Answer

Solution

1. 1

   Note on sign. The equation as printed has a positive sign on the right side, which gives a non-standard rule. The intended Keynes-Ramsey condition (from Blanchard-Fischer / Romer) is

2. 2
   $$\frac{d}{dt}\bigl[e^{-\rho t}U'(C)\bigr] = -e^{-\rho t}U'(C)\, r,$$
3. 3

   i.e.\ the present-value shadow price of consumption falls at rate $r$. Expanding the LHS:

4. 4
   $$-\rho e^{-\rho t}U'(C) + e^{-\rho t} U''(C)\dot C = -e^{-\rho t} U'(C) r,$$
5. 5
   $$U''(C)\dot C = U'(C)(\rho - r),$$
6. 6
   $$\dot C = \frac{U'(C)}{U''(C)}(\rho - r).$$
7. 7

   Since $U$ is strictly concave, $U''<0$, so $U'/U'' < 0$. Hence $\dot C > 0 \iff \rho - r < 0 \iff r > \rho$.

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