Colinearity in a two-households-model due to similar euler equations

Hi all!

I am trying to construct a very simple DSGE model consisting of a skilled and an unskilled household which both decide on consumption, leisure, and investment and a firm whose production function exhibits capital-skill complementarity. As of now, there is no stochastic component.
The resultant steady state unexpectedly has one household pile up a quite large amount of debt, but Dynare does not give any error and the rank condition is verified.

However, running model_diagnostics gives

MODEL_DIAGNOSTICS: The Jacobian of the static model is singular
MODEL_DIAGNOSTICS: there is 1 colinear relationships between the variables and the equations
Colinear variables:
CU
CS
U
S
K
KU
KS
wu
ws
Colinear equations
5 7

MODEL_DIAGNOSTICS: The singularity seems to be (partly) caused by the presence of a unit root
MODEL_DIAGNOSTICS: as the absolute value of one eigenvalue is in the range of ±1e-6 to 1.
MODEL_DIAGNOSTICS: If the model is actually supposed to feature unit root behavior, such a warning is expected,
MODEL_DIAGNOSTICS: but you should nevertheless check whether there is an additional singularity problem.
MODEL_DIAGNOSTICS: The presence of a singularity problem typically indicates that there is one
MODEL_DIAGNOSTICS: redundant equation entered in the model block, while another non-redundant equation
MODEL_DIAGNOSTICS: is missing. The problem often derives from Walras Law.

The two colinear equations are the euler-equations for the households. As I see it, one of them is redundant for the determination of the steady state, for each of them gives the steady state level of the interest rate, leaving me with infinitely many steady states.

When modelling one of the two households as non-ricardian i.e. not granting it access to the capital market, this issue disappears.

I have just started working with Dynare. How should I handle this? I cannot think of any non-redundant equation to replace the euler equation with.
rck.mod (1.6 KB)