I was given this problem: In order to investigate the hypothesis that greater pr

Question

I was given this problem:

In order to investigate the hypothesis that greater prestige leads to higher costs of attendance at higher educational institutions, a researcher randomly sampled national universities and liberal arts colleges from the U.S. News and World Report rankings. Based on the data collected, they estimated a model that sought to explain costs as a function of a number of institutional characteristics, with the following results (standard errors in parentheses):

Cost= 7,502 + 3,818Reputation – 0.21Size + 8,391Dpriv

    (1,277)    (0.13)     (2,386)

R2= .72           N = 92

where Cost is tuition and fees in dollars, Reputation is the index used in U.S. News and World Report (based on a survey of university presidents and chief academic officers), which ranges from 1 (“marginal”) to 5 (“distinguished”), Size is the number of undergraduate students, and Dpriv is a dummy variable for being a private institution.

In another model specification, the natural log of cost (lnCost) is used as the dependent variable. Explain how the sort of relationship between Cost and Reputation that is implied by this functional form differs from that implied by the use of Cost (not in log form) as the dependent variable.

My answer:

The relationship between Cost and Reputation when Cost is not a logarithm is: for every unit increase in the measure of an institution’s reputation, tuition costs will increase by a certain amount of dollars. When Cost is a logarithm, the relationship between Cost and Reputation is: when reputation increases by one unit, cost increases by a percent rather than a dollar amount.

Is my answer correct and sufficient or can there be further/more detailed explanation of the relationship of cost and reputation when using the natural log of cost? For example, is there anyway to determine the percent increase of cost that occurs per unit increase in reputation?

Explanation / Answer

Answer to the question)

The general tendency is when two variables do not share a linear relation, then the log of variables are taken and their relation is evaluated to obtain linearity.

Variables like wages, income and unemployement, are the variables that may increase exponentially and hence their relation cannot be expressed by a linear relation. Thus a log od wage , or log of income helps get a better picture of the relation between the variables

In this scenario too, as the reputation increases, there would be drastic increase in the cost and that is the reason why taking log helps in capturing the linear relation of cost and reputation

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