Your friend has a great business idea: food delivery service using autonomous drones. However, she has a problem: she has no savings to pay for the required initial investment, and she is asking you to finance the venture. In this problem, your task is to analyze whether and how to finance your friend's business. For simplicity, make the following assumptions. Both you and your friend are risk neutral with no time preferences, which implies that you are both satisfied with a risk-free interest rate of 0. The firm requires an investment of = 0.4 today, but your friend is broke and you would have to provide the full financing. The firm will have a deterministic (i.e. non-random) income in the next period depending on how hard your friend works. Your friend can choose her level of effort which will also be the firm's income (in essence, your friend can choose how much income the firm generates; surely this is a bit crazy assumption but it will keep things tractable). However, to achieve a higher level of income for the firm, your friend will have to work increasingly harder. This will result in a personal cost for her equal to 0.5^2. Hence, your friend's utility from operating the firm will be equal to

() = () 1 2 ^2

Here () denotes the income to the entrepreneur (hence the subscript ) which is a function of her chosen effort level . Note that the firm's total income is equal to . If your friend chooses not to start the firm her utility will be equal to 0. As the financier, you will of course want your share of the firm's income. Assume that financial markets are competitive (your friend has many other wealthy friends who could also finance the firm) so that you will just break even. This implies that the financier's income () should, in equilibrium, match the initial investment ().

a) Assume that you and your friend enter into a loan contract. You lend your friend the amount needed to finance the firm and she will pay you back the necessary amount when the firm has generated its income. What is your friend's optimal level of effort given the loan contract? Given this level of effort, should you provide the financing?

b) Now assume that, instead of debt financing, you agree to buy equity in your friend's firm. You pay your friend enough for the equity so that she can start the firm. This gives you an 100% ownership stake in the firm's equity. In essence this means that you will get share of the firm's income and your friend will get 1 share. Should you provide the financing in this case? What are the implications of your results for financing of entrepreneurial firms?

c) This final question does not directly relate to corporate finance but highlights how the incentive problem can arise in another, unfortunately timely, setting. Assume now that your friend had enough money to make the initial investment without having to resort to external financing or that the required investment equals zero. In any case, the firm is already established and debt-free and your friend holds 100% of its equity. All she has to do is choose her effort level, and the firm's pre-tax income, . The government now imposes a corporate income tax with the tax rate equal to . This implies that the firm pays in taxes and the after-tax income is (1 ). Given this setup, derive the amount of tax paid as a function of the tax rate