In the book the author claims that equation $(1)$ $$ f_x(x(t),y(t)) - f_y(x(t),y(t)) = a - b \hspace{10mm} (1) $$ where $f_x(\cdot)$ is the partial derivative of $f(\cdot)$ with respect to $x$ and $a,b$ are constants, along with condition $(2)$ $$ f_{xy}(x(t),y(t)) > 0 \hspace{10mm} (2) $$ implies that there is a one-to-one relationship between $x$ and $y$ of the form $$ y = \xi(x) $$ where $\xi(\cdot)$ is uniquely defined, strictly increasing and differentiable.

How would I go about seeing that? I understand that if I replace this generalized version for one using a Cobb-Douglas production function, for instance, I could see it more clearly, but I wanted to understand how it works in general.


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    $\begingroup$ Could you provide reference to the page where this appears? $\endgroup$ – 1muflon1 Oct 16 at 8:18
  • $\begingroup$ Yes. It appears on page 369, section 10.4. $\endgroup$ – Pedro Cunha Oct 16 at 9:22

I think this needs little bit context to be answered, because in your question you missed a whole host of background assumptions - this is not a result that will hold for arbitrary function.

The equations actually describe are derived from first order optimality condition for a steady state from a Hamiltonian

$$f_k(k(t), h(t)) − f_h(k(t), h(t)) = \delta_k − \delta_h,$$

where $f$ is production function, $k$ per capita capital and $h$ per capita human capital and $\delta_k$ and $\delta_h$ are the depreciations respectively. Furthermore, as stated in the first paragraph you omit host of many important assumptions about the production function.

These assumptions are too many to be listed here (the assumptions take several pages to explain in the textbook itself in ch 3.3 on pp 85 and following pages), but the main important assumptions (and their implications) are:

  • $f$ has constant returns to scale
  • $f$ is strictly concave in $k$ such that: $f (k^∗, h^∗)>f_k(k^∗, h^∗)k^∗ + f (0, h^∗) \implies f (k^∗, h^∗)>f_k(k^∗, h^∗)k^∗$
  • $f_k(k(t), h(t))>0, f_h(k(t), h(t))>0$ and $f_{kh}(k(t), h(t))> 0 $ implies function is monotonically increasing.
  • Inada conditions.

These imply that if you double the factors of production output will double and that one would always want to use human capital and capital together. This means that one would always want to increase usage of capital together with human capital rather than just usage of one factor.

Since whenever we increase use of $k$ we want to also increase use of $h$ and since difference between marginal productivity of both of them will be always constant, there should be some one-to-one mapping between $k$ and $h$ described by some function $k=\xi(h)$. Also this is why the textbook assumes that $\xi(\cdot)$ is strictly increasing, unique and differentiable. It has to be strictly increasing because the more human capital $h$ we use the more we will want to use regular capital $k$. It is unique given that from all conditions we impose on the model there will always be some unique equilibrium $(k^*,h^*)$ combination, and differentiable simply because this is obviously continuous function. Also this result would not hold for any arbitrary function $f$.

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