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A lump-sum profits tax is a fixed amount paid by the firm, irrespective of the quantity of output produced. Since this tax is a fixed cost, it does not affect the marginal cost or the marginal revenue of the firm. The profit-maximizing quantity of output is determined by the point at which marginal cost equals marginal revenue. Therefore, if a lump-sum profits tax does not affect the relationship between marginal cost and marginal revenue, the profit-maximizing quantity of output remains unchanged.

A proportional tax on profits is a tax rate applied to the firm's total profits. Since this tax rate is constant, the total tax paid by the firm increases proportionally as its profits increase. Similar to the lump-sum profits tax, a proportional tax on profits does not affect the relationship between the marginal cost and marginal revenue. This is because a proportional tax on profits is still based on the overall profitability of the firm, and not on the quantity of output. Therefore, like the lump-sum profits tax, a proportional tax on profits also does not affect the profit-maximizing quantity of output.

A tax assessed on each unit of output is a per-unit tax that the firm must pay when it produces each unit. This tax increases the cost of producing each additional unit. As this tax applies to every unit produced, this will increase the marginal cost of production. As a result, the firm will face a new, higher marginal cost curve. A higher marginal cost curve will intersect the marginal revenue curve at a different point. The new point of intersection will determine the profit-maximizing quantity of output for the firm under the new tax scenario. In general, when a tax is assessed on each unit of output, the firm's profit-maximizing quantity of output is likely to decrease. In conclusion, a lump-sum profits tax and a proportional tax on profits do not affect the profit-maximizing quantity of output for a firm. However, a tax assessed on each unit of output affects the marginal cost of production, which in turn influences the profit-maximizing output level, usually decreasing it.