Meaning of Additively Separable, Linear in X

Question

Often I see both in micro and macro two common terminology :

1. Additively separable.

2. Linear in price or linear in probability.

I understand exactly as they sound by looking at the functional form of the object.

But can someone provide why these structures or assumptions are sensible or unreasonable and why they are "convenient" or "useful"? The context can be anything consumer, producer, choice under uncertainty, game theory or GE. But trying to see why it is repeatedly coming up and why it is important or mathematically useful in many parts both in micro and macro.

Answer 1

A function is additively separable in its arguments if it has the form

$$f(x,y) = g(x) + h(y)$$

This means that the cross partials are zero, and so there is no "cross" effect of the one argument over the marginal effect that the other has on the value of the function. Since marginal effects are at the very heart of Economics (see here), assuming additive separability greatly simplifies the analysis. In dynamic problems, where the intertemporal utility function is assumed to be additively separable, it permits us to transform an infinite horizon problem into a recursive two-period one.

Functions that can be transformed into something additively separable (by usually considering their logarithms), are sometimes called "multiplicatively separable". The most famous example here is the Cobb-Douglas production function:

$$Q = K^aL^{1-a} \implies \ln Q = a\ln K + (1-a)\ln L$$

As for linearity, it is a unique (structurally) relationship, while non-linear relationships are many, perhaps too many. A mathematician once said that "the whole field of Analysis, is essentially the study of linear approximation of non-linear relations".

Again, mathematical tractability is the drive here, supported by the fact that a linearity assumption is a "first-order" approximation to the true relation (see Taylor expansion).

Answer 2

A utility function is additively separable if it can be written as: $U(x,y) = u(x) + v(y)$

Examples:

*$U(x,y) =ax + by$ is additively separable by inspection.

*$U(x,y) = ax + bx2 + cy$ is also.

*$U(x,y) = x^a y^b$ is additively separable, because you can write it as $U(x,y) = log(x^a)+log(y^b)=alog(x)+b log(y)= u(x) + v(y)$

*$U(x,y) = \frac{xy}{x+y}$ is not additively separable because there's no way to transform it into an independent sub-function of $x, y$. Even if you take logs, you're down to $U = log(x) + log(y) - log(x+y)$ - notice that the third term can't be 'split'. Generally, mixing addition, multiplication and exponentiation will destroy additive separability.

And so we can see, The actual definition of additive separability is:

A function $f(x_1,...,x_n)$ is AS if it can be rewritten as $f(x_1,...,x_n)=f_1(x_1)+...+f_n(x_n)$

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The assumption is usually one of mathematical convenience. For example, when utility is additively separable in $x,y$, then the marginal utility of $x$ does not depend on the level of $y$, and vice-versa. And so anything requiring the use of partial derivatives is made easier. Is it a reasonable assumption? Sometimes. For example, if your utility depends on apples and horses, we can probably assume additive separability. If Instead your utility depends on two closely related things, it might be a bad assumption.

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