<p>We have these following from the first order conditions:</p>

<p>${U_{LA} \over U_C}  = w_A \to MRS_{C,LA}=w_A$</p>

<p>${U_{LB} \over U_C} = w_B \to MRS_{C, LB}=w_B$</p>

<p>${U_{LA} \over U_{LB}} = {w_A \over w_B} \to MRS_{LB, LA} = {wA \over w_B}$</p>

<p>$C + w_A L_A + w_B L_B = (w_A + w_B)T + v_A + v_B$ </p>

<p>where $w_A, w_B$ are wage for person $A$ and $B$ respectively</p>

<p>$L_A, L_B$ are leisure for person $A$ and $B$ respectively</p>

<p>$C= C_A+C_B$ (total consumption) </p>

<p>And then it says that from the first three, we can derive the Marshallian demand functions for consumption and leisure:</p>

<p>$C*=C(w_A, w_B, v)$</p>

<p>$L_A* = L_A (w_A,w_B, v)$</p>

<p>$L_B* = L_B (w_A, w_B, v)$</p>

<p>My first question is, what do these mean exactly? What do these actually mean (being a marshallian demand function, how can I do this?), opened up? ($v$ is non-labor income).</p>

<p>Then there comes the marshallian labor supply, which probably is connected to the demand functions, they are given as:</p>

<p>$h_A * = h_A(w_A,w_B,v) = T - L_A (w_A, w_B, v)$</p>

<p>$h_B * = h_B(w_A,w_B,v) = T - L_B (w_A, w_B, v)$</p>

<p>These I can interpret as I get the answer to what are marshallian demand functions, as $T =$ time available to do work (but if someone has the time, maybe you could give a little info?)</p>

<p>Then there comes a Cobb-Douglas function s.t.</p>

<p>$U=C^\alpha L^\beta_A L^\gamma_B$ with $\alpha + \beta + \gamma = 1$</p>

<p>From these results, can be derived following results for the partial derivatives:</p>

<p>${d h_i \over d w_i} &gt; 0$</p>

<p>${d h_i \over d w_j} &lt; 0$</p>

<p>${d j_i \over d v} &lt; 0$</p>

<p>with $i, j = A, B$ and $ i \neq j$ and $v_A + v_B = v$</p>

<p>I dont even know what I need to derivate, but I'm supposed to proof the signs of the last derivatives. Everything is marked as in my source text. This is probably very easy, I'm just confused.</p>

CC BY-SA 3.0
2017-03-23T17:29:00.913Z
I'm confused too, about what you're trying to ask here. So it seems that you have problems understanding what Marshallian demand functions are and you've got bogged down by a bunch of messy math. Why don't you take a step back, and ask questions one by one.
Herr K.
CC BY-SA 3.0
2017-03-23T17:29:40.053Z
Also, what's all this math for? What is the context? Solving for general equilibrium? What are the utility functions and production function?
Herr K.
<cobb-douglas>
Unitary model with cobb-douglas function
repulsive23
# Unitary model with cobb-douglas function

(Asked by: username_0 on 2017-03-23)

We have these following from the first order conditions:

${U\_{LA} \over U\_C} = w\_A \to MRS\_{C,LA}=w\_A$

${U\_{LB} \over U\_C} = w\_B \to MRS\_{C, LB}=w\_B$

${U\_{LA} \over U\_{LB}} = {w\_A \over w\_B} \to MRS\_{LB, LA} = {wA \over w\_B}$

$C + w\_A L\_A + w\_B L\_B = (w\_A + w\_B)T + v\_A + v\_B$

where $w\_A, w\_B$ are wage for person $A$ and $B$ respectively

$L\_A, L\_B$ are leisure for person $A$ and $B$ respectively

$C= C\_A+C\_B$ (total consumption)

And then it says that from the first three, we can derive the Marshallian demand functions for consumption and leisure:

$C\*=C(w\_A, w\_B, v)$

$L\_A\* = L\_A (w\_A,w\_B, v)$

$L\_B\* = L\_B (w\_A, w\_B, v)$

My first question is, what do these mean exactly? What do these actually mean (being a marshallian demand function, how can I do this?), opened up? ($v$ is non-labor income).

Then there comes the marshallian labor supply, which probably is connected to the demand functions, they are given as:

$h\_A \* = h\_A(w\_A,w\_B,v) = T - L\_A (w\_A, w\_B, v)$

$h\_B \* = h\_B(w\_A,w\_B,v) = T - L\_B (w\_A, w\_B, v)$

These I can interpret as I get the answer to what are marshallian demand functions, as $T =$ time available to do work (but if someone has the time, maybe you could give a little info?)

Then there comes a Cobb-Douglas function s.t.

$U=C^\alpha L^\beta\_A L^\gamma\_B$ with $\alpha + \beta + \gamma = 1$

From these results, can be derived following results for the partial derivatives:

${d h\_i \over d w\_i} > 0$

${d h\_i \over d w\_j} < 0$

${d j\_i \over d v} < 0$

with $i, j = A, B$ and $ i \neq j$ and $v\_A + v\_B = v$

I dont even know what I need to derivate, but I'm supposed to proof the signs of the last derivatives. Everything is marked as in my source text. This is probably very easy, I'm just confused.

**Tags:** cobb-douglas
> **Comments:**
> **username_1**: I'm confused too, about what you're trying to ask here. So it seems that you have problems understanding what Marshallian demand functions are and you've got bogged down by a bunch of messy math. Why don't you take a step back, and ask questions one by one.  
> **username_1**: Also, what's all this math for? What is the context? Solving for general equilibrium? What are the utility functions and production function?  

---

## 0 answers

