Thank you for the help so far!

I would like to assign +/i signs for the function and solution such as (+)(+)(-) and therefore (-).

It is working for the first evaluation, λlower, but not for the second evaluation, λupper. I cannot see the difference - has it to do with "λi /. Solve" in both definitions? 

        (* Endogenous *)
        Clear[μ, ξ, β, γ, ri, si, rj, sj, λCri, λCsi , λlower, λupper, λlowereip, λip];
        (* Exogenous *)
        Clear[ ci, yp, λi, αi, ϵip, ηip, eip];
        ri = ϵip*λi + (1 - ηip)*(1 - λi);
        si = ηip*(1 - λi) + (1 - ϵip)*λi; 
        λCri = (ϵip*λi)/(ϵip*λi + (1 - ηip)*(1 - λi));
        λCsi = (ηip*(1 - λi))/(ηip*(1 - λi) + (1 - ϵip)*λi);
        λlower = λi /. 
        Solve[ri*(yp - eip - ci + αi*λCri) + si*(yp - eip + αi*(1 - λCsi)) == yp + αi*(1 - λi), {λi}];
        λupper = λi /.Solve[ri*(yp - eip - ci + αi*λCri) + 
      si*(yp - eip + αi*(1 - λCsi)) == yp + αi*λi - ci, {λi} ];
        Simplify[D[λlower, eip]]
        Simplify[Sign[D[λlower, eip]], αi > 0 && eip > 0 && ci > 0 && 0 < ϵip < 1 && 0 < ci < 1 && 0 < ηip < 1 && αi > ci && αi > ci + eip && ϵip + ηi < 1]
        Simplify[D[λupper, eip]]
        Simplify[Sign[D[λupper, eip]], αi > 0 && eip > 0 && ci > 0 && 0 < ϵip < 1 && 0 < ci < 1 && 0 < ηip < 1 && αi > ci && αi > ci + eip && ϵip + ηi < 1]