Lump-Sum Tax

A lump-sum tax is a fixed amount paid by a firm whenever it produces $q > 0$, independent of how much it produces. Because it does not enter marginal cost, it does not shift $MR = MC$ — the profit-maximising quantity and price are unchanged. It simply reduces the firm's profit by the tax amount. The same logic applies to a proportional tax on profit: scaling the objective $(1 - t)\Pi(Q)$ leaves the maximiser unchanged (the OPEC oil-tax example in Topic 2).

When to use

Reach for the lump-sum / profit-tax distinction whenever an exam asks "does the firm raise its price in response to this tax?". A lump-sum or profit tax → no. A Per-Unit Tax → yes, because it shifts $MC$ up by the tax rate.