We design a general set-up for the study of a generic economy whose development process is entirely driven by the spatio-temporal dynamics of capital accumulation. It allows us to take into account spatial heterogeneities in technological level and population distribution. The expression of the optimal dynamics of the system allows to simulate the behavior of the variables and, in particular, their optimal discounted long-run spatial distribution. The numerical results allow to study the two main effects at work when the space distributions of technology and population are heterogeneous: the core-periphery effect and the population effect.