This is “Relationship between Prices and Wages”, section 2.7 from the book Policy and Theory of International Trade (v. 1.0).
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The Ricardian model assumes that the wine and cheese industries are both perfectly competitive. Among the assumptions of perfect competition is free entry and exit of firms in response to economic profit. If positive profits are being made in one industry, then because of perfect information, profit-seeking entrepreneurs will begin to open more firms in that industry. The entry of firms, however, raises industry supply, which forces down the product price and reduces profit for every other firm in the industry. Entry continues until economic profit is driven to zero. The same process occurs in reverse when profit is negative for firms in an industry. In this case, firms will close down one by one as they seek more profitable opportunities elsewhere. The reduction in the number of firms reduces industry supply, which raises the product’s market price and raises profit for all remaining firms in the industry. Exit continues until economic profit is raised to zero. This implies that if production occurs in an industry, be it in autarky or free trade, then economic profit must be zero.
Profit is defined as total revenue minus total cost. Let ΠC represent profit in the cheese industry. We can write this as
where PC is the price of cheese in dollars per pound, wC is the wage paid to workers in dollars per hour, PCQC is total industry revenue, and wCLC is total industry cost. By rearranging the zero-profit condition, we can write the wage as a function of everything else to get
Recall that the production function for cheese is . Plugging this in for QC above yields
If production occurs in the wine industry, then profit will be zero as well. By the same algebra we can get