<p>This result is presented on page 14 of Foundations of Modern Macroeconomics. The key is to total differentiate both expressions.</p>
<p>Start with the supply of labor:</p>
<p><span class="math-container">\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}</span></p>
<p>To derive the expression for the demand of labor, one can follow a similar process (I will ommit the superscript <span class="math-container">$N^D$</span> for simplicity, but we know we are talking about labor demand). Recall that the FOC of the firm's problem is given by:</p>
<p><span class="math-container">\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}</span></p>

CC BY-SA 4.0
2021-01-16T20:51:17.477Z
2021-01-16T20:51:17.477Z
Alex Ruiz
<p>I would like to ask for your help to derive some expressions. Let the labor demand curve be described by:</p>
<p><span class="math-container">$N^D = N^D(W/P, \bar{K}), \quad N^D_{W/P}= \frac{1}{F_{NN}}&lt; 0, \quad N^D_K = -\frac{F_{NK}}{F_{NN}} &gt; 0$</span></p>
<p>and the labor supply curve by:</p>
<p><span class="math-container">$W/P^e = g(N^S)$</span></p>
<p>My question is, how can I express these two curves in terms of elasticities? I know that the result should be:</p>
<p><span class="math-container">$\frac{dN^D}{N^D} = \frac{d\bar{K}}{\bar{K}} - \varepsilon_D \Big(\frac{dW}{W}-\frac{dP}{P} \Big) \\
\frac{dN^S}{N^S} = \varepsilon_s \Big( \frac{dW}{W}-\frac{dP^e}{P^e}\Big)$</span></p>
<p>where <span class="math-container">$\varepsilon_D=-F_{NN}/(NF_{NN})$</span> and <span class="math-container">$\varepsilon_S = g(N)/(Ng_N)$</span> denote the wage elasticities of labor demand and labor supply, respectively.</p>
<p>Thank you for your help and time.</p>

CC BY-SA 4.0
2021-01-16T10:38:06.067Z
can you please first show us your attempt per our rules on self study questions? (for more on the rules see our help center)
1muflon1
CC BY-SA 4.0
2023-09-25T12:39:52.237Z
@Alex, I am reading Ben Heijdra too, cant find the detailed version in page 14, its just a short result at the end. Can you share me the link please?
Sanjida
<macroeconomics><microeconomics><mathematical-economics>
Labor demand and supply curves in terms of elasticities
Alex Ruiz
# Labor demand and supply curves in terms of elasticities

(Asked by: username_0 on 2021-01-16)

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.

**Tags:** macroeconomics, microeconomics, mathematical-economics

> **Comments:**
> **username_1**: can you please first show us your attempt per our rules on self study questions? (for more on the rules see our help center)  
> **username_2**: @Alex, I am reading Ben Heijdra too, cant find the detailed version in page 14, its just a short result at the end. Can you share me the link please?  

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## 1 answers

### Answer by username_0 on 2021-01-16. Score: 1 | Accepted answer

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}


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