If a production function is homothetic, the MRTS is the ratio of the inputs used. But is the converse also true?

Question

We know that when a production function is homothetic, the MRTS is constant along the ray through the origin (it is a ratio of the inputs used). But is it true that if the MRTS is ratio of the inputs used, then the production function is homothetic? Can anyone prove it?

Answer 1

Let me take the focus of a 2 goods consumption setting (which is analogues to the production setting) and assume that the indifference curves are convex and differentiable, so every point has a unique MRS.

Assume that we can write the $MRS$ as a function of the ratio of the two goods. In other words, at the point $(x,y)$ (with $x, y > 0$) we can write $MRS_{(x,y)} = M(x/y)$, where $M$ is some function $M$ from $\mathbb{R}_{++}$ to $\mathbb{R}_{++}$.

Let us for simplicity normalize prices $(p_x, p_y)$ such that the price of $y$ is unity. Then the demand functions are a function from the relative price $p_x/p_y = \omega$ and normalized income $z = m/p_y$. Let us denote by $D_x(\omega, z)$ and $D_y(\omega, z)$ be the demand functions for good $x$ and $y$ respectively.

Note that if we set the relative price $\omega = MRS_{(x,y)}$ and define the expenditure $z = \omega x + y$, then $x = D_x(\omega, z)$ and $y = D_y(\omega, z)$.

Let me first show that there is a function $M$ such that $MRS_{(x,y)} = M(x/y)$ if and only if the demand functions $D_x$ and $D_y$ are homogeneous of degree 1 (i.e. linear) in $z$.

• ($\to$) Consider $t > 0$, let $x = D_x(\omega, z)$ and $y = D_y(\omega,z)$ (i.e. $\omega = MRS_{(x,y)}$ and $\omega x + y = z$).

  We need to show that $t x = D_x(\omega, tz)$ and $ty = D_y(\omega, tz)$. This, however, follows immediately from the fact that: $$ MRS_{(tx, ty)} = M(tx/ty) = M(x/y) = MRS_{(x,y)} = \omega, $$ and $$ \omega (t x) + (t y) = t(\omega x + y) = t z. $$

• ($\leftarrow$) Assume that $D_x(\omega, z)$ and $D_y(\omega, z)$ are homogeneous of degree 1 in $z$. Fix $(x,y)$. We know that $x \in D_x(\omega, z)$ and $y \in D_y(\omega, z)$ when we take $\omega = MRS_{(x,y)}$ and $z = \omega x + y$.

  Consider any $(x^\ast, y^\ast)$ such that $x/y = x^\ast/y^\ast$. Let $t = y^\ast/y$. Note that $x^\ast = t x$ and $y^\ast = t y$.

  By homogeneity of degree of the demand functions in $z$, we have $tx = x^\ast = D_x(\omega, tz)$ and $ty = y^\ast = D_y(\omega, tz)$.

  This implies that, by definition, $MRS_{(x^\ast, y^\ast)} = \omega = MRS_{(x,y)}$. Conclude that $MRS_{(x,y)}$ is a function of $x/y$.

Given this, the question can be reformulated in the following way. Do linear (in income) demand functions, lead to homothetic utility functions.

The response is, no. But the answer is somewhat subtle.

It is known that utility functions that lead to linear (in income) demand functions (i.e. linear Engel curves) must be of the Gorman polar form. This means that the indirect utility functions are linear in income: $$ v(p_x, p_y, m) = a(p_x, p_y) + b(p_x, p_y) m. $$ for some functions $a$ and $b$ of $p_x$ and $p_y$. Using Roy's identity, the demand functions are of the form, $$ \begin{align*} &D_x(p_x, p_y, m) = \frac{a_x + b_x m }{b},\\ &D_y(p_x, p_y, m) = \frac{a_y + b_y m}{b}. \end{align*} $$ where $a_x, b_x$ and $a_y, b_y$ are the partial derivatives of $a$ and $b$ with respect to $p_x$ and $p_y$ respectively.

Homothetic utilities on the other hand have indirect utility functions of the form: $$ v(p_x, p_y, m) = b(p_x, p_y) m. $$ This gives rise to linear demand functions: $$ \begin{align*} &D_x(p_x, p_y, m) = \frac{b_x}{b} m,\\ &D_y(p_x, p_y, m) = \frac{b_x}{b} m. \end{align*} $$ So utility functions of the Gorman polar form are more general in the sense that they allow the Engel cruves to have a possible non-zero intercept $a_x/b$ and $a_y/b$. This means that there are non-homothetic utility functions that have linear Engel curves.

On the other hand, the fact that these Engel curves do not pass through the origin means that demand at zero income is not actual equal to zero (and consumption of some goods at that level become negative). In other words, demand functions consistent with the Gorman polar form are not ``well-defined'' for incomes near zero. Economists (better than myself) then argue that we should only look at regions in space far enough from the origin, which basically means you allow the Engel curves to become non-linear near the origin but they are linear when income is large enough.

tldr: If you restrict yourself to regions far enough from the origin, then there are non-homothetic utility functions that lead to $MRS$ that only depend on the ratio of the quantities. These are of the Gorman polar form. If you impose the ``MRS is function of the quantities ratio'' condition globally, then probably this indeed leads to homothetic utility functions.

Answer 2

No, the first statement, that the MRTS is the ratio of the inputs used is already false. A counterexample: the production function $$ F(K,L) = K + L $$ is homothetic, the absolute value of the MRTS is 1, and yet this does not tell us anything about the ratio of the inputs used, it does not have to be 1:1.

The above function is something of an edge case, but the statement is simple not true for asymmetric Cobb-Douglas and most other function types either.