Abstract
This chapter introduces the concept of scale as it is used in other disciplines, as an indicator of magnitude. It shows how economists can utilize this concept to add clarity, simplicity, and insight to their research. Applications to the incidental parameters problem, the Slutsky Equation, Mincer’s wage equation, mortality dynamics, and more elucidate the power of scale analysis, both theoretically and empirically.
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Notes
- 1.
This conclusion is stated in relative terms, but a similar conclusion also holds in absolute terms, assuming that a difference in demand elasticities of 0.1 matters little.
- 2.
Sampling variance is sometimes documented by the surveyor, or is easily estimated from the micro data. If the standard deviation of the N j individual observations in state*year cell j is σj, then the sampling variance in that unit is σ2 j/(N j-1); averaging these terms across all state*year cells yields the “average sampling variance.” As sampling error is independent of the random effect , this value and var(ξ) sum to the variance of the residual. Solon et al. (2015) mention some recent studies that reached erroneous conclusions because of the inappropriate use of WLS and present more general ways to address this issue.
- 3.
I don’t know if this has happened in practice, however, and I’m scared to look and see.
- 4.
One only because of a major data error (Swan, 1980)
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Food for Thought
Food for Thought
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1.
“In reasonably competitive markets, a second-order change in costs yields a first-order change in profit.” Treat costs in this statement as average accounting costs, which is the most likely way that a business professional would use the term in this context . Show this statement is true, always working with multiples of ten, and using a first order approximation for the typical business’s profit margin as a fraction of sales .
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2.
My grandfather is a profit maximizing farmer. One spring, after he has ordered his seed, insecticide, and fertilizer for the season, I give him $10, to be spent on seed, insecticide, or fertilizer: just one of the three. Which one should my grandfather spend this $10 on: seed, insecticide, or fertilizer?
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(a)
The correct answer is: it doesn’t matter. Explain why, employing the concept of scale.
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(b)
Assume a unit of seed, insecticide, and fertilizer each costs $1. Then write out a Taylor series expansion for the marginal product of ten units of each, to the second order. The answer in (a) assumes what about the relative magnitudes of the terms in the Taylor series expansion?
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(c)
Would the answer change if I gave my grandfather $1, instead of $10? What about $1000? Explain.
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(a)
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3.
The text refers to the unequal populations of the states or provinces within Brazil, India, and Indonesia, the nature of the heteroskedasticity that this gives rise to, and the problems that result by “overcorrecting” using population weights. Consider three analyses of the same phenomenon, estimated using Eq. (3.3): one using data from Brazil, another data from India, and another data from Indonesia.
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(a)
Intuitively, in which country should the “overcorrection” problem be largest? Smallest? You will have to look up state populations for each in order to answer.
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(b)
Can you articulate the heuristic(s) you used in answering part (a)? What statistics weighed most heavily in determining your answer?
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(a)
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4.
Table 3.1 lists relative automobile crash risks as a function of blood alcohol concentration . The relationship in this table can be described with a very simple and natural heuristic, expressed not in terms of blood alcohol concentration , but in “standard drinks.” Articulate this heuristic.
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5.
Table 3.1 confirms a truism about drinking drivers: though rare, they are far more likely to crash than sober drivers are. Using this fact, we can relate the incidence of traffic crashes to the extent of drinking and driving in a simple, clean way.
Define s and d as the number of miles driven by sober and drinking drivers, r as the average per-mile crash risk of sober drivers, and k as the average crash risk of drinking drivers relative to sober drivers. None of these are observed, only the total number of miles driven and the number of accidents involving sober and drinking drivers.
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(a)
Using scale analysis and a Taylor series approximation, derive a linear equation relating crashes per mile to r and the fraction of crashes involving drinking drivers.
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(b)
Does this relationship hold when k varies across time, or does it need to be generalized? Does it hold in first differences? Does it hold in Turkey, where very few road accidents involve alcohol? What about the country of Blottonia, where almost all do?
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(c)
In practice, no national repository records all the traffic accidents in the U.S. How, then, might we put this identity into practice with the data that is actually available?
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(a)
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Grant, D. (2018). Scale. In: Methods of Economic Research. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-01734-7_3
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DOI: https://doi.org/10.1007/978-3-030-01734-7_3
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