# A production with Automation (Lankisch et al. 2019)

We reproduce the main results from Lankisch et al., 2019, which investigates a production with “kind-of” endogenized automation progress. Automation technology directly competes with/substitutes for low-skilled workers.

Highlights

• The real wages of low-skilled workers have been shrinking in the US for decades.
• Per capita output and the wages of high-skilled workers have been increasing.
• We propose an economic growth model with automation to explain these trends.
• Automation has the potential to raise economic prosperity but also wage inequality.
• Investments in higher education can reduce the effect of automation on inequality.

## Production

Production is CES-type:

$Y = \left[ (1-\beta)L_s^\gamma + \beta(P+L_u)^\gamma \right]^{\frac{1-\alpha}{\gamma}} K^\alpha \ \ \ \ \ (1)$

where: ( $L_s, L_u$ ) are skilled and unskilled labor, ( P ) is automation technology, ( $\beta$ ) is the production weight of unskilled workers, ( $\gamma \in (-\infty, 1]$) is the substitutability between both types of workers.

Let $( s, s_K, \delta )$ be the saving rate, traditional capital investment fraction, depreciation rate, respectively.

Then:

$\dot{K} = s_K s Y - \delta K$ $\dot{P} = (1-s_K) s Y - \delta P$

Labor market equilibrium:

$L = L_s + L_u$

In terms of shares:

$l_s = \frac{L_s}{L_s + L_u} = \frac{L_s}{L} \\ l_u = \frac{L_u}{L_s + L_u} = \frac{L_u}{L}$

Hence:

$p = \frac{P}{L_s + L_u} = \frac{P}{L}$

To rewrite production in terms of per worker level, divide both sides of equation (1) to L:

\begin{aligned} & y \equiv \frac{Y}{L} = \frac{ \left[ (1-\beta)(l_s L)^\gamma + \beta(pL+l_uL)^\gamma \right]^{\frac{1-\alpha}{\gamma}} }{L} K^\alpha \\ & = \frac{ \left( \left[ (1-\beta)(l_sL)^\gamma + \beta (pL + l_u L)^\gamma \right]^\frac{1}{\gamma} \right)^{1-\alpha} }{L^{1-\alpha}} \frac{K^\alpha}{L^\alpha} \\ & = \left[ (1-\beta)l_s^\gamma + \beta(p+l_u)^\gamma \right]^{\frac{1-\alpha}{\gamma}} k^\alpha \end{aligned}

This shows that automation competes with low-skilled workers.

## Factor prices

### Capital

Taking the derivatives of Y wrt ($K \ \ \& \ \ P$) :

\begin{aligned} & r_K = \frac{\partial Y}{\partial K} = \alpha K^{\alpha-1} \left[ (1-\beta)L_s^\gamma + \beta(P+L_u)^\gamma \right]^{\frac{1-\alpha}{\gamma}} \\ & r_P = \frac{\partial Y}{\partial P} = K^\alpha \frac{1-\alpha}{\gamma} \beta\gamma(p+l_u)^{\gamma-1} \left[ (1-\beta)L_s^\gamma + \beta(P+L_u)^\gamma \right]^{\frac{1-\alpha}{\gamma} - 1} \end{aligned}

Applying the no-arbitrage condition (so that investors will invest in both traditional capital and automation technology):

\begin{aligned} & r_K = r_P \\ &\Leftrightarrow \alpha = K \frac{(1-\alpha)\beta(P+L_u)^{\gamma-1}}{\left[ (1-\beta)L_s^\gamma + \beta(P+L_u)^\gamma \right]} \\ &\Leftrightarrow K = \frac{ \alpha (P+L_u)^{1-\gamma} \left[ (1-\beta)L_s^\gamma + \beta (P+L_u)^\gamma \right]}{ (1-\alpha)\beta } \end{aligned}

In per-labor terms:

\begin{aligned} & k = \frac{K}{L} = \frac{ \alpha (p+l_u)^{1-\gamma} L^{1-\gamma} \left[ (1-\beta)l_s^\gamma L^\gamma + \beta (p+l_u)^\gamma L^\gamma \right]}{(1-\alpha)\beta L} \\ & = \frac{ \alpha(1-\beta)(p+l_u)^{1-\gamma}l_s^\gamma + \alpha\beta(p+l_u) }{(1-\alpha)\beta} \end{aligned}

### Wage

Taking the derivatives of Y wrt ($L_s \ \ \& \ \ L_u$ ) :

Skilled workers:

\begin{aligned} & w_s = \frac{\partial Y}{\partial L_s} = K^\alpha \frac{1-\alpha}{\gamma} \left[ \dots \right]^{\frac{1-\alpha}{\gamma}-1} (1-\beta)\gamma L_s^{\gamma - 1} \\ & = \frac{(1-\alpha)(1-\beta)Y}{L_s^{1-\alpha} \left[ (1-\beta)L_s^\gamma + \beta(P+L_u)^\gamma \right]}\end{aligned}

Unskilled workers:

\begin{aligned} & w_u = K^\alpha \frac{1-\alpha}{\gamma} \left[\dots\right]^{\frac{1-\alpha}{\gamma} - 1} \beta\gamma (P+L_u)^{\gamma-1} \\ & = \frac{(1-\alpha)\beta Y}{ (P+L_u)^{1-\gamma} \left[ (1-\beta)L_s^\gamma + \beta(P+L_u)^\gamma \right]} \end{aligned}
Written on January 18, 2021