Mathematical Economics Assignment on Arnold Harberger, “Monopoly
and Resource Allocation”
In his article, Harberger makes a seminal empirical attempt to assess the size
of the “allocative and welfare effects of monopoly” in the U.S. economy, just
before the Great Depression. Based on a number of simplifying assumptions,
Harberger determines an average monopoly price increase and the resulting
deadweight-loss from monopoly in the manufacturing sector as a percentage of
GNP. Deadweight-losses are since sometimes referred to as ‘Harberger-triangles’.
The assignment (specified in more detail below) is to critically review Arnold
Harberger’s classic article. For that purpose, you will first have to trace the
steps in his analysis. The questions below will help you to understand their
logic. The answers to the questions are tentative and meant to provide you
with key insights into the text containing the main ideas, which will help you
to write your essay on this paper, using mathematical economics.
1. What is a deadweight-loss, according to Cowen & Tabarrok, Chapter 6?
Harberger assumes that “marginal and average costs are the same” (page 77).
2. What assumption about manufacturing does this imply?
Harberger assumes “elasticity of unity” (page 79), which means that he takes
the own-price elasticity of demand for manufacturing goods to be equal to 1.
3. Which is the third and final main assumption that underlies Harberger’s
estimation approach?
4. What is assumed about the period from which Harberger takes his data,
1924-1928, that is important for interpreting the results?
5. How are “Excess Profits” in an industry calculated?
6. What is the meaning of negative numbers in Table 1 column (2)? How
do these constitute a positive “Welfare Cost” in column (4)?
7. Can the industry be assumed to be in long-run equilibrium, while there
are profits for some manufacturers and losses for others? (Hint: consult Cowen
& Tabarrok, Chapter 11.)
8. Discuss Harberger’s extrapolation of total transfers from Epstein’s industries
to the whole U.S. manufacturing sector. What is implicitly assumed?
9. On which other economist does Harberger base his concept of the ‘Harbergertriangle’?
(Hint: he has a line named after him.)
Let’s consider deadweight-losses a little more formally.
1
10. Consider footnote 2 on page 81 for each commodity–so drop the summation
sign. Also ignore the . Yet, confirm from the definition of “the welfare
loss” (let’s call it  )
  =
1
2

that it can be rewritten as
  =
1
2
()2


by using the definition of (positive) own-price elasticity of the demand for commodity

 =





11. By taking this definition of  , with 1
2 in it, what assumption does
Harberger implicitly make? Does it follow from the (other) assumption that the
elasticity of demand is unity?
12. What is the “formula” that returns the numbers in column (4) in Table
1 from those in the other columns as inputs? In your expression of this formula,
use 1, 2, 3 and 4 as the variables that relate to the columns 1 to 4.
13. What percentage of national income does Harberger find the deadweightloss
in manufacturing is?
14. How does Harberger interpret his finding? Is   a lot or little, he
thinks?
15. To what extent do the three assumptions in your answers to the questions
1-3 make Harberger’s estimate of deadweight-loss conservative?
Short essay question
In his article, Harberger concludes that: “Our economy emphatically does not
seem to be a monopoly capitalism in big red letters” (page 87). On the basis
of your understanding of Harberger’s analysis, you are to critically assess this
conclusion. Is it warranted, in that it follows logically from his study? To what
extent does the conclusion depend on the particular assumptions and comparisons
he makes? Also, is the estimate of the welfare loss really conservative,
as Harberger claims, or maybe also here and there rather downwardly biased
by his research approach? You are to explain your answers in a fluent review
article of the paper.
To help you organize your thinking, below are a few hints. It helps if you
visualize your analysis in a figure like Harberger’s Figure 1 on page 78.
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16. In Harberger’s calculation of “Excess Profits”, what if the whole manufacturing
industry makes large economic profits on average? Would that not
imply that the average is higher than a normal rate of return?
17. What percentage would   in manufacturing be of total sales in
manufacturing? Is it possible to recover total sales in manufacturing from Table
1 column (3)? Is a comparison within manufacturing not more accurate than
extrapolation to the whole national income?
18. It is insightful to study the root causes of Harberger’s finding a little
more formally. (N.B. In the model below, we drop the subscript , since the
analysis is obviously now for one hypothetical market only.)
Consider the (inverse) demand for a product given by the following function
 =  − 
 
where  is the price paid for the product,  is the total quantity demanded,
and ,  and  are positive parameters.
Let market supply be perfectly competitive and equal to
 =  + 
where  is the price received by the supplier(s),  is the total quantity supplied,
and  and  are positive parameters.
Initially, assume (with Harberger) constant unit cost (including a normal return
on capital).
19. Why does this assumption imply that  = 0?
Interpret Harberger’s assumption of a demand elasticity of unity as linear demand.
20. What does this assumption imply about ,  and ?
Consider uniform prices  =  = . Consider a  ( for cost) and  ( for
monopoly), so that   .
21. What is , given that it represents the price level in long run competitive
equilibrium?
22. Express   as a function of  − .
23. Show that   as a percentage of the total value of sales is:
 
 ()
=
( − )2
2 ( − )
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24. Suppose that  = 10 and  = 1.
24a. For what percentage monopoly price increase is the deadweight-loss
one-tenth of one percent of the total value of sales?
24b. Suppose we normalize  = 1. For what prices  as a function of  is
the deadweight-loss equal to one-tenth of one percent of the total value of sales?
For what value of  is the monopoly price increase 6%?
24c. Modern research shows that monopolized industries increase prices up
to 25%. In the normalized model ( = 1,  = 286), what is the ratio of  
to sales if prices are 25% higher in monopoly?
25*. Finally, let’s consider how conservative the assumption of unit elasticity
is. What if we allow a   1? How does the curvature of the demand function
affect the estimation of the deadweight-loss? For that, first consider the full
demand model. What is the general expression for   as an integral?
25a*. Express the relative deviation of actual   from  =1 in
linear approximation as a function of all parameters, in particular . That is,
derive:
 () =
  −  =1
 =1
25b*. Assess  (2). By how much does Harberger’s assumption of linear
demand underestimate actual deadweight-loss in this case?
25c*. Try some other values of : how does  () develop? How does Harberger
defend his view that his linear approximation is a conservative estimate?
* This part of the analysis is more advanced and some students may simply lack
the mathematical skills to do it. That is fine: the exercises are just meant to
give you a gist of the math that you can expect to learn in your further studies.
Try to follow the logic of the argument, for example drawing a graph, and use
it as far as you can in writing your assignment.

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