Derivation of Econ. Research paper's eqn: int[t,infty][e^{-int[t,tau]r(s)ds}pi(tau)

[Carlos2007]

Hello Everyone!

I hope you can help me. I am trying to derive equation 6' from equation 6 in the picture that I attached, but I could not do it. I think the picture may be enough, but I can provide more information, of course.

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The decision to produce a new specialized input depends on a comparison of the discounted stream of net revenue and the cost P_A of the initial investment in a design. Because the market for designs is competitive, the price for designs will be bid up until it is equal to the present value of the net revenue that a monopolist can extract. At every date t, it must be true that

. . . . .\(\displaystyle \large{\displaystyle \int_t^{\infty}\, e^{-\int_t^{\tau}r(s)ds}\, \pi(\tau)\, d\tau\, =\, P_A(t)\qquad (6)}\)

If P_A is constant (as it will be in the equilibrium described below), this condition can be put in a more intuitive form. Differentiating with respect to time t yields

. . . . .\(\displaystyle \large{\displaystyle \pi(t)\, -\, r(t)\,\int_t^{\infty}\, e^{-\int_t^{\tau}r(s)ds}\, \pi(\tau)\, d\tau\, =\, 0}\)

Substituting in the expression for P_A from Equation (6) yields

. . . . .\(\displaystyle \large{\displaystyle \pi(t)\, =\, r(t)\, P_A\qquad \qquad \qquad \qquad (6')}\)

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Thank you in advance!

Kind regards,
Carlos