There's a great article in the latest Harvard Business Review (December, 2015) titled "Why Overvalued Equity is a Problem," by Roger L. Martin and Alison Kemper. The article discusses how an overpriced stock can create bad incentives for managers. When a stock is overvalued, CEO's feel pressure to meet unrealistic market expectations in terms of sales and earnings growth (otherwise they can lose their jobs). Id.
CEO's and CFO's in this situation often end up making bad decisions such as: (1) engaging in deceptive or fraudulent accounting; (2) wasting money on non-productive R&D or capital intensive projects to create the false impression of assured future growth; or (3) making overpriced acquisitions of small, trending start-ups. Id. These moves can prop up a company's stock price over the short term, or at least for the length of management's tenure.
One comment here -- if a company's stock is overvalued then it's a great time for managers to use that stock to acquire an undervalued or fairly valued company through a stock swap. See post titled The Rational Management Checklist. Additionally, if a start-up could disrupt your business through a new market or low end business model then an acquisition that looks overpriced on the surface may be reasonable from a long term perpective -- it's usually far cheaper to acquire a disruptive start-up early rather than late. See Concepts page and discussion of Clayton Christensen.
I think the greater problem the HBR article suggests is that CEO's of overvalued companies are incentivized to either falsely prop up growth expectations or pursue unsustainable growth strategies. So the CEO of a high PE company might keep making acquisitions to drive ever-higher sales -- which keeps investors excited and may help keep the stock price up -- even though the acquired companies may not have meaningful earnings or a profitable business model (or a business model that could be made profitable through the acquirer's actions). Or the CEO of a high PE company may keep adding infrastructure to drive ever-higher sales even though additional, resulting profits are negligible/unattractive (i.e., the return on capital invested in new infrastructure is less than what an investor could get if he took the same corporate cash and invested it elsewhere).
Conversely, the CEO of a low PE company is arguably incentivized to improve earnings in a less wasteful, more sustainable way. Many low PE companies don't have much in the way of cash or earnings, and their stocks are too undervalued to make stock swap acquisitions attractive. A CEO in this situation has to carefully weigh the costs and benefits of any acquisition or capital improvement/expansion project, which means the CEO must consider whether the benefits are (1) short term and illusory or (2) long term and sustainable. The HBR article suggests that on the averages the CEO's of undervalued companies should be motivated to allocate capital more wisely.