# Are marginal utilities or utilities compared?

Consider decision making in an economic model. Suppose that there is some utility and it is a function of consumption and leisure where leisure is full-time work, part-time work, or full-time retirement. In such a model, the agent decides on part-time work if both of the following hold:

$$U^{PW} \color{gray}{− U^{FW}} > U^{FR} − \color{gray}{U^{FW}}$$

$$U^{PW} \color{gray}{− U^{FR}} > U^{FW} \color{gray}{− U^{FR}}$$

The left hand side in the first inequality is the marginal utility from working part-time instead of full-time work, and the right hand side is the marginal utility from full-time retirement instead if full-time work. The second inequality is interpreted similarly.

Now consider decision making in an econometric model, say, in the multinomial probit model (https://en.wikipedia.org/wiki/Multinomial_probit). In such a model, the agent decides on part-time work if both of the following hold:

$$U^{PW} > U^{FR}$$,

$$U^{PW} > U^{FW}$$.

In the economic model the comparisons are relative (to full-time work in the first inequality, to full-time retirement in the second inequality), while in the econometric model the comparisons are not relative. The question is how one should think of the two different types of comparisons made in the economic and in the econometric modelling. Why are they different? What is the mechanism that brings us from the comparison made in the economic model to the one made in the econometric model? What is the link between the two? The book from Winkelman and Boes (2006) mentions that "This raises the question of whether and how statistical models for binary dependent variables can be related to microeconomic models of choice, based on utility maximization subject to constraints. It turns out that such a link can in fact be made, and the class of discrete choice models has been developed to derive discrete probability models based on utility maximization (McFadden, 1974a, 1981)." I cannot find the answer to my question in the McFadden articles.

• They are mathemathically equivalent? – Giskard Jul 24 '19 at 18:14
• I do not think the question deserves a -1. If we forget about the comparisons made in the econometric model, why marginal utilities are compared and not just utilities in the economic model then? – Snoopy Jul 24 '19 at 19:09
• Can you please link to an article that uses the former (marginal utility) technique and claims that it is different from utility maximization? – Giskard Jul 24 '19 at 21:08
• I wouldn't consider $U^{PW}-U^{FW}>U^{FR}-U^{FW}$ a comparison of marginal utilities. PW, FW, and FR are discrete choices. Choosing one over another is not exactly making a choice "at the margin". – Herr K. Jul 24 '19 at 21:12
• Giskard please see "marginal utility of another year of retirement" under Eq. (5) in Burbidge and Robb (1980, Pensions and retirement behaviour). There is no part-time work in this paper. In the example I consider an extension. Besides this, neither the paper nor do I claim that "this is another technique than utility maximisation". – Snoopy Jul 24 '19 at 21:46