During an economic expansion, how do the ratios of Return on Equity and Return on Asset are affected? From my perspective, during an economic expansion, industrial production, employment, personal incomes and sales are increased excluding the inflation rate. To be more specific, companies buy new machines or build new factories to increase the production in this period and to expand the business, the company should finance the expansion either through debt or equity.

Therefore I thought Return on Equity (ROE) would be increased because companies borrow more debts, which leads the denominator of ROE formula, shareholders' equity, to be decreased and it results ROE to be increased. 

However if the company finance their expansion by the equity, not through debt financing, then denominator of ROE formula, shareholders' equity would be increased therefore the ROE will be decreased.

Then during an economic expansion, how does the ROE will going to be changed, decreased or increased?
If all else were equal, then using more equity financing would reduce return on equity. (This is assuming a normal situation where the return on assets is greater than the cost of interest. If the interest cost is greater than the return on assets, using debt for financing reduces profits. In this case, there are less profits, and a smaller equity base. The net effect on the return on equity depends upon the exact numbers.)

However, this tells us nothing about observed data during an expansion, it just tells us about the difference between two hypothetical scenarios (the scenarios being defined by the level of equity financing).

Aggregate profits are determined by many factors. The effects of those factors change every cycle, and may be unrelated to businesses' financing decisions.

For example, if the wage cost per unit of output (unit labour cost) is rising faster than selling prices, business profit margins will tend to fall (wages will represent a greater share of total income). Conversely, the profit share will generally rise if unit labour costs increase slower than selling prices. There is no law of economics that says which outcome will occur, and so we cannot be sure what is happening to profit margins.

Therefore, we have no way of answering the question of what happens to ROE during expansion, without pinning down what else is happening at the same time.