# How to Show That, on a Balanced Growth Path, Two Variables Grow at the Same Rate

If we are given that a variable is on a balanced growth path (for sake of argument we shall assume consumption), how do we show that another variable related to consumption (like capital or wealth) should grow at the same constant rate as consumption? I tried giving the verbal explanation below, but I was marked off for not being rigorous enough.

My verbal explanation was the following (in the problem there were only two variables, consumption and wealth and we were given that consumption was on a balanced growth path):

If consumption grows at a constant rate, wealth must also grow at the same constant rate. If the growth rate for wealth is greater than that of consumption, we will have positive wealth during the last time period, so we could not have been maximizing utility (because the agent gets no utility from holding wealth). If the growth rate for wealth is less than that of consumption, then a constant growth rate of consumption would not be sustainable over an infinite time horizon because the agent would eventually have less wealth than is necessary for the level of consumption prescribed by optimality.

My professor said that I could use FOC's to show that they must grow at the same rate, but I'm not sure how I would do so. Can anyone help me understand?

$\textbf{Edit:}$ The specific model for which I used this argument is the one laid out in the following question: Solution Method for Infinite-Horizon Maximization Problem. We end up getting steady state consumption (which is a form of a balanced growth path with the growth rate being 1). He said it was correct in this model that steady state consumption implied steady state wealth, but that my explanation was not rigorous enough.

• This is not true in all models. The production function in your model is probably a homogenous function of degree one. Otherwise the statement is not necessarily true. So perhaps use that also? And please specify your model in the question. Jul 20 '16 at 18:11
• @denesp I have added the model for which I used this argument. I did not think to apply homogeneity. However, my professor made it pretty clear that he was looking for first order conditions somehow proving that the growth rates are the same. He also gave the same explanation that you did saying that my argument would not always be correct (and that I just got lucky that it was on this question). That is why I am trying to learn the technique to show it rigorously Jul 20 '16 at 18:20
• Well, what's BGP? according to wikipedia en.wikipedia.org/wiki/Balanced-growth_equilibrium It means that all variables must grow at a constant rate... Aug 2 '16 at 20:22
• @Anoldmaninthesea. That is the definition we use, that all variables grow at a constant rate, however they need not grow at the same constant rate. The growth rates of different variables can be, and often are, different Aug 2 '16 at 20:25
• @DornerA you're right. my bad. ;) Aug 2 '16 at 20:30