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Page 127-128 of Romer's Advanced Macroeconomics states that

Equation (3.39) tells us that the profits at t are $\left[\frac{(1-\phi)}{\phi}\right]\left[\frac{(\bar{L}-L_{A})w(t)}{A(t)}\right]$. These profits grow at the growth rate of $w$, $\left[\frac{(1-\phi)}{\phi}\right]BL_{A}$, minus the growth rate of $A$, $BL_{A}$. They are discounted at rate $r$, $\rho + \left[\frac{(1-\phi)}{\phi}\right]BL_{A}$. The present value of the profits earned from the discovery of a new idea at time $t$ is therefore $$R(t)=\frac{\frac{(1-\phi)}{\phi} \frac{\bar{L}-L_{A}}{A(t)}w(t)}{\rho + BL_{A}}.$$

Note that since $L_{A}$ is constant, the real interest rate is constant.

However, it is unclear to me how the present value of profits was computed. I understand how the formula for the profits $\pi(t)$, interest rate $r(t)$, and the growth rates of wages $w(t)$ and ideas $A(t)$ are derived, but I don't know how they come together to form the equation above.

This is what I've been able to deduce so far. Since profit is given by $\pi(t)=\left[\frac{(1-\phi)}{\phi}\right]\left[\frac{(\bar{L}-L_{A})w(t)}{A(t)}\right]$, the growth rate of profits can be expressed as $\frac{\dot{\pi}(t)}{\pi(t)} = \frac{\dot{w}(t)}{w(t)} - \frac{\dot{A}(t)}{A(t)}$ and therefore $\frac{\dot{\pi}(t)}{\pi(t)} = \left[\frac{(1-\phi)}{\phi}\right]BL_{A}-BL_{A}$. Taking the difference between the interest rate $r(t) = \rho + \left[\frac{(1-\phi)}{\phi}\right]BL_{A}$ and the growth rate of profits yields \begin{equation} \begin{split} r(t) - \frac{\dot{\pi}(t)}{\pi(t)} & = \rho + \left[\frac{(1-\phi)}{\phi}\right]BL_{A} - \left(\left[\frac{(1-\phi)}{\phi}\right]BL_{A}-BL_{A} \right) \\ & = \rho + BL_{A}, \end{split} \end{equation} and so it follows that the equation for present value of profits can be re-expressed as $$ R(t) = \frac{\pi(t)}{r(t) - \frac{\dot{\pi}(t)}{\pi(t)}}, $$ but this equation doesn't make it clear for me in terms of understanding how present value of profits was determined. Why would we need to deduct the growth rate of profits from the interest rate and then divide the profits by this quantity to get the present value?

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The growth rate of $w$ is $\frac{1 - \phi}{\phi}BL_A$. So: $$ w(s) = w(t) e^{\left(\frac{1 - \phi}{\phi}BL_A\right)(s-t)} $$ Similar: $$ A(s) = A(t) e^{BL_A (s - t)}, $$ and the discount rate from $t$ to $s$ is $e^{-\left(\rho + \frac{1 - \phi}{\phi}BL_A\right)(s-t)}$.

Discounted profits are therefore: $$ R(t) = \int_t^\infty \frac{1 - \phi}{\phi} (\bar L - L_A) \frac{w(s)}{A(s)} e^{-\left(\rho + \frac{1 - \phi}{\phi}BL_A\right)(s-t)} ds. $$

So: $$ \begin{align*} R(t) &= \int_t^\infty \frac{1 - \phi}{\phi} (\bar L - L_A) \frac{w(t) e^{\left(\frac{1 - \phi}{\phi}BL_A\right)(s-t)}}{A(t) e^{BL_A (s - t)}} e^{-\left(\rho + \frac{1 - \phi}{\phi}BL_A\right)(s-t)} ds,\\ &= \frac{1 - \phi}{\phi} (\bar L - L_A)\frac{w(t)}{A(t)} \int_t^\infty e^{-(\rho + BL_A)(s-t)} \end{align*} $$ The integral evaluates to: $$ \frac{-1}{\rho + BL_A} \left[e^{-(\rho + BL_A)(s-t)}\right]_{s = t}^{s = \infty} = \frac{1}{\rho + BL_A}. $$

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  • $\begingroup$ This makes more sense than where I was initially headed. My intuition was inaccurate and had led me to a confusing conundrum, which is resolved here. Again, I am very grateful for your answer. Really appreciate it. Thanks so much! $\endgroup$
    – Tan2525
    Feb 27 at 10:44

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