Suppose now that the government gets smart and offers three plans, as follows:

Plan A: $15,000$ premium for full cover;

Plan B: $1,275$ premium for $17\%$ cover;

Plan C: $150$ premium for $5\%$ cover.

Show that this set of contracts satisfies the participation constraints $($all households choose to take out a plan$),$ incentive$-$compatibility constraints $($each household type takes out a different plan$)$ and is sustainable $($the government does not make a loss when offering them$).$