The World Health Organization (WHO, 2016) reports that 3.2 million deaths per year are attributable to physical inactivity, making it the fourth leading risk factor for global mortality. Physical inactivity is also a key risk factor for noncommunicable diseases such as cardiovascular disease, cancer, and diabetes (WHO, 2018). Globally, 1 in 4 adults is not active enough and, therefore, foregoes a myriad of health benefits associated with Physical Activity (PA; WHO, 2018). In the United States, only about 1 in 5 (21%) adults meet the 2008 Physical Activity Guidelines set by the Centers for Disease Control and Prevention (CDC, 2018). The CDC currently recommends adults engage in 150 min of moderate-intensity aerobic activity per week (CDC, 2018). Translated to steps, the recommendation can be met by taking 3,000 steps in 30 min, 5 days per week (Marshall et al., 2009). Physical inactivity is also a major contributor to obesity (WHO, 2018). According to the WHO (2018), worldwide prevalence of obesity almost tripled since 1975. In the United States, the medical costs of obesity were estimated to be $147 billion, or 10% of all medical spending (Finkelstein, Trogdon, Cohen, & Dietz, 2009). To combat the many problems associated with physical inactivity, the CDC (2015), the WHO (2018), and the American Heart Association (2018) prescribe increased PA. Furthermore, increased PA contributes to a variety of other health benefits, including a decreased risk for cardiovascular disease, type 2 diabetes, some cancers, as well as improved mental health, and increased life expectancy (CDC, 2018).

We present three chapters on theoretical issues of banking. These deal with bank runs, risk sharing, lending and profitability. In the first chapter, we examine the agency problem in the bank-depositor relationship. Depositors are the principals and banks are the agents. Banks choose investment portfolios and are subject to moral hazard in that they have incentive to take on more risk than desirable to depositors because they are residual claimants. We study an incentive-compatible mechanism that prompts banks to follow a safe investment policy. This mechanism leaves the bank a profit margin in a similar manner to a CEO being paid a bonus by a company. In the second chapter, we extend Allen and Gale (1998) by adding a long-term riskless investment opportunity to the original portfolio of a short-term liquid asset and a long-term risky illiquid asset. Through portfolio diversification, we identify the risk-sharing deposit contract in a three-period model that maximizes the ex-ante expected utility of depositors. Unlike Allen and Gale, there are no information-based bank runs in equilibrium. In addition, our model can improve consumers' welfare over the Allen and Gale model. I also show that the bank will choose to liquidate the cheaper investments, in terms of the gain-loss ratios for the two types of existing long-term assets, when there is liquidity shortage in some cases. Such a policy reduces the liquidation cost and enables the bank to meet the outstanding liability to depositors without large liquidation losses. In the third chapter, we study the role of banks in providing loans to borrower firms. This paper extends the theory of designing optimal loan contracts (for profits) in the Bolton and Scharfstein (1996) model to a setting where asymmetry of information exists. Based on the verifiability of information structure, we analyze complete and incomplete contracts. Through this analysis, optimal, incentive-compatible loan contracts that maximize the expected profit of the bank are characterized. Our analysis suggests that a bank could be induced to liquidate a borrower's project under specific conditions. Furthermore, we identify implementable mechanisms for the renegotiation game given the bargaining power between a borrower and a bank.