### Factor price inequality in neoclassical economic models

The following example demonstrates that in a neoclassical economic model
with the same Cobb-Douglas production function for all agents,
as well as rational behaviour of all agents and different start capital
the factor prices will not necessarily be equal at all times and therefore
it is not appropriate to assume a priori factor price equality in such models.

Suppose a model with 2 agents (2 groups) with equal production function y = k0.5 * l0.5
and with equal labour l1 = l2 = 1 however with different capital k1 = 1.5 and k2 = 0.5

y = (δy/δl) * l + (δy/δk) * k
y = 0.5*(k/l)0.5 * l + 0.5*(l/k)0.5 * k
y = w * l + r * k

w = 0.5*(k/l)0.5
r = 0.5*(l/k)0.5

Case 1: agents work autonomous
wry
agent 10.5*(1.5/1)0.5
0.61237
0.5*(1/1.5)0.5
0.40825
0.61237*1+0.40825*1.5
1.22474
agent 20.5*(0.5/1)0.5
0.35355
0.5*(1/0.5)0.5
0.70711
0.35355*1+0.70711*0.5
0.70711
sum  1.22474+0.70711
1.93185
Case 2: agents work together with fair share of wage and capital returns
wry
agent 10.5*((1.5+0.5)/(1+1))0.5
0.5
0.5*((1+1)/(1.5+0.5))0.5
0.5
0.5*1+0.5*1.5
1.25
agent 20.5*((1.5+0.5)/(1+1))0.5
0.5
0.5*((1+1)/(1.5+0.5))0.5
0.5
0.5*1+0.5*0.5
0.75
sum  1.25+0.75
2
Case 3: agent 2 works 60% autonomous and 40% for wage with agent 1
wry
agent 10.5*(1.5/(1+0.4))0.5
0.51755
0.5*((1+0.4)/1.5)0.5
0.48305
0.51755*1+0.48305*1.5
1.24212
agent 20.5*(0.5/(1-0.4))0.5
0.45644
0.5*((1-0.4)/0.5)0.5
0.54772
0.45644*0.6+0.51755*0.4+0.54772*0.5
0.75474
sum  1.24212+0.75474
1.99686

A rational agent 2 will choose case 3 because this leads to a higher income for agent 2.
For the chosen case 3 follows that the factor prices w and r are not equal as it is in case 2.