of L - 5. Assume that a risk-averse entrepreneur (i.e., the agent) wants to start a...

 

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of L - 5. Assume that a risk-averse entrepreneur (i.e., the agent) wants to start a business that requires an initial investment worth an amount 1 bar of gold. The entrepreneur has no money and thus must finance the cost of investment by borrowing it from a risk-neutral lender (i.e., the principal). The return on the business is random and equal to V 30 bars of gold with probability T₁ and V = 2 bars of gold with probability 1 Te, where e € {0,1} denotes the = effort level of the entrepreneur. In particular, let To π1 = 1/4 and 3/4 and suppose that exerting effort costs the entrepreneur 2. Suppose the entrepreneur receives utility from any money x received according to the function u(x) = x¹/³. Suppose the lender does not observe if the entrepreneur exerts effort or not. Finally, suppose the financial contract consists of repayments {(Z, Z)} to the lender, depending upon whether the business is successful or not. (Note that h(x) = u−¹(x) = x³.) - = (a) Write down the optimization problem of the lender where choice variables are repayments {(Z, Z)} to the lender such that he induces effort from the entrepreneur. Transform this problem to the one where choice variables are transfers {(t, t)} made to the entrepreneur. (b) Make necessary substitutions and show that the lender's opti- mization problem is a proper concave program. (c) Show that incentive compatibility and participation constraints of this transformed problem are binding. (d) Find the optimal repayments that solve the lender's problem. Does the lender find it profitable to lend money to the en- trepreneur? Explain. (e) Find the second-best cost of inducing effort and verify that the lender prefers not inducing effort rather than inducing it. of L - 5. Assume that a risk-averse entrepreneur (i.e., the agent) wants to start a business that requires an initial investment worth an amount 1 bar of gold. The entrepreneur has no money and thus must finance the cost of investment by borrowing it from a risk-neutral lender (i.e., the principal). The return on the business is random and equal to V 30 bars of gold with probability T₁ and V = 2 bars of gold with probability 1 Te, where e € {0,1} denotes the = effort level of the entrepreneur. In particular, let To π1 = 1/4 and 3/4 and suppose that exerting effort costs the entrepreneur 2. Suppose the entrepreneur receives utility from any money x received according to the function u(x) = x¹/³. Suppose the lender does not observe if the entrepreneur exerts effort or not. Finally, suppose the financial contract consists of repayments {(Z, Z)} to the lender, depending upon whether the business is successful or not. (Note that h(x) = u−¹(x) = x³.) - = (a) Write down the optimization problem of the lender where choice variables are repayments {(Z, Z)} to the lender such that he induces effort from the entrepreneur. Transform this problem to the one where choice variables are transfers {(t, t)} made to the entrepreneur. (b) Make necessary substitutions and show that the lender's opti- mization problem is a proper concave program. (c) Show that incentive compatibility and participation constraints of this transformed problem are binding. (d) Find the optimal repayments that solve the lender's problem. Does the lender find it profitable to lend money to the en- trepreneur? Explain. (e) Find the second-best cost of inducing effort and verify that the lender prefers not inducing effort rather than inducing it.

Answer rating: 100% (QA)

a Lenders Optimization Problem The lender wants to maximize his anticipated earnings considering the attempt desire of the entrepreneur Let Z be the reimbursement if the business is successful and Z b... View the full answer