Capital investment with two investing households

Dear all,

I have a model with three agents: two households with no liquidity constrains and one with liquidity constraints.

The two households without liquidity constraints differ in the distribution parameter and substitution elasticity for their nested CES consumption function. This is the only difference between the two household.

I wondered if I could assume that they have the same amount of capital stock and invest the same in capital goods ? It would simplify a lot the aggregation rule and the steady state equations.

  1. K = omega_1 * K_1 + omega_2 * K_2
  2. K = (omega_1+omega_2) * K_1 (or I can put similarly K_2)

At the end, the parameters that differ the two households do not end up in the capital Euler equation, though the shadow price on the budget constraint that depends on consumption levels (C/C(+1)) won’t be the same.

Thank you

Because the stochastic discount factor is time-varying and not identical across agents, I would presume that there is no simple aggregation like the one you envision.

Thank you @jpfeifer . That means the even this aggregation does not work ?

K = omega_1 * K_1 + omega_2 * K_2, the omégas being the share of each household in total population.

How do models manage to integrate heterogenous households with different liquidities then? I am already aware of non-Ricardian vs Ricardian set up, which is the one I have in an old version of the model and works perfectly.

Isn’t there a way to have two households that accumulate both capital ?

Many thanks

Maybe I misunderstood the question.
K = omega_1 * K_1 + omega_2 * K_2
is an identity that obviously need to hold.

Ah okay cool so I can do it via this identity but not the second one because of what you said.

Thank you @jpfeifer

There is the tricky part: finding K_1 and K_2 at SS. When having only a Ricardian and non Ricardian it’s straight forward. I tried to find K_1 and K_2 from the budget constraint of my household 1 and 2, but cannot find it at some point. With a Ricardian and non-Ricardian economy only, I usually have the analytical solution for the aggregate variables first, then I find the solutions for the non-Ricardian households. After that only I can find the the solutions for the Ricardian’s variables (e.g. Consumption of Ricardian = (Aggregate consumption - share of non Ricardian * Consumption of non Ricardian)/ share of Ricardian) . Doing that for for the two investing household economy is not possible. In my steady state model, I still find first the solutions for aggregate variables, then for the non-Ricardian household’s, but then finding the values for the two-investing households becomes tricky.

At SS, normally, both household 1 and 2, should have the same level of K investment/stock. I’d like them to depart from a same level of endowments. Note that these two households only differ in their preferences. They should have the same endowments. And I’m doing deterministic simulations.

I hope I was clear enough. Do you have any hint @jpfeifer ? I can send you the code.

Thank you

But what determines the split between the two agents in your model?

That’s the issue @jpfeifer . I do not know how to determine the split. I cannot find the slip from my Euler equations or just the budget constraint. Should I add another rule that determines this split ?

Usually, the rate of return on their investments need to be equal, which governs consumption-investment tradeoff and the overall investment (when combined with the budget constraints).

Hi @jpfeifer ,

When I follow what you said, a slight increase of the total capital demand leads to a decrease of the capital supply per household because their relative share in the economy increases through time. So for the following equation:

K = omega_1 * K_1 + omega_2 * K_2

when omegga_1 and omegga_2 increases, for small increase of K, K_1 and K_2 decreases and leads to decrease in C_1 and C_2. Note that omegga_1 and omegga_2 do not sum up to 1 because I have a third constrained household. When specifying this equation, should I rescale omegga_1 and omegga_2 and create one other parameter than would allow me to have:

K = omega_1_bis * K_1 + (1-omegga_1_bis) * K_2, where omegga_1_bis = omegga_1 / (omegga_1+omegga_2) ? Thsi would actually allow me to have K_1 and K_2 increasing and not decreasing.

But this would utlimately pose problems in the BC of these two households where everything in the constraint should be rescaled by omeggga_1.

What is your pick on that one ?

Thank you

I am not sure I understand the point. It seems your problem is the normalization and interpretation, not the working of the model per se.

Hi @jpfeifer ,

Thank you for the answer. I think the working model is okay, but I’m trying to understand why when the total share of savers in the economy increases, their capital stock decreases while the capital demand increase. I interpret that as the following : since the total capital demand from firms does not increase much, when having more and more savers in the economy, the capital supply per saver type is going to be lower than that if the share of savers would have remained constant throughout the years.

If I have one type of saver and not two in my model, this is the result I would have : K = omegga_saver*K_saver, if omegga_saver increases through time while the capital demand K does not increase much, then K_saver would actually decrease. I mean it all depends on the magnitude of evolution of omegga_saver and K of course.