Welfare Analysis: first-best versus competitive equilibrium

[davydiuk]

Hi all,

I am trying to estimate welfare loss a in competitive equilibrium relative to first-best allocation. The model is a standard RBC model with no labour: two agents - household with log utility - and firm. Firm has access to the decreasing returns to scale production technology AK^alfa. To fund investment in capital, it borrows from household. There is a government subsidy to capital - per each unit of capital investment, firm receives zetaK units of subsidy. Subsidy is financed by lump sum taxes. The first best allocation is characterized by

E[M(alfa*A*K^{alfa-1} + (1-delta)] = 1, where M = (C/C(-1))^{-1},

while the CE allocation is is characterized by

E[M(alfa*A*K^{alfa-1} + (1-delta)] = 1 - zeta.

Basically, firm perceives capital less expensive than it is in reality and as a result overinvests in capital. As far as I understand, the highest welfare should be achieved in FB. But when I calculate E[V] in FB and CE, I get the opposite ranking
V = log(C) + beta*V(+1).

Can somebody explain why this is the case? or where is my mistake? I attache the two model setups. Btw, I have also tried second-order approximation.

Thank you,
Tetiana

[jpfeifer]

I would need to see the actual model-setup. There might still be something wrong here, but my guess is the following:
You are looking at steady state welfare. In the economy with the capital subsidy, you induce a higher steady state capital stock, which is associated with permanently higher consumption (and higher investment to maintain this capital stock). This must be welfare improving, because the household loves consumption.
Usually without distortions, you are at the Golden rule level of capital. Any change should bring the marginal product below/above the effective depreciation rate (time preference + depreciation). That does not seem to happen here. The reason seems to be that you consider the steady state equilibrium and do not look at transition paths that might optimally arise under policies when you alter the margin in the Euler equation.
In the extreme, consider a case where you use the subsidy to equalize the marginal product of capital to the depreciation rate (you effectively counter the time preference rate of agents with the tax). In that case, you increase the capital stock to its highest sustainable level and steady state welfare will be at its maximum. This is the highest steady state welfare, but not the best policy for this economy. Given the initial high steady state capital, keeping the subsidy would not be the optimal policy. Due to time preference, the agents would like to consume a big chunk of capital immediately at the expense of permanently lower consumption in the future. Welfare along this transition from the high steady state capital to the "First Best Economy" in the other mod-file would be higher than when permanently staying in the CE economy.