We contribute to the literature on optimal growth in two-sector models by solving a Ram- sey problem with a concave utility function. The unique possible steady-state is independent of initial conditions and of the instantaneous utility function, but not of the discount rate, and is characterized by a wage-rental ratio depending solely on the technology of the capital sector. For an initially low-capital economy, we show that the wage-rental ratio increasingly converges to its balanced value during transition. If the consumption sector is relatively capital-intensive, the relative price of capital increases during transition. If the investment sector is relatively more capital-intensive, it decreases. We also prove that a negative shock on the subjective rate of impatience, that makes the social planner more patient, leads to an immediate positive jump in asset prices.