Part 1 / Section 2: The Cobb--Douglas technology Section 2.2

Substitution and scale effects

So far we have held output fixed. That isolates substitution, but it leaves out another important response: cheaper production can lower the output price, increase sales, and raise the scale of production. We now add that channel (and will return to it later on in the course). Suppose the market has isoelastic product demand

\[\begin{equation} Y \;=\; Bp^{-\varepsilon},\qquad \varepsilon>0, \tag{2.13} \end{equation}\]

The key question is:

What happens to the demand for inputs when capital becomes cheaper?

To answer this question we need to know how the output price responds to changes in input prices. We derive the unit cost from the cost minimization problem in (1.4):

\[\begin{equation} C(Y;r,w) \;=\; Y\underbrace{ \frac{1}{z} \left(\frac{r}{\alpha}\right)^\alpha \left(\frac{w}{1-\alpha}\right)^{1-\alpha} }_{\text{per-unit cost of production }c(r,w)}. \tag{2.14} \end{equation}\]

Total cost is proportional to output, so \(c(r,w)=C(1;r,w)=C(Y;r,w)/Y\).

We now hold \(w\), \(z\), \(\alpha\), and the demand parameters fixed and change \(r\). Taking logs of unit cost makes its response transparent:

\begin{align}

\frac{\partial\log c}{\partial\log r}&\;=\;\alpha. \tag{2.15} \end{align}

Since \(p\,=\,c(r,w)\), market demand becomes \(Y\,=\,Bc(r,w)^{-\varepsilon}\). Taking logs and applying the chain rule gives

\[\begin{align} \log Y \;=\; \log B\,-\,\varepsilon\log c(r,w) \quad \longrightarrow\quad \frac{\partial\log Y}{\partial\log r} \;=\; -\varepsilon\frac{\partial\log c}{\partial\log r} \;=\; -\varepsilon\alpha. \tag{2.16} \end{align}\]

A one-percent fall in \(r\) therefore lowers unit cost and the output price by approximately \(\alpha\) percent, increasing output by \(\varepsilon\alpha\) percent.

Finally, the optimality condition for labor coming from cost minimization gives

\[\begin{equation} w\;=\;(1-\alpha)c\frac{Y}{L} \quad\longrightarrow\quad \log L\;=\;\log(1-\alpha)\,+\,\log c\,+\,\log Y\,-\,\log w. \tag{2.17} \end{equation}\]

Since \(w\) is fixed, differentiating gives the total labor response:

\[\begin{equation} \frac{\partial\log L}{\partial\log r} \;=\; \frac{\partial\log c}{\partial\log r} \;+\;\frac{\partial\log Y}{\partial\log r}\\ \;=\; \underbrace{\alpha}_{\text{substitution}} \;-\;\underbrace{\varepsilon\alpha}_{\text{scale}} \;=\; \alpha(1-\varepsilon). \tag{2.18} \end{equation}\]

The first term is the substitution effect derived above: at a fixed level of output, cheaper capital reduces labor demand. The second term is the scale effect: cheaper production expands output and raises demand for inputs. If \(\varepsilon>1\), the scale effect is larger, so a fall in \(r\) raises labor demand. If \(\varepsilon<1\), substitution dominates and labor demand falls.