In the absence of the new invention, total welfare is the present value of the sum over time of the difference between consumer benefits and producer costs:
On the other hand, if there is an invention which is granted a patent of length T welfare is:
where AB are the benefits accruing to new consumers who consume the good at the new lower price.
The first term in (7.10) is the cost of the new invention. The second term is the present value of the surplus that consumers continue to enjoy during the life of the patent. The third term is the present value of the firm's profit as a result of being able to sell the good at (just below) the current market price P but produce at a marginal cost of zero during the life of the patent. The final term is the present value of consumer benefits from being able to consume the good at a price of zero once the patent has expired.
Equation (7.10) can be simplified to:
New inventions are not always welfare-improving, even if they make consumers better off - innovation only improves welfare if its benefits exceed its costs. Therefore, welfare is higher from the invention only if:
Equation (7.12) states that for a patent of length T it is efficient for the invention to take place if the present value of the sunk costs is less than the present value of the reduction in marginal production costs, plus the present value of additional consumption benefits. The sum on the right-hand side is therefore a measure of the benefits of innovation, whereas the term on the left-hand side is the cost of innovation. For the remainder of this section, we assume that there is a positive value T such that the condition in (7.12) holds.