Some Pedantic Things (that probably nobody cares)

Some of those below are not necessarily mistakes, but it is better to be conceptually discriminating.

Constant returns to scale and strict concavity are incompatible!!!

I never imagined that some basic concepts could confuse so many people, until I recently noticed certain mistakes appearing repeatedly. Perhaps this is because some classic references propagate these errors, and others simply adopt them without critically examining the logic.

A common error occurs in macroeconomic notes, where constant returns to scale and strict concavity are simultaneously assumed for $F(K,L)$. However, this is incorrect.

Take $x \neq 0$ (where $x$ and $0$ can be vectors if you like), so $x \neq 2x$. Then take $\lambda = 1/2$.

Then $(1/2)x + (1/2)2x = (3/2)x$. If $h$ were strictly concave, it must follow that $h((3/2)x) > (1/2)h(x) + (1/2)h(2x)$.

However, by constant returns to scale, $h((3/2)x) = (3/2)h(x)$,

while
$(1/2)h(x) + (1/2)h(2x)$, using constant returns to scale, equals $(1/2)h(x) + h(x)$, which is $(3/2)h(x)$ again, leading to a contradiction (Since only the weak inequality, rather than the strict one, can be guarenteed).

Actually, you can find exercises explicitly asking for a proof of this incompatibility in textbooks on mathematical economics, such as Sydsæter, K., Seierstad, A., & Strøm, A. (2008)., which I learned from a lot.

It is okay to state that $F(K,L)$ is constant returns to scale and concave (but not strictly concave), or that $F(K,L)$ is constant returns to scale and $f(k) = \frac{F(K,L)}{L}$ is strictly concave.

However, the assumption of constant returns to scale and strict concavity for $F(K,L)$ is fundamentally wrong.

Second Order Stochastic Dominance

One common misundertanding of those who rely entirely on MWG (although I think MWG is already make it clear that the assumption of the same mean is to make the comparison of risks more explicit, rather than an inherent requirement in the definition) about Second Order Stochastic Dominance is they thought this concept applies only to lotteries with the same expectation. No, it applies also to lotteries with different expections as well.

They are two criterions in this category of comparing risk, increasing concave order and concave order. Increasing concave order is associated with no-lower-mean, while concave order is associated with the equal mean.

Given two lotteries $F$ and $G$, if the expectaion of any concave function under $F$ is higher than the the expectaion of this same concave function under $G$, then we say that $F$ dominates $G$ in the concave order.

Given two lotteries $F$ and $G$, if the expectaion of any increasing concave function under $F$ is higher than the the expectaion of this same concave function under $G$, then we say that $F$ dominates $G$ in the increasing concave order.

Notes that since the set increasing concave functions is only a proper subset of that of concave functions, if $F$ dominates $G$ in the concave order, it also dominates $G$ in the increasing concave order.

Concave order is mean-preserving spread, while increasing concave order is second order stochastic dominance. Since the common applications of this concept is for utility functions, where the marginal utility is positive, it is not to difficult to accept that what we are usually talking is the increasing concave order, which is weaker than the concave order.

So

• in general, mean-preserving spread is not equivalent to second order stochastic dominance;

• second order stochastic domiance can be applied to compare lotteries with different expectations, although mean-preserving spread cannot.

Check Gollier (2001) [The economics of risk and time].

In fact, I acknowledge the fact that some scholars define SOSD as the one with equal mean, while others define it as the one with no less mean. However, if you define SOSD with requiring equal mean, then the utility functions being equivalent to the equal mean will be the concave functions, no matter increasing or decreasing. Which is, $F$ SOSD $G$ according to the integral criterion and the expectation of $F$ and $G$ are the same, if and only if every DM with concave utility functions prefer $F$ to $G$ (with possibly negative marginal utilities.) And if you want to find the necessary and sufficient condition for that every DM with increasing concave utility functions prefer $F$ to $G$, that will be only the integral criterion, without requiring the expectation of $F$ and $G$ are the same (but as an implication, the expectation of $F$ will be no less than that of $G$).