Revisiting development of Solow Model with no population growth and no technical progress

Question

Consider an economy descried by the production function: 𝑌 = 𝐹(𝐾, 𝐿) = 𝐾^0.3𝐿^0.7.
a) Derive the per-worker production function.
b)Assuming no population growth or technological progress, find the steady-state capita stock per worker, output per worker and consumption per worker as a function of the saving rate and depreciation rate.

Answer 1

You can use the following results provided below to try and answer the problem yourself.

1. Consider a Constant Returns to Scale(CRS) production function $F:\mathbb{R}_+^2\to \mathbb{R}$
   Let $Y=F(K,L)$ denote the level of output produced using $K$ units of capital and $L$ units of labor given the production function $F$.

   By CRS, $F$ must satisfy $: \quad F(\lambda K, \lambda L)=\lambda F(K,L)$ where $\lambda \geq 0$ is some constant

   Since the above holds for any positive $\lambda$, we can set $\lambda=\frac{1}{L}$ and let $k=\frac{K}{L}$ and $y=\frac{Y}{L}$.

   This gives us: $$\begin{eqnarray} & F\left(\frac{K}{L},1\right)=\frac{F(K,L)}{L} \\ & F(k,1)=\frac{Y}{L} \\ & y=f(k) && \text{where }F(k,1)\equiv f(k) \tag{*} \end{eqnarray}$$

   The expression $\boxed{y=f(k)}$ is known as the intensive form (per-worker production) function (which is what you are asked to find in part a)

2. Consider a Solowian Economy with the following features:
   $\bullet$ Time is continuous and there is no distinction between the labor force and the population at any given period of time. Labor (or population) growth rate $\frac{\dot{L}}{L}=n$ , where $\dot L= \frac{dL}{dt}$ is exogeneous
   $\bullet$ In each period a constant proportion of income $s\in(0,1)$ is saved and capital depreciates by $\delta$ in each period
   $\bullet$ There is no technological progress and output in any period is given by $Y_t=F(K_t,L_t)$ with the assumption that $F$ satisfies all the properties of a neo-classical production function

   Denoting the growth rate of any object $x$ (say) as $g_x$ let us use the above data to solve for the steady state.

   The law of motion of capital is given by: $\begin{eqnarray} &\dot K = I-\delta K\\ \implies &\dot K= sY-\delta K \tag{**} \end{eqnarray}$

   we defined per-capita capital as $k=\frac{K}{L}$ and at steady state we must have $\frac{\dot k}{k}=g_k=0$ i.e., the growth rate of per-capita capital must be zero at the steady state. Let us find an expression for $\frac{\dot k}{k}$ or $g_k$ as a function of $k$ in order to find steady state per-capita capital. $$\begin{align} k&=\frac{K}{L} \\ \ln k&= \ln K -\ln L & \text{taking log on both sides} \\ \frac{\dot k}{k}&=\frac{\dot K}{K}-\frac{\dot L}{L} & \text{taking time derivative on both sides} \\ &=\frac{sY-\delta K}{K}-n & \because \; \frac{\dot L}{L}=n \text{ and using (**)}\\ &=s\frac{Y/L}{K/L}-(n+\delta) \\ &=\frac{sy}{k}-(n+\delta)=\frac{sf(k)-(n+\delta)k}{k} & \text{using (*)} \end{align}$$

   $$\therefore \quad g_k(k)=\frac{sf(k)-(n+\delta)k}{k}$$

   The steady-state per-capita capital is given by $k^*$ such that $g_k(k^*)=0$. Which can be obtained as follows: $$\begin{eqnarray} &g_k(k^*)\overset{set}{=}0 \\ &\boxed{sf(k^*)=(n+\delta)k^*}\tag{1} \end{eqnarray}$$

   A solution to $(1)$ will give you the steady state $k$ which in this case must be unique because of the assumption we made about production technology $F$ being neo-classical.

   Once you have found out the steady-state capital per-capita $k^*$, we can use $(^*)$ to find the steady-state output per worker and consumption per worker.

   steady-state output per worker: $\boxed{y^*=f(k^*)}$
   steady-state consumption per worker: $\boxed{c^*=(1-s)f(k^*)}$

Answer 2

To find the per-worker production function we need to divide the given production function with L i.e. $\frac{F}{L}$ = $$\frac{K^{0.3}L^{0.7}}{L}$$ $$y=\frac{K^{0.3}}{L^{0.3}} = k^{0.3}; k =\frac{K}{L}$$ This completes first part of the question.
For the second part, we have to recall the key equation of the solow model i.e.,
$$\dot{k}= sk^{\alpha}-(\delta+n+g)k$$ As stated in the question growth rate of population and technology is 0, $$\therefore n = g = 0$$ Which transforms our equation into $$\dot{k}=sk^{\alpha}-{\delta}k ; {\alpha} = 0.3$$ We know at steafdy state $\dot{k}=0$ therefore the above equation trandforms into $$sk^{0.3}={\delta}k$$ $$\Rightarrow k^{0.7}=\frac{s}{\delta}$$ $$\Rightarrow k= (\frac{s}{\delta})^ \frac{1}{0.7}$$ and since output per worker i.e. y = k^0.3 $$\therefore y = (\frac{s}{\delta})^{\frac{0.3}{0.7}}$$ and since consumption per worker $$c=(1-s)y$$ $$\therefore c= (1-s)(\frac{s}{\delta})^{\frac{0.3}{0.7}}$$