Time consistency and the soft budget constraint:- 

A firm is run by a risk-neutral entrepreneur with wealth A, and has a fixed-size project with investment cost I. The project, if undertaken at date 0, will deliver a verifiable income, y ∈ {yL, yH} in the case of success and 0 in the case of failure, at date 2, provided that one worker is employed in the firm. The project yields nothing if it is interrupted (the worker is laid off). y = yH with probability ρ and y = yL

(One will assume that B is small enough that it is worth inducing the entrepreneur to behave in the case of continuation.) The (risk-neutral) worker is paid w in the case of continuation and 0 otherwise. He obtains unemployment benefit paid by the

state wu

(i) Write the firm's NPV depending on whether the firm continues (x = 1) or stops (x = 0) when productivity is low (y = yL).

Show that x∗ = 1. Assuming a perfectly functioning capital market at date 1, what is the amount of liquidity that is needed to complement capital market refinancing?

(ii) Introduce a government that can at date 1 bring a subsidy s  0 to the firm (it is a pure subsidy: the government takes no ownership stake in exchange). The shadow cost of public funds is λ, and so the cost of subsidy s for the taxpayers is (1 + λ)s. The government maximizes total welfare (entrepreneur, investors, worker, taxpayers). Assuming that

and that the government selects its subsidy at date 1 (having observed the realization of y), what is the liquidity L chosen by entrepreneur and investors at date 0? How would the government (contingent) choice of s be affected if the government could commit to s at date 0, before the investors and the entrepreneur write their contract?