productive services used by the industry. (This assumption will be dropped later.)

  1. Draw the new supply curve and the demand curve of the industry (Fig. 3). Read off the new equilibrium price.
  2. Draw the new cost curves and the demand curve of the individual firm (Fig. 4). Explain the construction of these graphs (Fig. 3-4).
  3. Why can the price not remain as low as $15?
  4. Why can the price not rise to and remain at $19?
  5. Precisely what would happen if the price remained for a time at $16?
  6. At precisely what level would the price become temporarily stable? What does it mean to say that this is an equilibrium level?
  7. Suppose the short-run equilibrium price to be $17. How would you answer the query:

“I don’t see why every firm should produce 15 units per day when the price is $17. It would make just as much if it produced only 14, for the 15thunit adds just as much to expenses as it adds to revenues.” Precisely what would happen if some firms produced 14 units per day and others 15 units?

  1. Would short-run equilibrium be reached at a higher or lower price (and with larger or smaller output) if the elasticity of demand were lower (less than unity? If it were higher (greater than unity)?
  2. What would happen if demand had an elasticity of zero? An elasticity of infinity?

 

Part iii

As Figure 4 will reveal, the new minimum average cost is $19. The short-run equilibrium price was $17; hence this industry becomes unattractive as an investment, relative to other industries. As plants are worn out, therefore, they will not be replaced; plants will be junked sooner; and even maintenance will be reduced. To simplify the problem, we assume:

  1. Each plant has a life of 1,000 weeks.
  2. The plants in the industry are staggered so that, at the time the tax was imposed, there is one plant 1 week old, one plant 2 weeks old, etc.
  3. At the time the tax was imposed, 20 plants were so near completion that it is impossible to divert them to other uses. These are completed at one-week intervals.

Hence for 20 weeks the price will stay at $17, and then rise gradually as entrepreneurs fail to replace worn-out plants.

  1. What will the situation be at the end of the 25thweek? (Answer in terms of “greater than” or “less than.”)
  2. When 120 weeks have passed (900 plants left), will the price be above or below $18? Explain carefully.
  3. How many weeks must pass (how many plants must be scrapped) before the price rises to $18? Explain precisely.
  4. Will the output per plant increase or decrease as the number of plants declines?
  5. When 220 weeks have passed (800 plants left), will the price be above or below $19?
  6. How many plants must be scrapped before the price rises precisely to $19?
  7. What would the price be if the number of plants declined to 750? What would be the output per plant? What would happen to the number of plants?
  8. What happens to the short-run supply curve of the industry as the number of plants diminishes? Draw, on the same graph (Figure 5), the supply curve when there are 1,000 firms and 800 firms. Compute elasticities of supply for these two curves at a given price.
  9. How could the process of adjustment, and the final equilibrium, be different.
    1. If the elasticity of demand were greater than unity?
    2. If the elasticity of demand were less than unity?
      (The significant points are: (1) price, (2) output per plant immediately after the tax is imposed, and (3) number of plants and total output at the new long-run equilibrium).

 

Part iv (Optional)

Finally, the prices of the productive services will be affected by the purchases of the industry. Some of the services will be specialized: Larger quantities can be secured only at higher prices, and smaller quantities can be secured at lower prices. Assume that all of these services are “fixed”, and that all variable services are unspecialized (i.e., any quantity can be secured by the industry at a constant price).

  1. Will the short-run effects of the tax be any different than they were in Part 2? Explain in detail.
  2. How will the long-run adjustment differ? Will the final price be more or less than $19, and the daily output more or less than 13,421? Again explain in detail.
  3. Suppose that a special and scarce kind of land is required for production of the taxed commodity, and that this land is not used (or within practicable limits usable at all) in the production of any other commodity, and that all other resources are completely unspecialized. What is likely to be the effect of the tax on the price of the use of such land (on its rent)?
  4. Suppose that this special and scarce land is also used in one other industry. Will the rent of this land fall more or less, if the demand for the product of this second industry is elastic or inelastic?

 

Source:   Duke University. David M. Rubenstein Rare Book and Manuscript Library. Economists’ Papers Archives. Papers of Martin Bronfenbrenner, Box 26, Folder “Micro-econ & Distribution, 1958-67, n.d. 2 of 2”.

Image Source:  Henry Calvert Simons portrait at the University of Chicago Photographic Archive, apf1-07613, Special Collections Research Center, University of Chicago Library.