A simple model of intellectual property rights
This section considers these issues in more detail by analysing a model of innovation in which spending on research and development by a firm results in that firm being able to lower its marginal cost of production. We assume, however, the new innovation, once produced, can be easily copied by other firms (in the real world this need not always be the case). We also assume that firms sell their goods on a competitive market, so allowing other firms to copy the innovation drives down prices and benefits consumers. However, if firms could copy the innovation as soon as it becomes public, then the innovating firm would be unable to capture the full private benefits of its invention.
A patent confers a temporary monopoly right on the innovating producer, allowing it to produce the good at the lower marginal cost for a particular period of time. The idea is to allow the firm to capture some economic profits temporarily (for the length of the patent), which cover the sunk cost of innovation and thereby inducing the firm to innovate in the first place. Then, after the term of the patent has expired, the firm's competitors to gain access to the new technology and drive down
prices, thus benefiting consumers. The optimal patent length balances these two opposing objectives.
Consider a competitive market for a single good. Consumer benefits of the good are equal to B, suppose that the current market price of the good is P, which is equal to marginal cost in the industry. Suppose that there is a sunk, upfront cost of K > 0, which is the cost of innovation. Innovation drives the marginal production cost to zero. Thus, in a competitive market in which all firms had access to the new technology, price would be driven to zero. At the new lower price, we assume that additional consumers enter the market. Thus, in addition to existing consumers enjoying the benefits of the good at a lower price, there are additional consumers who now consume the good but who previously did not.
Finally, we assume that benefits that are to be received in the future (and costs that are to be incurred in the future) are discounted at the common discount rate r > 0, with continuous compounding. Thus, the value today of a dollar to be received at time T in the future is e~rT.