2 Capital as a Signaling Device In the economy there are many different entrepreneurs each with an idea of a project they wish to pursue. To make things easy we will assume that there are only two types of projects, good and bad with each being equally likely. Good projects generate a return of 1 with certainty. Bad projects return 1 with probability .3 and 0 with 7. Entrepreneurs know the quality of their project but they need to borrow money from lenders to implement them. Specifically entrepreneurs need to borrow an amount equal to 0.8. Lenders do not know the quality of a project upfront but know the percentage of good versus bad projects and they know the probability that a bad project pays off. We will assume that lenders could take their potential investment and invest it elsewhere at a return of 10% if they decline to invest in an entrepreneur. So the reservation utility of the lender is equal to .8*(1+.1) = 0.88 where 10% is the default return on capital investment in the economy. If they expect to receive an amount less than this in expectation they would clearly not invest in the entrepreneur's project and would instead invest in their outside option. An important element to this arrangement is that an entrepreneur only pays back the lender if the project is successful. If successful, the entrepreneur must pay some rate of return, r, on top of the original principal so the total they pay back to the lender is . 8*(1+r). If the project fails, then the entrepreneur pays back nothing to the lender and the lender loses his entire investment of 0.8. In this model r is the price of borrowing money a. Let's first examine a simpler case in which both entrepreneurs and lenders can identify the quality of a project. Would it be possible for entrepreneurs with good projects to get lenders to agree to accept a different rate of return, r, than those with bad projects? Would it be possible for either or both types of projects to be funded? Explain and derive the interest rates that support 1 your answers.(Hint: If a type of project will be funded, find the range of interest rates that could be agreed to. If a type of project will not be funded, show why.) b. Now go back to the original version and assume that lenders can not identify the type of project but the entrepreneurs know the type. Would it be possible for entrepreneurs with good projects to get lenders to agree to accept a different rate of return, r, than those with bad projects? Would it be possible for either or both types of projects to be funded? Explain and derive the interest rates that support your answers. c. So far we have assumed that all of the capital required comes from the lenders. What if the entrepreneur has some stock of capital which they could invest themselves, k, meaning that they then only borrow 0.8 k from the lender. If entrepreneurs with different types of projects can invest at different rates from each other, can this help solve any problems in the market you found in part b? Explain the idea intuitively. Then propose a rigorous framework for an equilibrium of this game that would support your story. (Hint: for any amount the entrepreneur invests in his own project, k, you have to account for the opportunity cost of not investing that capital elsewhere which is k(1+.1) as the entrepreneur could always loan out that money himself. So you still have to think about the amount he borrows from himself costing k(1+.1). ) 2 Capital as a Signaling Device In the economy there are many different entrepreneurs each with an idea of a project they wish to pursue. To make things easy we will assume that there are only two types of projects, good and bad with each being equally likely. Good projects generate a return of 1 with certainty. Bad projects return 1 with probability .3 and 0 with 7. Entrepreneurs know the quality of their project but they need to borrow money from lenders to implement them. Specifically entrepreneurs need to borrow an amount equal to 0.8. Lenders do not know the quality of a project upfront but know the percentage of good versus bad projects and they know the probability that a bad project pays off. We will assume that lenders could take their potential investment and invest it elsewhere at a return of 10% if they decline to invest in an entrepreneur. So the reservation utility of the lender is equal to .8*(1+.1) = 0.88 where 10% is the default return on capital investment in the economy. If they expect to receive an amount less than this in expectation they would clearly not invest in the entrepreneur's project and would instead invest in their outside option. An important element to this arrangement is that an entrepreneur only pays back the lender if the project is successful. If successful, the entrepreneur must pay some rate of return, r, on top of the original principal so the total they pay back to the lender is . 8*(1+r). If the project fails, then the entrepreneur pays back nothing to the lender and the lender loses his entire investment of 0.8. In this model r is the price of borrowing money a. Let's first examine a simpler case in which both entrepreneurs and lenders can identify the quality of a project. Would it be possible for entrepreneurs with good projects to get lenders to agree to accept a different rate of return, r, than those with bad projects? Would it be possible for either or both types of projects to be funded? Explain and derive the interest rates that support 1 your answers.(Hint: If a type of project will be funded, find the range of interest rates that could be agreed to. If a type of project will not be funded, show why.) b. Now go back to the original version and assume that lenders can not identify the type of project but the entrepreneurs know the type. Would it be possible for entrepreneurs with good projects to get lenders to agree to accept a different rate of return, r, than those with bad projects? Would it be possible for either or both types of projects to be funded? Explain and derive the interest rates that support your answers. c. So far we have assumed that all of the capital required comes from the lenders. What if the entrepreneur has some stock of capital which they could invest themselves, k, meaning that they then only borrow 0.8 k from the lender. If entrepreneurs with different types of projects can invest at different rates from each other, can this help solve any problems in the market you found in part b? Explain the idea intuitively. Then propose a rigorous framework for an equilibrium of this game that would support your story. (Hint: for any amount the entrepreneur invests in his own project, k, you have to account for the opportunity cost of not investing that capital elsewhere which is k(1+.1) as the entrepreneur could always loan out that money himself. So you still have to think about the amount he borrows from himself costing k(1+.1). )