Consumer Maximization Problem Doesn't Work in Wolfram Mathematica

Question

I'm trying to solve a problem that can be seen in the paper "The Role of the Structural Transformation in Aggregate Productivity". The problem is:

Manually it is difficult to find the solution, but this problem has an analytical solution. So I'm using Wolfram Mathematica. I created the following code:

ClearAll["Global`*"]

(*Define the utility function*)
Utility = 
  a*Log[ca - abar] + (1 - a)*(1/rho)*
    Log[b*cm^rho + (1 - b)*(cs + sbar)^rho];

(*Define constraint*)
constraint = pa*ca + pm*cm + ps*cs - w *l;

(*Define the Lagrangian and its derivatives*)
Lagrangian = Utility - \[Lambda]*constraint;

Foc1 = D[Lagrangian, ca];
Foc2 = D[Lagrangian, cm];
Foc3 = D[Lagrangian, cs];
Foc4 = D[Lagrangian, \[Lambda]];

(*Solve problem*)
sols = Simplify[
  Solve[{Foc1 == 0, Foc2 == 0, Foc3 == 0}, {ca, cm, cs},
   Assumptions -> {pa > 0, pm > 0, ps > 0,
      ca > 0, cm > 0, cs > 0,
      abar > 0, sbar > 0,
      w > 0, l > 0,
      0 < a < 1 , 0 < b < 1, rho < -1}
   ]]

However, he does not find a symbolic solution to my problem. Is there something wrong in my code? What can I do to simplify or force Wolfram to find the solution?