We introduce the Human Development Index (HDI) in a growth model à la Lucas (1988), where human capital has an additional positive effect on social welfare through the quality of individual health and education. In a simple economy with a Cobb-Douglas technology and logarithmic preferences, we provide the explicit trajectories for human capital, consumption and the HDI, which correspond to the Balanced Growth Path (BGP). Using a two-step maximization strategy, we compute the optimal initial value of the control variable, in this case, the initial optimal labor supply. In other words, we prove the optimality of the BGP. We high light a HDI crossing property: the propensity to consume has a positive effect on the HDI in the short run, but negative in the long run. Finally, both the growth rate and overall welfare are proven to decrease in the propensity to consume.