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#+TITLE: The Solow model
#+DATE: February 2022
#+AUTHOR: Stéphane Adjemian
#+EMAIL: stephane.adjemian@univ-lemans.fr
#+PROPERTY: header-args:python :python /tmp/blog-solow-model/bin/python

#+BEGIN_QUOTE
This note offers an alternative presentation of the Solow model. The
presentation is more general than the one given in the lectures. For
instance, the form of the production function is not postulated (in the
lectures we mostly worked with a Cobb-Douglas production function). In this
more general setting we establish the existence and the global stability of
the steady state, and study the dynamics of the speed of adjustment towards
the steady state.
#+END_QUOTE

* The production function

We already know that the properties of the Solow model essentially follow
from the properties of the production function. We therefore begin by
describing the production technology in general terms. It determines the
quantity of a homogeneous good produced from the quantities of physical
capital and labour. We assume that there exists a continuous function $F$
from $\mathbb R_+^2$ to $\mathbb R_+$ of class $\mathcal C^2$:

\[
Y = F(K, L)
\]

where $Y$, $K$ and $L$ are respectively the quantities of the good, of
physical capital and of labour.

We assume that $F$ is a neoclassical production function, that is, it
satisfies the following properties[fn:1: We follow the definition given in
Barro and Sala-i-Martin]:

\\

[[color:red][(N1)]] The function $F$ is homogeneous of degree 1.


[[color:red][(N2)]] The partial derivatives $\frac{\partial F}{\partial K}$ and $\frac{\partial F}{\partial L}$ are positive.


[[color:red][(N3)]] The second derivatives $\frac{\partial^2 F}{\partial K^2}$ and $\frac{\partial^2 F}{\partial L^2}$ are negative.


[[color:red][(N4)]] The Inada conditions: $\lim_{K\rightarrow 0}\frac{\partial F}{\partial K}=\lim_{L\rightarrow 0}\frac{\partial F}{\partial L}=\infty$ and $\lim_{K\rightarrow\infty}\frac{\partial F}{\partial K}=\lim_{L\rightarrow\infty}\frac{\partial F}{\partial L}=0$.

\\


The homogeneity of degree 1 of the production function formalises the
assumption of constant returns to scale. The function $F$ is homogeneous of
degree 1 if and only if for every $\lambda>0$:

\[
F(\lambda K, \lambda L) = \lambda F(K, L)
\]

If the quantities of production factors are doubled, then output must be
exactly doubled. We shall see below that this property is indispensable for
rewriting the technology in intensive form (that is, for expressing output
per worker as a function of the stock of physical capital per worker), and
that it has consequences for the marginal products and for the profit of
firms.

The condition on the partial derivatives, that is on the marginal products,
tells us that, other things being equal, if the quantity of a factor
increases then output must increase.

The condition on the second derivatives tells us that the marginal products
are decreasing. Other things being equal, when the quantity of physical
capital increases, the marginal product of capital decreases.

Finally, the Inada conditions impose boundary restrictions on the marginal
products. We shall see below that they are essential to guarantee the
existence of a non-trivial (that is, positive) steady state in the Solow
model.

If a production function $F$ satisfies the conditions [[color:red][(N1)]]-[[color:red][(N4)]], then the
following properties can be derived.

#+BEGIN_property
The production factors are essential, in the sense that $F(0,L)=F(K,0)=0$ for all $(K,L)\in\mathbb R_+^2$.
#+END_property

This property tells us that it is not possible to produce in the absence of
one of the two production factors. The production function must therefore
"go through" the origin.

#+BEGIN_proof
Let us show that labour is essential to production. Start by noting that the
average product of capital can be written as a monotonically increasing
function of $L/K$, using the homogeneity of degree 1 of the production
function [[color:red][(N1)]]:

\[ \frac{Y}{K} = \frac{F(K,L)}{K} = F\left(1,\frac{L}{K}\right) \]

If we manage to show that $F(1,0)$ is equal to zero, then we will have shown
that labour is essential to production (indeed if $F(1,0)=0$ then we also
have $F(K,0)=0$ for all $K\geq 0$ by [[color:red][(N1)]]). From the last expression of the
average product of capital, we know that for any level of the population $L$
the average product of capital must tend to $F(1,0)$ as $K$ tends to
infinity. Moreover, by l'Hôpital's rule we know that the asymptotic
behaviour of the average product is the same as the asymptotic behaviour of
the marginal product:

\[
\lim_{K\rightarrow\infty} \frac{F(K,L)}{K} = \lim_{K\rightarrow\infty} \frac{\partial F(K,L)}{\partial K}
\]

which must be equal to $F(1,0)$. By the Inada conditions [[color:red][(N4)]], the marginal
product tends to 0 as $K$ tends to infinity, so we have $F(1,0)=0$ and hence
$F(K,0)=0$ for all $K\geq 0$, which shows that labour is an essential
production factor. One shows in the same way that physical capital is an
essential factor, by looking at the average product of labour.
#+END_proof

Before turning to the next properties, we recall and prove a theorem
characterising the properties of homogeneous functions (Euler's theorem).

#+BEGIN_theorem
Let $g(x,y)$ be a differentiable function from $\mathbb R^2$ to $\mathbb R$, homogeneous of degree $k$ with respect to $x$ and $y$. Then:

\[
k g(x,y) = g_x(x,y)x+g_y(x,y)y
\]

where $g_x$ and $g_y$ are the partial derivatives with respect to $x$ and
$y$. These partial derivatives are themselves homogeneous of degree $k-1$
with respect to $x$ and $y$.
#+END_theorem

#+BEGIN_proof
Since the function is homogeneous of degree $k$ we have:

\[
g(\lambda x, \lambda y) = \lambda^k g(x,y)
\]

for all $\lambda$. Differentiating with respect to $\lambda$, we get:

\[
g_x(\lambda x, \lambda y)x + g_y(\lambda x,\lambda y)y = k \lambda^{k-1} g(x,y)
\]

In particular, for $\lambda=1$ we therefore have:

\[
k g(x,y) = g_x(x,y)x+g_y(x,y)y
\]

To show that the partial derivative $g_x$ is homogeneous of degree $k-1$, we
differentiate the first equation with respect to $x$ (rather than $\lambda$):

\[
g_x(\lambda x, \lambda y) \lambda = \lambda^k g_x(x,y)
\]

\[
\Leftrightarrow g_x(\lambda x, \lambda y) = \lambda^{k-1} g_x(x,y)
\]

One shows in the same way that the partial derivative $g_y$ is homogeneous of
degree $k-1$ (by differentiating with respect to $y$).
#+END_proof

#+BEGIN_property
The Hessian matrix of $F$ is negative semi-definite for all $(K,L)\in\mathbb
R_+^2$. Denoting by $\mathcal H_F(K,L)$ the matrix of second derivatives, the
determinant and the trace of the Hessian matrix satisfy $|\mathcal
H_F(K,L)|=0$ and $\mathrm{tr}\bigl(\mathcal H_F(K,L)\bigr) < 0$ for all
$(K,L)\in\mathbb R_+^2$.
#+END_property

#+BEGIN_proof
The Hessian matrix is:

\begin{equation*}
\mathcal H_F(K,L) =
\begin{pmatrix}
\frac{\partial^2 F(K,L)}{\partial K^2} & \frac{\partial^2 F(K,L)}{\partial K\partial L}\\
\frac{\partial^2 F(K,L)}{\partial L\partial K} & \frac{\partial^2 F(K,L)}{\partial L^2}
\end{pmatrix}
\end{equation*}
since $F\in\mathcal C^2$ this matrix is symmetric (by [[https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives][Young's theorem]]) and the determinant is given by:

\begin{equation*}
|\mathcal H_F(K,L)| = \frac{\partial^2 F(K,L)}{\partial K^2}\frac{\partial^2 F(K,L)}{\partial L^2}-\left(\frac{\partial^2 F(K,L)}{\partial K\partial L}\right)^2
\end{equation*}
Since $F$ is homogeneous of degree 1, by Theorem 1 we have:

\[
F(K,L) = \frac{\partial F(K,L)}{\partial K} K + \frac{\partial F(K,L)}{\partial L} L
\]
Differentiating with respect to $K$ or $L$ yields:

\begin{cases}
\frac{\partial F(K,L)}{\partial K} &= \frac{\partial^2 F(K,L)}{\partial K^2}K+\frac{\partial F(K,L)}{\partial K}+\frac{\partial^2 F(K,L)}{\partial L\partial K} L \\
\frac{\partial F(K,L)}{\partial L} &= \frac{\partial^2 F(K,L)}{\partial L^2}L+\frac{\partial F(K,L)}{\partial L}+\frac{\partial^2 F(K,L)}{\partial K\partial L} K
\end{cases}

or equivalently (taking into account the equality of the cross second
derivatives):

\begin{cases}
\frac{\partial^2 F(K,L)}{\partial K^2} &= -\frac{\partial^2 F(K,L)}{\partial K\partial L} \frac{L}{K} \\
\frac{\partial^2 F(K,L)}{\partial L^2} &= -\frac{\partial^2 F(K,L)}{\partial K\partial L} \frac{K}{L}
\end{cases}

In passing, note that the cross second derivative must be positive (since the
marginal products are decreasing). Thus when the quantity of labour (capital)
increases, the marginal product of capital (labour) increases. Substituting
into the expression of the determinant we obtain:

\begin{equation*}
|\mathcal H_F(K,L)| = \left(-\frac{\partial^2 F(K,L)}{\partial K\partial L} \frac{L}{K} \right)\left(-\frac{\partial^2 F(K,L)}{\partial K\partial L} \frac{K}{L}\right)-\left(\frac{\partial^2 F(K,L)}{\partial K\partial L}\right)^2 = 0
\end{equation*}

The trace of the Hessian matrix is the sum of the diagonal elements. The
trace is negative since the marginal products are decreasing by [[color:red][(N3)]]. The
Hessian matrix therefore has one zero eigenvalue and one negative eigenvalue.
#+END_proof

If $F$ is a neoclassical production function it must be concave and hence
quasi-concave. This property is useful for characterising the optimal
behaviour of firms (see below).

#+BEGIN_property
The marginal products depend only on the ratio $k=K/L$.
#+END_property

#+BEGIN_proof
By Theorem 1, since the production function is homogeneous of degree one, we
know that the marginal products are homogeneous of degree zero. Thus we have:

\begin{cases}
F_K(K,L) &= F_K\left(\frac{K}{L}, 1\right)\\
F_L(K,L) &= F_L\left(\frac{K}{L}, 1\right)
\end{cases}

where $F_K$ and $F_L$ are the partial derivatives with respect to $K$ and $L$.

#+END_proof

#+BEGIN_property
The marginal rate of substitution between the factors is a decreasing function of $k$.
#+END_property

#+BEGIN_proof
The marginal rate of (technical) substitution is defined, in absolute value,
as the ratio of the marginal products (which depend only on $k$):

\[
\textrm{MRS}(k) = \frac{F_K(k,1)}{F_L(k,1)}
\]

up to the sign, it is the slope of an isoquant at $(K,L)$. Its derivative is:

\[
\frac{\mathrm d \textrm{MRS}(k)}{\mathrm d k} = \frac{F_{KK}F_L-F_{KL}F_K}{F_L^2}=\frac{F_{KK}F_L+F_{KK}kF_K}{F_L^2} < 0
\]

because the second derivatives are negative by [[color:red][(N3)]].
#+END_proof

#+BEGIN_property
The elasticities with respect to the production factors are positive and sum to one.
#+END_property

#+BEGIN_proof
Let $\varepsilon_{Y/K}$ and $\varepsilon_{Y/L}$ denote the elasticities of $Y$ with respect to $K$ and $L$. By definition, we have:

\[
\varepsilon_{Y/K} = \frac{\frac{\partial Y}{\partial K}}{\frac{Y}{K}}\quad\text{and}\quad \varepsilon_{Y/L} = \frac{\frac{\partial Y}{\partial L}}{\frac{Y}{L}}
\]

The elasticity reads as the ratio of a marginal product to an average
product. Clearly these quantities must be positive, by [[color:red][(N2)]]. By Theorem 1, we
have:

\[
Y = \frac{\partial Y}{\partial K} K + \frac{\partial Y}{\partial L} L
\]

dividing both sides by $Y$, we obtain:

\[
1 = \varepsilon_{Y/K} +\varepsilon_{Y/L}
\]
#+END_proof

#+BEGIN_property
In a perfectly competitive economy, where the production factors are paid
their marginal products, firms earn zero profit.
#+END_property

#+BEGIN_proof
Let $R$ and $w$ denote the real factor payments (that is, in terms of the
homogeneous good produced in the economy); profit is defined by:

\[
\Pi = Y - R K - w L
\]

Expressing output in terms of the marginal products, by Theorem 1, and
knowing that the payments $R$ and $w$ are respectively equal to
$\frac{\partial Y}{\partial K}$ and $\frac{\partial Y}{\partial L}$, one
indeed obtains that profit is zero.
#+END_proof

#+BEGIN_property
The technology can be written in intensive form, that is, by expressing
output per worker $y=Y/L$ as a function of the stock of physical capital per
worker, $k = K/L$.
#+END_property

#+BEGIN_proof
Since the production function $F$ is homogeneous of degree 1, we have:

\[
\lambda Y = F(\lambda K, \lambda L)
\]

for all $\lambda\geq 0$. In particular, for $\lambda = L^{-1}$ we have:

\[
y = F(k, 1)
\]

In what follows we set $f(k) = F(k,1)$, the intensive production function.
#+END_proof

#+BEGIN_property
The intensive production function, $f(k)$, inherits the properties of the
production function $F(K,L)$. We have:
1. $f(0) = 0$,
2. $f'(k)\geq 0$,
3. $f''(k)\leq 0$,
4. $\lim_{k\rightarrow 0}f'(k)=\infty$ and $\lim_{k\rightarrow \infty}f'(k)=0$,
5. $F_K(K,L) = f'(k)$, and
6. $F_L(K,L) = f(k)-f'(k)k$.
#+END_property

#+BEGIN_proof
(1) By definition of the intensive technology, we have $f(0)=F(0,1)$, so we
directly get $f(0)=0$ by Property 1. (2)-(5) By construction of the intensive
technology we directly have $f'(k)=F_K(K,L)$, or $f'(k)=F_K(k,1)$ since the
marginal product is homogeneous of degree zero. The positivity of $f'(k)$ and
its decrease, that is $f''(k)\leq 0$, follow directly. The Inada conditions
on $f$ follow just as directly from the Inada conditions on $F$ (for the
marginal product of capital). (6) We use Theorem 1 again, according to which:

\[
Y = F_K(K,L)K + F_L(K,L)L
\]

dividing both sides by $L$:

\[
y = F_K(K,L)k + F_L(K,L)
\]

or equivalently:

\[
y = F_K(k,1)k + F_L(K,L)
\]

since the marginal product of capital depends only on $k$ (Property 3), and
hence:

\[
f(k) = f'(k)k + F_L(K,L)
\]

\[
 F_L(K,L) = f(k) -f'(k)k
\]
#+END_proof

Before moving on to the next sections, where we describe the behaviour of
households and firms, then the equilibrium of the Solow model, we discuss two
quantities which will characterise the production technology: the elasticity
of output with respect to capital, which we have already met, and the
elasticity of substitution between the factors.

The elasticity of output with respect to capital is defined as the ratio of
the marginal product of capital to the average product of capital. We know
that the marginal product depends only on the capital stock per worker
(Property 3). The average product of capital also depends only on the
capital stock per worker: indeed, since the function $F$ is homogeneous of
degree one we have $\frac{Y}{K} = F(1,k^{-1})$. Thus the elasticity of
output with respect to physical capital is a function of the capital stock
per worker. By analogy with the usual notation in the Cobb-Douglas case, we
denote this elasticity by $\alpha(k)$, and we have:

\[
\alpha(k) =  \frac{f'(k)k}{f(k)}
\]

The elasticity of substitution between the factors characterises the
curvature of the production function. It is, along an isoquant, the ratio of
the growth rate of the labour/capital ratio to the growth rate of the ratio
of the marginal products (that is, of the marginal rate of substitution).
Since the marginal products depend only on $k$, this elasticity depends
exclusively on $k$. We write:

\[
\sigma(k) = \frac{\frac{\mathrm d \frac{L}{K}}{\frac{L}{K}}}{\frac{\mathrm d\frac{F_K}{F_L}}{\frac{F_K}{F_L}}}
\]

omitting, to lighten the notation, the dependence on $K$ and $L$ of the
marginal products (partial derivatives) $F_K$ and $F_L$. We now seek a more
explicit and operational expression of this elasticity. To this end, we
express $\mathrm d \frac{F_K}{F_L}$ in terms of the (first and second order)
partial derivatives of the function $F$. Considering the total differential
of $\frac{F_K}{F_L}$, we get:

\[
\mathrm d\frac{F_K}{F_L} = \frac{\partial \frac{F_K}{F_L}}{\partial K}\mathrm d K + \frac{\partial \frac{F_K}{F_L}}{\partial L} \mathrm d L
\]

where, applying the well-known differentiation rules, we have:

\[
\frac{\partial \frac{F_K}{F_L}}{\partial K} = \frac{F_{KK}F_L-F_{KL}F_K}{F_L^2}
\]

and

\[
\frac{\partial \frac{F_K}{F_L}}{\partial L} = \frac{F_{KL}F_L-F_{LL}F_K}{F_L^2}
\]

Since along an isoquant we have $\frac{\mathrm d L}{\mathrm d K} = -\frac{F_K}{F_L}$,
we can eliminate $\mathrm d K$ from the total differential. We therefore
have:

\begin{equation*}
\begin{split}
\mathrm d\frac{F_K}{F_L} &=
\left(\frac{\partial \frac{F_K}{F_L}}{\partial L} - \frac{\partial \frac{F_K}{F_L}}{\partial K}\frac{F_L}{F_K} \right)\mathrm dL \\
&= \left(\frac{F_{KL}F_L-F_{LL}F_K}{F_L^2}F_K  - \frac{F_{KK}F_L-F_{KL}F_K}{F_L^2} F_L \right)\frac{\mathrm dL}{F_K} \\
&= \frac{F_{KL}F_LF_K-F_{LL}F_K^2-F_{KK}F_L^2+F_{KL}F_KF_L}{F_L^2}\frac{\mathrm dL}{F_K}\\
&= \frac{2F_{KL}F_KF_L - F_{LL}F_K^2 - F_{KK}F_L^2}{F_L^2}\frac{\mathrm dL}{F_K}
\end{split}
\end{equation*}

Moreover, the total differential of $\frac{L}{K}$ is:

\[
\mathrm d \frac{L}{K} = \frac{K\mathrm d L - L\mathrm d K}{K^2}
\]

expressing $\mathrm d K$ in terms of $\mathrm d L$ (and of the partial
derivatives):

\[
\mathrm d \frac{L}{K} = \frac{K F_K + L F_L}{K^2}\frac{\mathrm d L}{F_K}
\]

Substituting into the definition of the elasticity, we get:

\begin{equation*}
\begin{split}
\sigma(k) &= \frac{K F_K + L F_L}{F_K K^2}\frac{F_L^2 F_K}{2F_{KL}F_KF_L - F_{LL}F_K^2 - F_{KK}F_L^2}\frac{K}{L}\frac{F_K}{F_L}\\
&= \frac{K F_K + L F_L}{KL}\frac{F_L F_K}{2F_{KL}F_KF_L - F_{LL}F_K^2 - F_{KK}F_L^2}
\end{split}
\end{equation*}

Since the marginal products are positive, the sign of the elasticity is
given by the sign of $2F_{KL}F_KF_L - F_{LL}F_K^2 - F_{KK}F_L^2$. As the
second derivatives $F_{LL}$ and $F_{KK}$ are negative and the cross second
derivative is positive (see the proof of Property 2), we conclude that the
elasticity of substitution between capital and labour must be positive. The
expression of the elasticity can be simplified by noting that, by Euler's
theorem, we have $KF_K+LF_L=Y$ and also:

\begin{equation*}
\begin{split}
\sigma(k) &= \frac{Y}{KL}\frac{F_L F_K}{2F_{KL}F_KF_L + F_{KL}\frac{K}{L}F_K^2 + F_{KL}\frac{L}{K}F_L^2}\\
&= \frac{Y F_K F_L}{F_{KL} \left(2KLF_KF_L + K^2F_K^2 + L^2F_L^2\right)}\\
&= \frac{Y F_K F_L}{F_{KL} \left(KF_K + LF_L\right)^2}\\
&= \frac{Y F_K F_L}{F_{KL} Y^2}
\end{split}
\end{equation*}

We therefore have:

\[
\sigma(k) = \frac{F_K F_L}{Y F_{KL}}
\]

which we can finally express in terms of the intensive technology (see
Property 8)

\[
\sigma(k) = -\frac{f'(k)\left(f(k)-f'(k)k\right)}{k f(k) f''(k)}
\]

noting that $\frac{\partial^2 F}{\partial K\partial L} = \frac{\partial f'\left(\frac{K}{L}\right)}{\partial L} = -\frac{k}{L}f''(k)$ and that $Y = Lf(k)$. Clearly, the two elasticities are related; one can equivalently write:

\[
\sigma(k) = -\frac{f'(k)(1-\alpha(k))}{k f''(k)}
\]

* Behaviour of households

The economy is populated by a continuum of households $m\in[0,1]$. We assume
that a household $m$ has an infinite lifetime; one sometimes speaks of a
dynasty rather than a household. At instant $t$ the size of the household,
that is the number of heads in the household, is denoted $L(t,m)>0$. We
assume that the size of household $m$ grows at the constant rate $n>0$. If
the initial size of household $m$ is $L(0,m) = L_0(m)$, at instant $t$ we
therefore have $L(t,m) = e^{nt} L_0(m)$. We denote by $L(t) =
\int_0^1 L(t,m)\mathrm d m = e^{nt}\int_0^1 L_0(m)\mathrm d m \equiv e^{nt}L_0$
the total population at instant $t$, and we normalise the initial population
by setting $L_0=1$. The parameter $n$ is interpreted as the population
growth rate of our economy. At every instant $t$, each household
inelastically supplies one unit of labour per head on a perfectly
competitive market and receives in return a wage $w(t)$ (per unit of
labour).

Moreover, each household initially holds physical capital in quantity
$K_0(m)$. The stock of physical capital depreciates at the constant rate
$\delta>0$ and obeys the following law of motion:

\[
\dot K(t,m) = I(t,m) - \delta K(t,m)
\]

where $I(t,m)$ is investment in physical capital at instant $t$. This
differential equation simply tells us that the stock of physical capital
increases, $\dot K(t,m)>0$, if and only if investment exceeds depreciation.
In intensive terms, the dynamics of the stock of physical capital per head
is given by:

\[
\dot k(t,m) = i(t,m) - (n+\delta) k(t,m)
\]

The household rents the services of this capital to the firms, which use it
to produce the homogeneous good (through the production technology described
above, see the next section). In return for renting out its capital,
household $m$ receives $r(t)$ per unit of capital.

In our economy there is only one good, the output of the production
technology described in the first section; it can be directed towards
capital accumulation (investment) or consumed. The prices $w(t)$, for
renting labour, and $r(t)$, for renting capital, are real, that is expressed
in units of the homogeneous good. For instance, if the household supplies
$L(t,m)$ units of labour, it receives $L(t,m)w(t)$ units of the homogeneous
good as wages.

Finally, we assume that households own the firms producing the homogeneous
good. We could describe how ownership titles are distributed among
households and hence how profits are shared. We could also consider a market
where households would trade these ownership titles on the firms. But
profits are zero at every instant in this model, where firms operate in a
perfectly competitive environment and the production technology has constant
returns to scale, see Property 6 in the previous section, so this
distribution is inconsequential. We shall ignore profits in what follows.

A household $m\in[0,1]$ uses its income to consume or invest. Its binding
budget constraint[fn:2: In this model, the behaviour of the household is
exogenous, so the budget constraint is directly written as an equality.]
reads:

\[
C(t,m) + I(t,m) = w(t)L(t,m) + r(t)K(t,m)
\]

or in intensive terms:

\[
c(t,m) + i(t,m) = w(t) + r(t)k(t,m)
\]

where the right-hand side, the sum of labour and capital income, is the real
income (that is, expressed in units of the homogeneous good) of the
household. We denote by $y(t,m)$ the income per head of household $m$ at
instant $t$. In a closed economy this income must equal output. On the
left-hand side, the split between consumption and investment is determined
by the following linear rule:

\begin{cases}
c(t,m) &= (1-s)y(t,m)\\
i(t,m) &= sy(t,m)
\end{cases}

The saving rate $s$ is here assumed constant across individuals and over
time.

* Behaviour of firms

At instant $t$, the economy is populated by a continuum of firms $e\in[0,1]$.
Firms hire labour and rent capital on perfectly competitive markets, to
produce the same homogeneous good with a common technology. They sell their
output to households on a perfectly competitive market. We denote by
$K(t, e)$ and $L(t,e)$ the factor demands (capital and labour) of firm $e$ at
instant $t$. The profit of firm $e\in[0,1]$ then reads:

\[
\Pi(t,e) = F\left(K(t,e),L(t,e)\right) - w(t)L(t,e) - r(t)K(t,e)
\]

Each firm chooses its factor demands so as to maximise its profit. If the
production function is neoclassical, the objective is well defined and the
programme of firm $e$ has a unique interior solution, which must satisfy:

\begin{cases}
F_K(K(t,e),L(t,e)) &= r(t)\\
F_L(K(t,e),L(t,e)) &= w(t)
\end{cases}

At the optimum of firm $e$ the marginal products must equal the factor
prices. Suppose, for instance, that the marginal product of labour exceeds
the wage rate $w(t)$. In that case, the firm would gain by demanding more
labour, since the increase in quantity would yield more than it costs. As
the marginal product is decreasing by assumption, $F_{LL} < 0$, the increase
in the quantity of labour brings the marginal product closer to the wage
rate. Conversely, if the marginal product of labour is below the wage rate,
the firm gains by reducing its labour demand. At the optimum of the firm the
marginal products must therefore equal the factor prices.

Since the production function is neoclassical, we know that the marginal
products are homogeneous of degree zero. In other words, the marginal
products depend only on the ratio $X(t,e) = K(t,e)/L(t,e)$. We have:

\begin{cases}
F_K(K(t,e),L(t,e)) &= F_K(X(t,e),1)\\
F_L(K(t,e),L(t,e)) &= F_L(1,X(t,e)^{-1})
\end{cases}

The marginal product of capital is therefore decreasing in the
capital/labour ratio, since $F_{KK} < 0$, whereas the marginal product of
labour is increasing in the same ratio. Clearly, since the same prices,
$r(t)$ and $w(t)$, apply to all firms, the capital/labour ratio must be
unique and independent of the index $e$:

\[
X(t,e) = X(t)\quad \forall e\in[0,1]
\]

For the programme of a firm $e\in[0,1]$ to have a unique solution, there
must exist a unique capital/labour ratio $X(t)$ such that:

\begin{cases}
r(t) &= F_K(X(t),1)\\
w(t) &= F_L(1,X(t)^{-1})
\end{cases}

Expressing the marginal products in terms of the intensive technology, we
must therefore have:

\begin{cases}
r(t) &= f'(X(t))\\
w(t) &= f(X(t))-X(t)f'(X(t))
\end{cases}

which implicitly imposes a constraint on the factor prices $w(t)$ and $r(t)$
for them to be consistent with a competitive equilibrium. Inverting the
first equation[fn:3: The function $f'$ is a bijection from $\mathbb R^+$ to
$\mathbb R^+$, since it is monotonically decreasing, so the existence of the
inverse function $f'^{-1}$ is guaranteed.], that is expressing the ratio
$X(t)$ as a function of $r(t)$, then substituting into the second, we obtain:

\[
w(t) = f\left(f'^{-1}\left(r(t)\right)\right) - r(t)f'^{-1}\left(r(t)\right) \triangleq \mathcal F\left(r(t)\right)
\]

The function $\mathcal F: \mathbb R^+\setminus\{0\}\rightarrow\mathbb
R^+\setminus\{0\}$ is commonly called the /factor price frontier/. It is a
continuous and differentiable function on $\mathbb R^+\setminus\{0\}$, and one
can show that it is monotonically decreasing. Its derivative is given by:

\begin{equation*}
\begin{split}
\mathcal F'(r) & = f'\left(f'^{-1}(r)\right)\left(f'^{-1}(r)\right)'-f'^{-1}(r)-r \left(f'^{-1}(r)\right)'\\
& = -f'^{-1}(r)\\
& = -X
\end{split}
\end{equation*}

For the factor prices to be consistent with a competitive equilibrium (and
hence with the optimal behaviour of firms), if the price of capital rises by
$\mathrm dr$ then the price of labour must fall by $X\mathrm dr$. The
decreasing factor price frontier tells us that it is not possible (in this
model /and for a given technology/) to observe a simultaneous increase in
both factor prices. One can also show that the factor price frontier is a
convex function[fn:4: Indeed, we have:
$\mathcal F''(r) = -\left(f'^{-1}(r)\right)'=-\frac{1}{f''\left(f'^{-1}(r)\right)}>0$ since $f''(k) < 0$.]
and that $\lim_{r\rightarrow 0}\mathcal F(r) = \infty$ and $\lim_{r\rightarrow\infty}\mathcal F(r) = 0$.

#+BEGIN_SRC bash :results silent :exports none :async t
python3 -m venv /tmp/blog-solow-model
source /tmp/blog-solow-model/bin/activate
pip install numpy matplotlib
#+END_SRC

#+begin_src python :session solow-model-en :exports none :results none
  def production(k, alpha, sigma, A):
      """ Returns output per worker associated with the level of physical capital per worker k, for a
      CES production function.
      """
      psi = (sigma-1)/sigma
      if abs(psi)<1e-6:
          return A*k**alpha
      else:
          return A*(alpha*k**psi+1-alpha)**(1/psi)

  def marginalproductivity(k, alpha, sigma, A):
      """ Returns the marginal product of capital associated with the level of physical capital per worker k, for a
      CES production function.
      """
      psi = (sigma-1)/sigma
      if abs(psi)<1e-6:
          return alpha*A*k**(alpha-1)
      else:
          return alpha*A*(alpha+(1-alpha)*k**(-psi))**((1-psi)/psi)

  def fpf(r, alpha, sigma, A):
      """ Returns the real wage associated with the interest rate r along the factor price frontier, for a
      CES production function.
      """
      psi = (sigma-1)/sigma
      if abs(psi)<1e-6:
          return (1-alpha)*A*(alpha*A/r)**(alpha/(1-alpha))
      else:
          k = (1/(1-alpha))**(-1/psi)*((r/(alpha*A))**(psi/(1-psi))-alpha)**(-1/psi)
          return production(k, alpha, sigma, A) - k*marginalproductivity(k, alpha, sigma, A)
#+end_src

Figure [[fig:fpf][1]] shows the factor price frontier for a CES production function
(which in general is not a neoclassical production function) for different
values of the elasticity of substitution between labour and physical
capital. The red curve corresponds to the Cobb-Douglas case, the green curve
to a CES with factors more substitutable than in the Cobb-Douglas case
($\sigma=1.2$), the blue curve to a CES with factors less substitutable than
in the Cobb-Douglas case ($\sigma=0.8$).

#+begin_src python :session solow-model-en :exports none :results none
import numpy as np
import matplotlib.pyplot as plt
plt.figure(1)
r = np.linspace(0.05, 2, 1000)
w = fpf(r, .3, .8, 1.0)
plt.plot(r, w, 'b', label=r'$\sigma=0.8$')
w = fpf(r, .3, 1.2, 1.0)
plt.plot(r, w, 'g', label=r'$\sigma=1.2$')
w = fpf(r, .3, 1, 1.0)
plt.plot(r, w, 'r', label=r'$\sigma=1$')
plt.legend()
plt.xlabel("Real interest rate (r)")
plt.ylabel("Real wage rate (w)")
plt.savefig("fpf.svg", transparent=True)
#+end_src

Each point $(r,w)$ along a frontier is associated with a capital/labour ratio
$X$ (the absolute value of the slope), and with a distribution of income
between capital and labour payments. If the wage rate is high, and hence the
return to capital is low, since there is a decreasing relation between these
two prices, the capital/labour ratio will be high because it is then optimal
for the firm to employ more capital than labour. All the more so as it is
easy to substitute labour and capital, which is why the factor price
frontier is higher the larger the elasticity of substitution.

\\

#+CAPTION: *Factor price frontier with a CES function.*
#+LABEL: fig:fpf
[[file:fpf.svg]]

Note that the optimality conditions of firm $e$ determine the ratio of the
production factors, but not the "size" of the firm (the levels of $L(t,e)$
and $K(t,e)$ are not determined). Note also that these same conditions,
through Euler's theorem, imply:

\[
r(t)X(t,e)+w(t) = f(X(t,e))
\]

and hence zero profit:

\[
\Pi(t,e) =
L(t,e)\biggl(f(X(t,e))-r(t)X(t,e)-w(t)\biggr)
\]


* Market clearing

The labour market clears if and only if:

\[
\underbrace{\int_0^1 L(t,e)\mathrm de}_{\text{Labour demand}} = \underbrace{e^{nt}\int_0^1\mathrm dm}_{\text{Labour supply}}
\]

\[
\Leftrightarrow \int_0^1 L(t,e)\mathrm de = L(t)
\]

that is, if the labour demand of firms equals the labour supply of
households.

\\

The capital market clears at instant $t$ if and only if:

\[
\underbrace{\int_0^1 K(t,e)\mathrm de}_{\text{Capital demand}} = \underbrace{e^{nt}\int_0^1k(t,m)\mathrm dm}_{\text{Capital supply}}
\]

\[
\Leftrightarrow \int_0^1 K(t,e)\mathrm de = K(t)
\]

where $K(t)$ is the aggregate capital stock in the economy, that is, if the
capital demand of firms equals the capital supply of households.


* General equilibrium

We can now bring households and firms together, define the equilibrium and
show its existence and uniqueness. Households and firms meet on three
markets: the labour market, the capital market and the market for the
homogeneous good.

#+BEGIN_definition
The equilibrium of the decentralised economy is an intertemporal allocation
$\bigl\{\bigl(k(t,m),c(t,m),i(t,m)\bigr)_{m\in[0,1]},
\bigl(K(t,e),L(t,e)\bigr)_{e\in[0,1]}\bigr\}_{t\in\mathbb R^+}$ and price
paths $\bigl\{w(t),r(t)\bigr\}_{t\in\mathbb R^+}$ such that:

1. Given the prices $\bigl\{w(t),r(t)\bigr\}_{t\in\mathbb R^+}$, $\bigl\{k(t,m),c(t,m),i(t,m)\bigr\}_{t\in\mathbb R^+}$ is consistent with the behaviour of households for all $m\in[0,1]$.

2. Given the prices $\bigl\{w(t),r(t)\bigr\}_{t\in\mathbb R^+}$, $\bigl\{K(t,e),L(t,e)\bigr\}_{t\in\mathbb R^+}$ is consistent with the behaviour of firms for all $e\in[0,1]$.

3. The factor markets clear at every instant.
#+END_definition

Note that this definition makes no reference to the market for the
homogeneous good (where households buy the good produced by firms). This is
Walras' law: in an economy with $n$ markets, if $n-1$ markets clear, then the
last market ($n$) is necessarily in equilibrium (simply because an excess
supply or demand on this market would have to be offset by a disequilibrium
on at least one other market; this is Walras' law).

\\

Let us start by characterising the intertemporal equilibrium. We know, from
the optimality conditions of firm $e\in[0,1]$, that we must have:

\[
K(t,e) = X(t) L(t,e)\quad\forall e\in[0,1]
\]

Summing over $e$, we obtain the following relation between aggregate capital
demand and aggregate labour demand:

\[
\int_0^1 K(t,e)\mathrm de = X(t) \int_0^1L(t,e)\mathrm de
\]
Since the factor markets must clear, we must have the following relation
between the aggregate supplies of capital and labour:

\[
K(t) = X(t) L(t)
\]

that is:

\[
X(t) = k(t)
\]

The capital-labour ratio of firms must equal the capital stock per head in
the economy. Substituting into the optimality conditions of firms, the factor
prices can be written as functions of the capital stock per head:

\begin{cases}
r(t) &= f'(k(t)) \triangleq r(k(t)) \\
w(t) &= f(k(t)) - k(t)f'(k(t)) \triangleq w(k(t))
\end{cases}

Note that $r'(k) = f''(k) < 0$ and $w'(k)=-f''(k)k>0$. The real interest rate
is a decreasing function of the capital stock per head, reflecting the
assumption of decreasing returns. The real wage is an increasing function of
the capital stock per head. Noting that $w'(k)$ is proportional to the cross
second derivative, $F_{KL}(K,L) = w'(k)/L$, one understands that the increase
of the wage with $k$, which also echoes the decreasing factor price
frontier, reflects the complementarity of the factors (when the quantity of
capital increases, the productivity of labour increases).

Summing the budget constraints of households over $m\in[0,1]$, we get:

\[
\int_0^1 C(t,m)\mathrm dm + \int_0^1 I(t,m)\mathrm dm = r(t)\int_0^1 K(t,m)\mathrm dm + w(t)L(t)
\]

\[
\Leftrightarrow C(t) + I(t) = r(t)K(t) + w(t)L(t)
\]

or in intensive terms:

\[
c(t) + i(t) = r(t)k(t) + w(t)
\]

Since $f(k)=rk+w$ by Euler's theorem, we finally have:

\[
c(t) + i(t) = f(k(t))
\]

the resource constraint of the economy.

Summing the accumulation rule of the capital stock per head over $m\in[0,1]$:

\[
\int_0^1 \dot k(t,m)\mathrm dm = \int_0^1 i(t,m)\mathrm dm - (n+\delta)\int_0^1 k(t,m)\mathrm dm
\]

\[
\Leftrightarrow \dot k(t) = i(t) - (n+\delta)k(t)
\]

Aggregating the decisions of households, we get:

\[
\int_0^1 i(t,m)\mathrm dm = s\int_0^1 y(t,m)\mathrm dm
\]

\[
\Leftrightarrow i(t) = s y(t)
\]

Hence finally:

\[
\dot k(t) = s f(k(t)) - (n+\delta)k(t)
\]

\[
\Leftrightarrow \dot k(t) = G\left(k(t)\right)
\]

$G:\mathbb R_+\rightarrow\mathbb R$ is a continuous function, differentiable
on $\mathbb R_+^\star$; it goes through zero, $G(0)=0$; it is increasing then
decreasing (from $\bar k$ such that $G'(\bar k)=0$, the level of capital per
head at which the marginal product of capital equals $\frac{n+\delta}{s}$).
The derivative $G'(k) = sf'(k)-(n+\delta)$ is not bounded on $\mathbb R_+$, by
the Inada condition [[color:red][(N4)]] it tends to infinity as $k$ tends to zero, but $G$ is
of class $\mathcal C^1$, hence locally Lipschitz, on $\mathbb R_+^{\star}$. By
the Cauchy-Lipschitz theorem the differential equation therefore has a unique
solution for any strictly positive initial condition. In plain words: given
an initial endowment $k(0)>0$ there exists a unique path for the stock of
physical capital $\left(k(t)\right)_{t\in\mathbb R_+}$. There is therefore
also a unique path for the other aggregates (output, consumption and
investment) and for the prices $w(t)$ and $r(t)$, which are functions of the
capital stock per head.


#+BEGIN_proposition
For any initial distribution $\bigl(k(0,m)\bigr)_{m\in[0,1]}$, an
intertemporal equilibrium exists. The allocation of production across firms
is not determined, but the equilibrium is unique with respect to the
aggregates and the allocations of households. Capital per head is determined
by a first-order differential equation:
\[
\dot k(t) = G\left(k(t)\right)
\]
for all $t\in\mathbb R_+$ and $k(0) = \int_0^1k(0,m)\mathrm d m$ given, with:
\[
G(k) = sf(k)-(n+\delta)k
\]
The real equilibrium prices are given by:
\[
r(t) = f'(k(t))
\]
and
\[
w(t) = f(k(t)) - k(t)f'(k(t))
\]
#+END_proposition

Note that the law of motion characterising the equilibrium dynamics of the
capital stock per head is identical to the one seen in the lectures for the
centralised version of the model (that is, without markets). This is not
surprising, given that the economy is here assumed perfectly competitive
(see the welfare theorems in the microeconomics course).

Figure [[fig:solow][2]] shows, for a Cobb-Douglas production function, the function
$G$ and the dynamics in the $(k,\dot k)$ plane. The change in the capital
stock is increasing in $k$ for small values of $k$, then monotonically
decreasing. The function $G$ vanishes at 0 and at $k^{\star}$. These two
points are steady states. The first is unstable and the second seems
globally stable since $G>0$ if and only if $k < k^{\star}$. In what follows
we characterise the dynamics of the capital stock per head, starting by
showing the existence, uniqueness and stability of the steady state.

#+begin_src python :session solow-model-en :exports none :results none
  def getkstar(alpha, sigma, A, s, n, delta):
      """ Returns the steady state of the capital stock per head, for a CES production function.
      """
      psi = (sigma-1)/sigma
      if abs(psi)<1e-6:
          return (A*s/(n+delta))**(1/(1-alpha))
      else:
          if ((psi<0) & (s*A*alpha**(1/psi)>(n+delta))) | ((psi>0) & (s*A*alpha**(1/psi)<(n+delta))):
              return (((1-alpha)*(s*A)**psi)/((n+delta)**psi-alpha*(s*A)**psi))**(1/psi)
          else:
              raise NameError("No non-trivial steady state.")

  def getkbar(alpha, sigma, A, s, n, delta):
      r""" Returns the value of k maximising \dot{k}, for a CES production function.
      """
      psi = (sigma-1)/sigma
      if abs(psi)<1e-6:
          return (alpha*A*s/(n+delta))**(1/(1-alpha))
      else:
          return ((1-alpha)**(1/psi))*(((n+delta)/(s*alpha*A))**(psi/(1-psi))-alpha)**(-1/psi)

  alpha = 0.30
  sigma = 1.00
  delta = 0.02
  s = 0.20
  n = 0.02
  A = 1.00

  kbar = getkbar(alpha, sigma, A, s, n, delta)
  kstar = getkstar(alpha, sigma, A, s, n, delta)

  kdotatkbar = s*production(kbar, alpha, sigma, A)-(n+delta)*kbar

  with open('solow-transition.tex', 'w') as f:
      f.write("\\documentclass[10pt,tikz]{standalone}\n")
      f.write("\\usepackage{tikz,pgfplots}\n")
      f.write("\\usetikzlibrary{patterns, intersections, arrows, decorations.pathreplacing, decorations.markings, calc}\n")
      f.write("\\pgfplotsset{plot coordinates/math parser=false}\n")
      f.write("\\begin{document}\n")
      f.write("  \\begin{tikzpicture}\n")
      f.write("    \\draw[xscale=.3,->] (0, 0) -- (15, 0) node [right] {{\\footnotesize $k$}}\n")
      f.write("      [postaction={decorate, decoration={markings,mark=between positions 0.15 and 0.65 step 0.05 with {\\arrow[red]{>};}}}, postaction={decorate, decoration={markings,mark=between positions 0.74 and 0.95 step 0.05 with {\\arrow[red]{<};}}}];\n")
      f.write("    \\draw[yscale=10,->] (0, -.1) -- (0, 0.2) node [left] {{\\footnotesize $\\dot k$}} ;\n")
      f.write("    \\draw[xscale=.3,yscale=10,domain=0:14, smooth, variable=\\x, blue, samples=1000] plot ({{\\x}}, {{{0}*\\x^{1}-{2}*\\x}}) node[right] {{{{\\footnotesize $G(k)$}}}};\n".format(s,alpha,n+delta))
      f.write("    \\draw[dashed,xscale=.3,yscale=10] (0,{0}) -- ({1},{0}) -- ({1},0) node[below] {{{{\\footnotesize $\\bar k$}}}};\n".format(kdotatkbar, kbar))
      f.write("    \\draw[xscale=.3] ({0},0) node[below]{{{{\\footnotesize $k^{{\\star}}$}}}};\n".format(kstar))
      f.write("  \\end{tikzpicture}\n")
      f.write("\\end{document}\n")

#+end_src

#+BEGIN_SRC shell :session solow-sh-en :exports none :results none
  pdflatex solow-transition
  pdf2svg solow-transition.pdf solow-transition.svg
  rm solow-transition.aux solow-transition.log solow-transition.pdf solow-transition.tex
#+END_SRC

#+CAPTION: *Changes in the capital stock per head.*
#+LABEL: fig:solow
#+ATTR_HTML: :width 500px
[[file:solow-transition.svg]]



* Steady state

The non-trivial steady state, $k^{\star}>0$, if it exists, must be such that
investment per head equals the depreciation of the capital stock per head:

\[
sf(k^{\star}) = (n+\delta)k^{\star}
\]

\[
\Leftrightarrow \frac{f(k^{\star})}{k^{\star}} = \frac{n+\delta}{s}
\]

Let $\varphi(k)$ denote the average product of capital. This function from
$\mathbb R_+$ to $\mathbb R_+$ is continuous and differentiable on
$\mathbb R_+^{\star}$. It is monotonically decreasing:

\[
\varphi'(k) = \frac{k f'(k) - f(k)}{k^2} = \frac{f(k)}{k^2}\bigl(\alpha(k)-1\bigr) < 0
\]

because the elasticity of output with respect to capital, $\alpha(k)$, is
less than one. Moreover, by the Inada conditions and l'Hôpital's rule, we
have:

\[
\lim_{k\rightarrow 0}\varphi(k) = \infty \quad\text{ and }\quad \lim_{k\rightarrow \infty}\varphi(k) = 0
\]

There thus exists a unique $k^{\star}$ such that investment per head equals
the depreciation of the capital stock per head.

One can show that this non-trivial steady state is globally stable. To this
end, we define a function measuring the distance between $k(t)$ and
$k^{\star}$: $\zeta(k) = (k-k^{\star})^2$. By construction, $\zeta(k^{\star})=0$
and $\zeta>0$ otherwise. One can also show that this distance is
monotonically decreasing:

\begin{equation*}
\begin{split}
\dot \zeta(k) &= 2 \dot k (k - k^{\star})\\
&= 2 (k - k^{\star}) \bigl(sf(k)-(n+\delta)k\bigr)\\
&= 2 (k - k^{\star}) \left(\frac{s}{n+\delta}\frac{f(k)}{k}-1\right) (n+\delta)k\\
&= 2 (k - k^{\star}) \left(\frac{\varphi(k)}{\varphi(k^{\star})}-1\right) (n+\delta)k\\
\end{split}
\end{equation*}

Since $\varphi$ is a monotonically decreasing function, one checks that the
first term $2(k-k^{\star})$ and the second term,
$\varphi(k)/\varphi(k^{\star})-1$, never have the same sign. As the last
term, $(n+\delta)k$, is always positive, we can conclude that the change in
the distance $\zeta$ is strictly negative (it is zero only if $k=0$ or
$k=k^{\star}$). Thus, whatever the initial condition $k(0)>0$, the capital
stock must converge in the long run to the non-trivial steady state
$k^{\star}$.

* Characterisation of the dynamics

We define the speed of convergence towards the steady state as the rate of
decay of the growth rate[fn:5: See [[https://stephane-adjemian.fr/en/posts/simulating-the-solow-model/][here]] for an alternative definition based
on the dynamics of the distance to the steady state rather than on the
dynamics of the growth rate.]. In what follows we are interested in the
speed of convergence of output per head. Knowing that the growth rate of
output per head equals the growth rate of the capital stock per head
multiplied by the elasticity of output with respect to capital, we obtain
the following expression of the growth rate of output per head:

\[
g_y = sf'(k) - (n+\delta)\frac{kf'(k)}{y}
\]

We know that this growth rate is positive if and only if the economy lies
below its steady state, and that the growth rate approaches zero as the
economy approaches the steady state. The change in the growth rate is:

\[
\dot g_y = sf''(k)\dot k - (n+\delta)\frac{\left[f''(k)\dot k k + f'(k)\dot k\right]y - f'(k)k\dot y}{y^2}
\]

\[
\Leftrightarrow \dot g_y = sf''(k)\dot k - (n+\delta)\left[\frac{f''(k)\dot k k}{y} + (1-\alpha(k))\frac{\dot y}{y}\right]
\]

\[
\Leftrightarrow \dot g_y = -(n+\delta)(1-\alpha(k))g_y + f''(k)\dot k \left[s-(n+\delta)\frac{k}{y}\right]
\]

\[
\Leftrightarrow \dot g_y = -(n+\delta)(1-\alpha(k))g_y + \frac{f''(k)k}{f'(k)}\frac{f'(k)k}{y}g_k\left[s\frac{y}{k}-(n+\delta)\right]
\]

\[
\Leftrightarrow \dot g_y = -(n+\delta)(1-\alpha(k))g_y + \frac{f''(k)k}{f'(k)}g_y\left[s\frac{y}{k}-(n+\delta)\right]
\]

that is, using the relation between the elasticity of output with respect to
capital and the elasticity of substitution unveiled at the end of the first
section:

\[
\Leftrightarrow \dot g_y = -(n+\delta)(1-\alpha(k))g_y - \frac{1-\alpha(k)}{\sigma(k)}(n+\delta)g_y\left[\frac{s}{n+\delta}\frac{y}{k}-1\right]
\]

The speed of convergence, defined as the opposite of the ratio of the change
in the growth rate to the growth rate, is therefore given by:

\[
\beta (k) = (n+\delta)(1-\alpha(k))\left[1+\frac{1}{\sigma(k)}\left(\frac{s}{n+\delta}\frac{f(k)}{k}-1\right)\right]
\]

as a function of the capital stock per head. Using the definition of the
steady state (the average product of capital must equal $(n+\delta)/s$), we
also have:

\[
\beta (k) = (n+\delta)(1-\alpha(k))\left[1+\frac{1}{\sigma(k)}\left(\frac{f(k)/f(k^{\star})}{k/k^{\star}}-1\right)\right]
\]

One easily checks that in the Cobb-Douglas case, that is with $\sigma(k)=1$
and $\alpha(k)=\alpha$ for all $k$, we obtain:

\[
\beta(k) = (n+\delta)(1-\alpha)\left(\frac{k}{k^{\star}}\right)^{-(1-\alpha)}
\]

As $k$ converges to $k^{\star}$, we recover the speed of convergence we would
obtain by linearising the Solow model (see [[https://stephane-adjemian.fr/en/posts/simulating-the-solow-model/][here]]). The speed of convergence
is monotonically decreasing over time if and only if the capital stock is
below its long-run level; otherwise the speed of convergence is monotonically
increasing:

\[
\dot \beta (k) = - \frac{n+\delta}{k^{\star}}(1-\alpha)^2\left(\frac{k}{k^{\star}}\right)^{\alpha-2}\dot k
\]

Under the assumption of a Cobb-Douglas production function, observing a
decrease in the speed of convergence over time suggests that the initial
endowment of the economy is below its long-run endowment (it is initially
"poor").

\\

For a CES production function, that is with $\sigma(k)=\sigma$ for all $k$,
the direction of change of the speed of convergence still depends on the
position of the initial condition, but also on the level of the elasticity
of substitution between the factors. Suppose there exists a non-trivial
steady state[fn:6: This is not guaranteed with a CES production function,
since it is not a neoclassical production function: the Inada conditions are
not satisfied. For this steady state to exist, the saving rate must be high
enough when the factors are less substitutable than in the Cobb-Douglas case,
or low enough when the factors are more substitutable than in the
Cobb-Douglas case.] $k^\star$. The speed of convergence is:

\[
\beta (k) = (n+\delta)(1-\alpha(k))\left[1+\frac{1}{\sigma}\left(\frac{f(k)/f(k^{\star})}{k/k^{\star}}-1\right)\right]
\]

Specifying the following production function:

\[
f(k) = \left(a k^\frac{\sigma-1}{\sigma} + (1-a)\right)^{\frac{\sigma}{\sigma-1}}
\]

with $a\in[0,1]$, and noting that the average product can be expressed in
terms of the elasticity of output with respect to capital:

\[
\frac{f(k)}{k} = a^\frac{\sigma}{\sigma-1}\alpha(k)^\frac{\sigma}{1-\sigma}
\]

the speed of convergence can be rewritten as follows:

\[
\beta (k) = (n+\delta)(1-\alpha(k))\left[1+\frac{1}{\sigma}\left(\left(\frac{\alpha(k)}{\alpha(k^\star)}\right)^\frac{\sigma}{1-\sigma}-1\right)\right]
\]

Clearly, if $\sigma>1$ then the speed of convergence (when the steady state
exists) is always positive and monotonically decreasing in $k$[fn:7: Indeed,
in that case $\beta$ is a monotonically decreasing function of $\alpha$, and
one can show in general that the elasticity of $y$ with respect to $k$ is
increasing if and only if the elasticity of substitution between the factors
is greater than 1:
\[
\alpha'(k) = \frac{\left(f'(k)+kf''(k)\right)f(k)-kf'(k)^2}{f(k)^2}
= \frac{kf''(k)f(k)\left[1-\sigma(k)\right]}{f(k)^2}
\]
since the marginal return to capital is decreasing.]. In that case, the speed
of convergence decreases along the transition if and only if the economy
approaches the steady state from below (the initial condition is below the
long-run level). If $\sigma < 1$, one can obtain, for high values of $k$, a
negative speed of convergence! Because the average product of capital tends
to 0. And the speed of convergence is no longer a monotonic function of the
capital stock per head... Figure [[fig:speed][3]] illustrates this case with
$\sigma=0.2$: the speed of convergence (green curve, left axis) vanishes
around $1.25\,k^{\star}$ and becomes negative beyond, while the elasticity of
output with respect to capital (blue curve, right axis) decreases towards
zero.


#+BEGIN_SRC python :session solow-model-en :exports none :results none
  def elasticity(k, alpha, sigma, A):
        """ Returns the elasticity of output with respect to capital as a function of the capital stock per head, for a
        CES production function.
        """
        psi = (sigma-1)/sigma
        if (abs(psi)<1e-6):
              return alpha
        else:
              return k*marginalproductivity(k, alpha, sigma, A)/production(k, alpha, sigma, A)

  def speed(k, alpha, sigma, A, n, delta, s):
        """ Returns the speed of convergence as a function of the capital stock per head, for a
        CES production function.
        """
        psi = (sigma-1)/sigma
        if abs(psi)<1e-6:
              return (n+delta)*(1-alpha)*(k/getkstar(alpha, 1.0, A, s, n, delta))**(-(1-alpha))
        else:
              e = elasticity(k, alpha, sigma, A)
              y = production(k, alpha, sigma, A)
              kstar = getkstar(alpha, sigma, A, s, n, delta)
              estar = elasticity(kstar, alpha, sigma, A)
              ystar = production(kstar, alpha, sigma, A)
              # return (n+delta)*(1-e)*(1+(1/sigma)*((e/estar)**(sigma/(1-sigma))-1))
              return (n+delta)*(1-e)*(1+(1/sigma)*(y*kstar/(k*ystar)-1))
#+end_src

#+BEGIN_SRC python :session solow-model-en :exports none :results none
  import numpy as np
  sigma = .2
  s = .2
  kstar = getkstar(alpha, sigma, A, s, n, delta)
  kgrid = np.linspace(.1*kstar, 2*kstar, 10000)
  spval = speed(kgrid, alpha, sigma, A, n, delta, s)
  egrid = elasticity(kgrid, alpha, sigma, A)
  fig, ax1 = plt.subplots()
  ax2 = ax1.twinx()
  ax1.plot(kgrid, spval, 'g-')
  ax2.plot(kgrid, egrid, 'b-')
  ax1.set_xlabel('k')
  ax1.set_ylabel(r'$\beta(k)$', color='g')
  ax2.set_ylabel(r'$\varepsilon(k)$', color='b')
  fig.savefig("speed.svg", transparent=True)
#+END_SRC

#+CAPTION: *Speed of convergence and elasticity of output with respect to capital ($\sigma=0.2$).*
#+LABEL: fig:speed
[[file:speed.svg]]

* Footnotes