# Unitary model with cobb-douglas function

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.

• 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. Mar 23 '17 at 17:29
• 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. Mar 23 '17 at 17:29