We study models that display growth with financial deepening and increasing inequality along the way to a perpetual if distant steady state. In particular, Townsend (1978) and Greenwood and Jovanovic (GJ, 1990) describe a model with financial intermediaries, essentially a complete markets model but with transaction costs. These consist of a one-time entry fee plus a variable cost. Here, we use the dynamic GJ model as a benchmark, and characterize the transitional dynamics, both analytically and numerically, for a wider class of utility functions. This necessitates proofs of the existence of an optimal program and its equivalence with the value function approach. We present new proofs for stochastic dynamic program- ming for the case of unbounded return functions and perpetual growth. Also, a seemingly non-convex technology of participation is shown to be convexified by the optimal choice of portfolio shares among risky and safe assets. These analytical results also enable us to study transitional dynamics further with numerical methods. In particular, we calibrate the model and report quantitative predictions for Thai- land 1976-1996, an emerging economy in a phase of economic expansion with uneven financial deepening and increasing inequality. To examine the explanatory power of the model, we carefully construct con- ditional confidence intervals. We find a discrepancy between the model prediction and the actual data. We suspect that barriers to financial deepening cause this discrepancy, and we evaluate quantitatively the distribution of the welfare gains from their elimination. JEL Classification Numbers: G21, G28, 016, 041