No, there cannot be a positive level of output that a profit-maximising firm produces in a competitive market at which the market price is not equal to the marginal cost.

The fundamental condition for profit maximization is that profits are maximum at the level of output for which Marginal Revenue (MR) is equal to Marginal Cost (MC). This equality is necessary for the firm to be in equilibrium.

In a perfectly competitive market, firms are price-takers. For a price-taking firm, Marginal Revenue is equal to the Market Price. This relationship holds because the demand curve faced by the firm is perfectly elastic; the firm can sell any quantity at the prevailing market price, meaning the revenue from each additional unit sold is exactly the market price.

Therefore, substituting the market price for marginal revenue in the profit-maximization condition ($MR = MC$), we find that the firm’s profit-maximizing output is the level of output at which $P = MC$. The provided text confirms that if there is a positive level of output at which a firm’s profit is maximised, the first condition that must hold is that the market price equals the marginal cost ($p = SMC$ in the short run and $p = LRMC$ in the long run).

Consequently, at any positive profit-maximizing level of output, the market price must necessarily be equal to the marginal cost.