Labor demand and supply curves in terms of elasticities I would like to ask for your help to derive some expressions. Let the labor demand curve be described by:
$N^D = N^D(W/P, \bar{K}), \quad N^D_{W/P}= \frac{1}{F_{NN}}&lt; 0, \quad N^D_K = -\frac{F_{NK}}{F_{NN}} &gt; 0$
and the labor supply curve by:
$W/P^e = g(N^S)$
My question is, how can I express these two curves in terms of elasticities? I know that the result should be:
$\frac{dN^D}{N^D} = \frac{d\bar{K}}{\bar{K}} - \varepsilon_D \Big(\frac{dW}{W}-\frac{dP}{P} \Big) \\
\frac{dN^S}{N^S} = \varepsilon_s \Big( \frac{dW}{W}-\frac{dP^e}{P^e}\Big)$
where $\varepsilon_D=-F_{NN}/(NF_{NN})$ and $\varepsilon_S = g(N)/(Ng_N)$ denote the wage elasticities of labor demand and labor supply, respectively.
Thank you for your help and time.
This result is presented on page 14 of Foundations of Modern Macroeconomics. The key is to total differentiate both expressions.
Start with the supply of labor:
\begin{gather}
W/P^e = g(N^S) \\
d(W/P^e) = d\Big( g(N^S) \Big) \\
\frac{1}{P^e} dW - \frac{W}{(P^e)^2}dP^e = g_NdN^S \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{1}{g_N}\frac{W}{P^e}\frac{dP^e}{P^e} \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e}, \quad \text{where} \quad W/P^e = g(N^S) \\
dN^S = \frac{1}{g_N}\frac{g(N^S)}{W}dW - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e} \\
dN^S = \frac{g(N^S)}{g_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg] \\
\frac{dN^S}{N^S} = \frac{g(N^S)}{N^Sg_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg]
\end{gather}
To derive the expression for the demand of labor, one can follow a similar process (I will ommit the superscript $N^D$ for simplicity, but we know we are talking about labor demand). Recall that the FOC of the firm's problem is given by:
\begin{gather}
PF_N(N,\bar{K}) = W \\
F_N(N,\bar{K}) = \frac{W}{P} \\
d\Big( F_N(N,\bar{K}) \Big) = d\Big( W/P \Big) \\
F_{NN}dN + F_{NK}d\bar{K} = \frac{1}{P}dW - \frac{W}{P^2}dP \\
F_{NN}dN + F_{NK}d\bar{K} = F_N\frac{dW}{W} - F_N\frac{dP}{P} \\
F_{NN}dN = F_N \bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] - F_{NK}d\bar{K} \\
dN = -\frac{F_{NK}}{F_{NN}}d\bar{K} + \frac{F_N}{F_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = -\frac{F_{NK}}{N^DF_{NN}}d\bar{K} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = \frac{d\bar{K}}{\bar{K}} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg], \quad \text{where} \quad KF_{NK} = -NF_{NN}
\end{gather}