Notice that for a zero risk-free rate the equal consumption approach still leads to price equality the same way as in the model with initial consumption.
Therefore, an equal consumption distribution can be proxied by an equal skill distribution.
It remains to show that there is a portfolio rule that makes equal consumption possible.
Notice that with consumption only at (t=1), equal consumption does not lead to equal SDFs, but rather to proportional SDFs.
I show that the portfolio rule that ensures equal consumption is the same as the one in Kruschwitz and Löffler (2009).
Noting that the production of services has to equal consumption, we find that: l_{text{n}} = c_{text{n}} + n_{text{n}}.
Alternatively to Eq. (88), one can also use the expression in Proposition 4 rearranged for (mathbf {n}_{i}^{*prime }), which yields exactly the same portfolios weights, which, in turn, imply equal consumption in both economies.
Since consumption at the different dates is the only variable argument in the SDFs of the agents, it is clear that equal consumption of agents in both economies leads to equal SDFs.
With equal initial portfolios and wealth, i.e., agents have the same initial characteristics in both economies, equal consumption means that an optimum in the no-tax economy is equivalent to an optimum in the tax economy.
Under equal consumption of heating energy and cooling air conditions, two types of actively managed thermal loading with different allocation ratios of heating energy (ϕ = 0.1 and ϕ = 0.2) in the outer and inner surface of disk were considered in the process of optimization.
The portfolio rule for risky assets is the same as for the case with the zero risk-free rate and equal consumption and it is the same as the one presented in Kruschwitz and Löffler (2009) for the constant absolute risk aversion case.
