2025-10-08

Topics about the Growth Model.

The typical question is that, no matter how the initial values of different countries are, they will all converge to the same steady state eventually. Or will the countries with lower initial capital per worker keep lower than those with higher initial capital per worker forever?

The set up of the model

Preferences

$$\sum {\beta }^{t}u(c_t,1-h_t)$$

We won’t consider the market clear, we do not care about it.

It’s weird that since there is only one household in the Growth Model, but we still want to mention the pareto efficiency.

First, we find the partial derivative of the labor supply function with respect to the wage, w:

$\frac{\partial h}{\partial w}=\frac{\partial }{\partial w}\left(\frac{\alpha T}{\alpha +1}-\frac{y}{w(\alpha +1)}\right)=\frac{y}{w^2(\alpha +1)}$ Now, we can find the elasticity: $ϵ=\left(\frac{y}{w^2(\alpha +1)}\right)\frac{w}{h}=\frac{y}{w(\alpha +1)}\frac{1}{\frac{\alpha wT-y}{w(\alpha +1)}}=\frac{y}{\alpha wT-y}$ To find the compensated wage elasticity, ${ϵ}^c$, we use the Slutsky equation in elasticity form from the slides[cite: 135]: $ϵ={ϵ}^c+w\frac{\partial h}{\partial y}$ We first need the partial derivative of labor supply with respect to unearned income, $y$: $\frac{\partial h}{\partial y}=\frac{\partial }{\partial y}\left(\frac{\alpha wT-y}{w(\alpha +1)}\right)=-\frac{1}{w(\alpha +1)}$ Now, we rearrange the Slutsky equation to solve for ${ϵ}^c$:

${ϵ}^c=ϵ-w\frac{\partial h}{\partial y}=\frac{y}{\alpha wT-y}-w\left(-\frac{1}{w(\alpha +1)}\right)$

${ϵ}^c=\frac{y}{\alpha wT-y}+\frac{1}{\alpha +1}$ To combine these two parts, we find a common denominator, which is $(\alpha wT-y)(\alpha +1)$: ${ϵ}^c=\frac{y(\alpha +1)+1(\alpha wT-y)}{(\alpha wT-y)(\alpha +1)}$ Now, we expand the numerator: ${ϵ}^c=\frac{\alpha y+y+\alpha wT-y}{(\alpha wT-y)(\alpha +1)}$ The $+y$ and $-y$ terms in the numerator cancel, leaving: ${ϵ}^c=\frac{\alpha y+\alpha wT}{(\alpha wT-y)(\alpha +1)}$ Finally, we can factor out $\alpha$ from the numerator to get the final expression for the compensated wage elasticity: ${ϵ}^c=\frac{\alpha (wT+y)}{(\alpha +1)(\alpha wT-y)}$

${ϵ}^{\prime }{|}_{h^{\prime }\to 0}={\lim }_{h^{\prime }\to 0}\left(\frac{\partial h^{\prime }}{\partial w^{\prime }}\frac{w^{\prime }}{h^{\prime }}\right)$Since $\frac{\partial h^{\prime }}{\partial w^{\prime }}$ at $w^{\prime }=w$ is a positive constant and $w>0$, the limit is:${ϵ}^{\prime }{|}_{h^{\prime }\to 0}=\left(\frac{y+w{h}^{\ast }}{w^2(\alpha +1)}\right)\cdot {\lim }_{h^{\prime }\to 0}\left(\frac{w}{h^{\prime }}\right)=\infty$