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I have a few mathematical problems with the paper Moral hazard in teams 1982. How do I get from (2) to (1) by integrating and why is the qualification necessary "for almost all"

$$g(y,a)=h_i(y,a_{-i})*p_i(T_i(y),a) \qquad (1) $$

$$\frac{g_{ai}(y_1,a)}{g(y_1,a)}=\frac{g_{ai}(y_2,a)}{g(y_2,a)} \qquad \text{for almost all} ~ y_1,y_2 \in \{y \vert T_i(y)=T_i \} \qquad (2) $$

How to integrate (2) to get (1) or how to use the partial derivative to get from (1) to (2)?

Thank you in advance and sorry for my bad english. i am not a native speaker. if there are any ambiguities with the question please let me know :)

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  • $\begingroup$ In (1) the function's name is $g$. What is $g_{ai}$ ? Is it a second derivative wrt $a$ and $y_i$? Who are the authors of the paper, or what is the exact reference of the paper? $\endgroup$
    – Bertrand
    Aug 16, 2023 at 7:00
  • $\begingroup$ The distribution of y as a function of a is given by $G(y, a)$ , with density $g(y, a)$. assumed that the derivative of $ g$ with respect to $ai$, denoted $g_{ai}$ exists for all {i}. reference: Holmstrom, B. (1982). Moral hazard in teams. The Bell journal of economics, 324-340. $\endgroup$ Aug 16, 2023 at 9:34
  • $\begingroup$ Could the following approach be correct: $$g(y,a)=h(y,a_{-i})p(T(y),a) \\ \rightarrow \ln(g(y,a))=\ln (h(y,a_{-i}) \cdot p(T(y),a)) \\ \rightarrow \ln(g(y,a))=\ln (h(y,a_{-i}))+ \ln (p(T(y),a))$$ derivative wrt $a_i$: $$\frac{g_{ai}(y,a)}{g(y,a)}=\frac{p_{ai}(T(y),a)}{p(T(y),a)}$$ $\endgroup$ Aug 16, 2023 at 9:49

2 Answers 2

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The fact that $$ \frac{g_{a_i}(y,a)}{g(y,a)}, $$ is the same for all $y$ for which $T_i(y) = T_i$ implies that the fraction can be written as a function of the value of $T_i(y)$ and $a$ alone.

So there is some function $P_i(T_i(y),a)$ such that: $$ \frac{g_{a_i}(y,a)}{g(y,a)} = P_i(T_i(y),a). $$ Now the left hand side equals the derivative of $\ln(g(y,a))$ with respect to $a_i$, so $$ \frac{\partial \ln(g(y,a))}{\partial a_i} = P_i(T_i(y),a). $$ We can integrate both sides with respect to $a_i$, $$ \ln(g(y,a)) = K_i(T_i(a),a) + C_i(y,a_{-i}), $$ where $K_i(T_i(y),a) = \int P_i(T_i(y),a) d a_i$ and where $C_i(y,a_{-i})$ is the constant of integration, which can depend on both $y$ and $a_{-i}$ (but not on $a_i$ as this is the variable you are integrating over). So, $$ g(y,a) = e^{K_i(T_i(y),a)} e^{C_i(y,a_{-i})}. $$ Now denote $p_i(T_i(y),a) = e^{K_i(T_i(y),a)}$ and $h_i(y,a_{-i}) = e^{C_i(y,a_{-i})}$ to get to the result.

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  • $\begingroup$ Thanks a lot! I think you forgot to write $e^{ln(g(y,a))}=g(y,a)$ in the last equation. Do you have an idea why the qualification "for almost all" is necessary ? $\endgroup$ Aug 16, 2023 at 12:28
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    $\begingroup$ Thanks, corrected it. The for all, means (I think) that the condition can be violated on a set of measure zero. This set will not impact the shape of the density function. $\endgroup$
    – tdm
    Aug 16, 2023 at 13:08
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I add to the great answer of tdm some specifications about the term for almost all, because it is very important in mathematical analysis and in probability theory, and it is very likely to find it in texts or papers. $$\;$$

why is the qualification necessary "for almost all".

for almost all in mathematics means the same as almost everywhere (usually abbreviated as a.e.).

This is a concept that relates to measure theory. Measure theory is a branch of mathematics which, roughly speaking, has the aim of establishing the 'extension' of a set, you may think it as a generalization of the concept of area or of length of a segment.

Measure theory defines on some subsets a function, a measure, which assigns to the subsets a real number or $+\infty$, (that is an element of the extended real line).

Formally, a (real) measure m is defined as a set function as

$$ m: \Omega \rightarrow [0, +\infty]$$ where $\Omega$ is usually a so-called sigma-algebra of subsets, such that $m(\emptyset)=0$, and the function $m$ has the property of sigma-additivity. $$\;$$

The term almost everywhere defines a property that is defined on a set, except on a subset of zero measure. More formally:

For a measurable set $E$, we say that a property holds almost everywhere on $E$, or it holds for almost all $x\in E$, provided that there is a subset $E_0$ of $E$ for which $m(E_0)=0$ and the property holds for all $x\in E \sim E_0$.$^1$

For instance, equalities between mathematical entities can be defined almost everywhere, meaning that they are equal except over a subset of zero measure: two functions can be equal a.e., and this kind of equality is very important in real analysis. It is a part of the theory of measure necessary to build the Lebesgue integral, which is a more general concept of integral with respect to Riemann integral.

There are many types of measures. Usually, when not otherwise specified, one refers to Lebesgue measure.

There are many important results of mathematical analysis linked to sets of zero measure. One of the most important is the

Theorem (Lebesgue) Let $f$ be a bounded function on the closed, bounded interval $[a,b]$.Then $f$ is Riemann integrable over [a,b] if and only if the set of the points in [a,b] at which $f$ fails to be continuous has measure zero.$^2$

$$\;$$

Theory of measure is used in probability theory, sometimes probability theory is considered a part of measure theory.$^3$

I cannot read the article you mentioned, but I suppose that it uses theory of measure and Lebesgue integral.


$^1$ Royden H. L., Fitzpatrick P. M., Real Analysis, Prentice Hall, 2010, p. 45. the sign $\sim$ here is the set difference.

$^2$ Ibid. p. 104. Don't think of zero measure as a synonymous of small, for instance in the sense of cardinality of a set. Infinite sets also can be of measure zero: an infinitely countable set has measure zero.

$^3$ A standard reference is Billingsley, Probability and Measure, Wiley, 1995.

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