<p>You could set up the profit function as suggested by @Kitsune , but here I take the expenditure minimization approach (which is mathematically equivalent).</p>

<p>Set up the minimization:
$$\min_{L,K} wL + rK \\
s.t. \quad 25LK=Q$$</p>

<p>Using the Lagrange method:
$$\mathcal{L}=(-wL - rK) + \lambda (25LK-Q)$$
Giving us the First-Order Conditions:
$$(1)\quad \frac{\partial\mathcal{L}}{\partial L}=-w + \lambda 25K=0$$
$$(2) \quad \frac{\partial\mathcal{L}}{\partial Q}=-r + \lambda 25L=0$$
$$(3) \quad \frac{\partial\mathcal{L}}{\partial \lambda}=25LK-Q=0$$</p>

<p>Taking (1) and (2):
$$(4) \quad K=\frac{w}{r}L$$
From (3):
$$(5) \quad K=\frac{Q}{25L}$$</p>

<p>From (4) and (5):
$$\frac{w}{r}L=\frac{Q}{25L}$$ </p>

<p>Solving for L:
$$L=\sqrt{\frac{rQ}{25w}}$$</p>

CC BY-SA 3.0
2016-12-06T04:56:37.037Z
2016-12-06T04:56:37.037Z
Walrasian Auctioneer
<p>I have been stuck on this for a few hours (actually a few days). I can't find much useful information on the web that has made it any clearer.</p>

<p>Suppose that a firm’s production function of output Q is a function of two inputs, labour (L) and capital (K) and can be written Q = 25LK.  </p>

<p>Letting the wage rate for labour be w and the price of capital be r, what is the equation for the firm’s demand for labour?</p>

<p>I understand that the answer is L=√(rQ/25w) but I don't know how to get there.</p>

CC BY-SA 3.0
2016-12-05T15:34:37.427Z
Set up a profit function and maximize it with respect to L and K.
Kitsune Cavalry
<microeconomics>
The equation for the firm's demand for labour
Mike
# The equation for the firm's demand for labour

(Asked by: username_0 on 2016-12-05)

I have been stuck on this for a few hours (actually a few days). I can't find much useful information on the web that has made it any clearer.

Suppose that a firm’s production function of output Q is a function of two inputs, labour (L) and capital (K) and can be written Q = 25LK.

Letting the wage rate for labour be w and the price of capital be r, what is the equation for the firm’s demand for labour?

I understand that the answer is L=√(rQ/25w) but I don't know how to get there.

**Tags:** microeconomics
> **Comments:**
> **username_1**: Set up a profit function and maximize it with respect to L and K.  

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## 1 answers

### Answer by username_2 on 2016-12-06. Score: 1 | Accepted answer

You could set up the profit function as suggested by @Kitsune , but here I take the expenditure minimization approach (which is mathematically equivalent).

Set up the minimization:
$$\min\_{L,K} wL + rK \\
s.t. \quad 25LK=Q$$

Using the Lagrange method:
$$\mathcal{L}=(-wL - rK) + \lambda (25LK-Q)$$
Giving us the First-Order Conditions:
$$(1)\quad \frac{\partial\mathcal{L}}{\partial L}=-w + \lambda 25K=0$$
$$(2) \quad \frac{\partial\mathcal{L}}{\partial Q}=-r + \lambda 25L=0$$
$$(3) \quad \frac{\partial\mathcal{L}}{\partial \lambda}=25LK-Q=0$$

Taking (1) and (2):
$$(4) \quad K=\frac{w}{r}L$$
From (3):
$$(5) \quad K=\frac{Q}{25L}$$

From (4) and (5):
$$\frac{w}{r}L=\frac{Q}{25L}$$

Solving for L:
$$L=\sqrt{\frac{rQ}{25w}}$$


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