Labor demand and supply curves in terms of elasticities

Question

I would like to ask for your help to derive some expressions. Let the labor demand curve be described by:

$N^D = N^D(W/P, \bar{K}), \quad N^D_{W/P}= \frac{1}{F_{NN}}< 0, \quad N^D_K = -\frac{F_{NK}}{F_{NN}} > 0$

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.

Answer 1

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}