Book excerpt: This dissertation consists of three essays in monetary economics. Although the topic of each chapter differs, the approach is shared: I extend a random matching model of money by augmenting the set of money holdings, and compute socially desirable allocations in the spirit of mechanism design analysis. The augmentation is not just technically improving the model, but making the model rich enough to think about the economic problem that each chapter delves into. I document some interesting properties of the desirable allocations, and highlight the differences generated by the extension.Chapter 1. "A beneficial role of government bonds"I study a random matching model of money to show that the existence of bonds can be beneficial to a society, compared to having only money. In the model, anonymous agents randomly meet in pairs to produce and consume, hence money becomes essential. I compare two identical economies except the availability of bonds, in the sense that people can use any available assets as payments. Following the mechanism design approach, I define implementable allocations and the optimum. Under the notion of the implementability, social planner can devise trading mechanisms that induce people to hold both assets without exogenously given advantages of money as means of payment. I find that having both bondsand money in the economy can improve social welfare over having only money. This role of bonds is associated with a beneficial effect of inflation produced by lump-sum transfers, and it is achieved differently from the previously documented mechanism.Chapter 2. "Optimal intervention in a random-matching model of money" (joint with Wataru Nozawa)Wallace [2014] conjectures that there generically exists an inflation-financed transfer scheme that improves welfare over no intervention in pure-currency economies. We investigate this conjecture in the Shi-Trejos-Wright model with different upper bounds on money holdings. The choice of an upper bound affects the results as some potentially beneficial transfer schemes cannot be studied under small upper bounds. Numerical optima are computed for different degrees of discounting rate and risk aversion. As the upper bound on money holdings increases, optima are more likely to have positive money creation (and inflation),and this result is in line with the conjecture.Chapter 3. "Optimal inflation in a model of inside money: A further result" (joint with Wataru Nozawa)We extend the Deviatov and Wallace [2014] model of inside money in which they find some examples where inflation is beneficial. Their model is restrictive in that it cannot address policies that provide interests on cash (Friedman rule). With a higher upper bound on money holdings than what they use, such policies can be engineered without inflation and resulting allocations are potentially better than what they find, in which case positive inflation is not a property of good allocation. We investigate this possibility and confirm their results in a more generalized setting for some parameters. At optima for the examples, interest on cash is not provided and positive inflation arises in a similar manner to their work. Welfareat optimum increases monotonically with respect to discount factor and public monitoring capacity of a society, but other variables change in a more complex way.