<p>Consider <span class="math-container">$F(K, L) = \min(2L + K, L + 2K)$</span>. Check that conditional factor demands satisfy :</p>

<p><span class="math-container">$K^*(Q = 9, w = 1, r = 1) = L^*(Q = 9, w = 1, r = 1) = 3$</span>
and
<span class="math-container">$K^*(Q = 9, w = 2, r = 3) = L^*(Q = 9, w = 2, r = 3) = 3$</span></p>

<p>Therefore, the statement is false.</p>

CC BY-SA 4.0
2019-03-15T07:35:55.593Z
I’m having a bit of trouble solving for K* and L* since MPL/MPK is constant. Can you provide any guidance?
taurus
CC BY-SA 4.0
2019-03-15T19:39:39.230Z
Taurus, if $\frac{MP_L}{MP_K}$ is constant, then you are dealing with the perfect substitutes case, in which the optimal bundle will always contain either only labor or only capital. You need to consider two cases, when isocost lines are steeper than isoquants ($\frac{MP_L}{MP_K}<\frac{w}{r}$) and when isoquants are steeper than the isocost lines ($\frac{MP_L}{MP_K}>\frac{w}{r}$). In the former case, the cost-minimizing input bundle will contain only capital, while in the latter it will contain only labor.
dlnB
CC BY-SA 4.0
2019-03-15T19:41:44.683Z
If we are in case 1 ($\frac{MP_L}{MP_K}<\frac{w}{r}$), we immediately know $L^*=0$. Plug this into the production function will allow you to solve for $K^*$. In case 2, ($\frac{MP_L}{MP_K}>\frac{w}{r}$), we immediately know $K^*=0$. Plug this into the production function will allow you to solve for $L^*$. It helps to draw the isoquant map to better understand this.
dlnB
CC BY-SA 4.0
2019-03-15T06:48:39.723Z
2019-03-15T06:48:39.723Z
Amit
<p>I’ve been given this true false question: Consider the minimization of wL + rK given F(K, L) <span class="math-container">$\geq$</span> Q with F(K, L) strictly increasing in K and L. The conditional factor demands K*(Q, w, r) and L*(Q, w, r) would definitely change if w doubles and r triples. </p>

<p>I’m inclined to say this is false, because I can imagine a scenario where a firm far prefers labor to capital, so the amount of labor employed stays the same after the change, but I’m not sure how to find a function with those properties. </p>

<microeconomics><producer-theory><firm>
Changing Constant Factor Demands
taurus
# Changing Constant Factor Demands

(Asked by: username_0 on 2019-03-15)

I’ve been given this true false question: Consider the minimization of wL + rK given F(K, L) $\geq$ Q with F(K, L) strictly increasing in K and L. The conditional factor demands K\*(Q, w, r) and L\*(Q, w, r) would definitely change if w doubles and r triples.

I’m inclined to say this is false, because I can imagine a scenario where a firm far prefers labor to capital, so the amount of labor employed stays the same after the change, but I’m not sure how to find a function with those properties.

**Tags:** microeconomics, producer-theory, firm


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## 1 answers

### Answer by username_1 on 2019-03-15. Score: 0 | Accepted answer

Consider $F(K, L) = \min(2L + K, L + 2K)$. Check that conditional factor demands satisfy :

$K^\*(Q = 9, w = 1, r = 1) = L^\*(Q = 9, w = 1, r = 1) = 3$
and
$K^\*(Q = 9, w = 2, r = 3) = L^\*(Q = 9, w = 2, r = 3) = 3$

Therefore, the statement is false.

> **Comments:**
> **username_0**: I’m having a bit of trouble solving for K* and L* since MPL/MPK is constant. Can you provide any guidance?  
> **username_2**: username_0, if $\frac{MP_L}{MP_K}$ is constant, then you are dealing with the perfect substitutes case, in which the optimal bundle will always contain either only labor or only capital. You need to consider two cases, when isocost lines are steeper than isoquants ($\frac{MP_L}{MP_K}<\frac{w}{r}$) and when isoquants are steeper than the isocost lines ($\frac{MP_L}{MP_K}>\frac{w}{r}$). In the former case, the cost-minimizing input bundle will contain only capital, while in the latter it will contain only labor.  
> **username_2**: If we are in case 1 ($\frac{MP_L}{MP_K}<\frac{w}{r}$), we immediately know $L^*=0$. Plug this into the production function will allow you to solve for $K^*$. In case 2, ($\frac{MP_L}{MP_K}>\frac{w}{r}$), we immediately know $K^*=0$. Plug this into the production function will allow you to solve for $L^*$. It helps to draw the isoquant map to better understand this.  

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