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Analyze credit market equilibrium under conditions of symmetric and asymmetric information, focusing on loan contracts, interest rates, and the behaviors of different borrower types. Use the provided scenarios involving safe and risky farmers, a monopolist lender, and an uninformed lender, to derive and graph expected incomes, profits, and determine optimal interest rates under various informational settings. Incorporate equations, expected utility calculations, and graphical analysis to answer questions about borrower incentives, lender profits, and interest rate decisions, culminating in a comprehensive understanding of asymmetric information impacts in lending markets.

Paper For Above instruction

Introduction

The functioning of credit markets depends heavily on the information available to lenders about borrowers. When lenders possess complete and symmetric information, they can tailor loans appropriately, minimizing adverse selection and moral hazard. Conversely, asymmetric information complicates the lending process, leading to issues such as adverse selection, moral hazard, and the potential for market failure. This paper examines the intricacies of credit market equilibrium under these conditions through two core scenarios: one with symmetric information and another with asymmetric information. The analysis focuses on how interest rates are determined, the incentives of different borrower types, and the resulting profits for lenders, encapsulated within graphical models and mathematical expressions.

Part 1: Credit Market Under Symmetric Information

The first scenario involves Ram, a monopolist lender, who knows precisely whether farmers are SAFE or RISKY. Both types seek a \$200 loan, but their income prospects differ. SAFE farmers have certain income at harvest, earning \$400 with 100% probability, and RISKY farmers have uncertain outcomes, earning \$600 with a 50% probability or \$0 otherwise. Since Ram has perfect information, he can offer different contracts tailored to each farmer type.

Expected income for SAFE farmers (denoted as Ì‘¦s) is straightforward:

  • Expected income: E(Ì‘¦s) = 400 (since they always earn \$400).

Expected income for RISKY farmers (Ì‘¦r) accounts for their probabilistic outcomes:

  • Expected income: E(Ì‘¦r) = 0.5 × 600 + 0.5 × 0 = \$300.

Ram’s profit functions depend on the interest rate i and the amount loaned (principal + interest). For a loan of \$200, the repayment if harvest is good (with certainty for SAFE farmers, probabilistically for RISKY farmers) is \$200(1 + i).

Expected profit from lending to SAFE farmers (Îœs) is derived as:

  • Expected profit: E(Îœs) = p × (200(1 + i) - 200) = p × 200i, where p is the probability that the farmer's harvest is good (which is 1 for SAFE farmers).

Similarly, for RISKY farmers (Îœr):

  • Expected profit: E(Îœr) = 0.5 × (200(1 + i) - 200) + 0.5 × (0 - 0) = 0.5 × 200i.

Graphically, plotting Ì‘¦s, Ì‘¦r, Îœs, and Îœr as functions of interest rate i over 0 to 3 provides visual insights into the relationships, as depicted in Figure 1.

Questions about maximum sustainable interest rates and profit maximization are answered by analyzing these functions. The highest interest rate SAFE farmers are willing to pay corresponds to the rate at which their expected income precisely covers their reservation utility (zero in this case), which is when the repayment equals their certain revenue. For SAFE farmers, this occurs at i = 1. RISKY farmers, with their probabilistic outcomes, are willing to pay interest up to the rate where their expected income minus repayment remains non-negative, also roughly i = 1.

Ram, as a monopolist, maximizes profit by setting interest rates where borrowers are willing to lend, leading to equilibrium interest rates that balance profitability and borrower acceptance. For SAFE borrowers, this occurs at i = 1, with expected profit 200 × 1 = \$200. For RISKY borrowers, the profit is half that, 0.5 × 200i, reaching a maximum at the same interest rate.

Part 2: Lending Under Asymmetric Information

The second scenario involves Ali, who faces asymmetric information identical to Ram's earlier setting but lacks perfect knowledge of farmer types. He can only offer a single interest rate to all farmers, with half being SAFE and half RISKY. Since Ali does not know individual types, he must choose an interest rate that makes borrowing attractive to both types, or risk adverse selection if the rate is too high.

The maximum interest rate Ali can charge without discouraging SAFE farmers is derived from their expected utility, which must be at least their reservation utility (zero here). Given their guaranteed income of \$400, and paying a \$50 premium with a potential repayment of \$200(1 + i), the interest rate ceiling is where the SAFE farmers just tolerate the loan—specifically, when repayment equals their certain income minus the premium:

  • imax = (400 - 50 - 200) / 200 = 1.75.

At interest rates above this threshold, SAFE farmers prefer not to borrow, leading to adverse selection against high interest rates. Conversely, RISKY farmers are willing to accept higher interest rates because their expected income under risky outcomes remains positive at rates up to the point where the expected repayment exceeds their potential earnings, causing their participation constraint to fail beyond certain thresholds.

Ali's expected profit is maximized when the interest rate is set at the highest point where both types are willing to borrow, which, due to participation constraints, occurs at the same interest rate as for SAFE farmers (i ≈ 1.75), with expected profit:

  • Expected profit: 0.5 × (200 × 1.75 - 50) + 0.5 × (200 × 1.75 - 50) = (200 × 1.75 - 50) = \$300.

Graphical analysis (Figure 2) illustrates expected profit curves, showing how profit declines once the interest rate exceeds the threshold where SAFE or RISKY farmers drop out.

Increasing the interest rate beyond this threshold results in the market failing to clear, with only RISKY farmers remaining, possibly leading to adverse selection and a subsequent decrease in expected profit.

Part 3: Risk Preferences and Insurance Decisions

Regarding risk preferences, Rachel, Phoebe, and Monica exhibit different attitudes towards risk, influencing their activity choices. Each has a distinct utility function:

  • Rachel: U(χ) = 0.05 × Ï‡²
  • Monica: U(χ) = 20χ - 0.05χ²
  • Phoebe: U(χ) = 0.5 × Ï‡

The expected values for activities (full-time farming, construction, part-time farming) are computed by considering the probabilities and payoffs described, then utility functions are applied to derive certainty equivalents (CE), risk premiums (RP), and expected utilities (EU). For instance, for full-time farming with an expected income of \$200 (good harvest) or \$40 (bad harvest), the CE is that income level corresponding to the same utility as the risky prospect, calculated via the inverse utility function.

Each individual’s activity choice results from comparing the CE and RP across options. Rachel, being less risk averse (utility quadratic with small coefficient), prefers risky activities with higher expected returns, while Monica’s linear utility indicates risk neutrality. Phoebe, with moderate risk aversion, balances potential income with risk, likely favoring safer or insured activities.

Insurance contracts offered by Joey—who charges a premium of \$50 and pays an indemnity of \$100 in case of a bad harvest—alter expected income and utility, affecting individual choices. Joey’s expected profit calculation involves the premiums collected minus indemnity paid, with the expected payout weighted by the probability of a bad harvest (50%).

When individuals choose activities considering Joey’s insurance, their expected utility enhances when risk is mitigated, influencing their activity decisions, as shown in the decision table.

Conclusion

The interplay between asymmetric information, borrower types, interest rates, and risk preferences significantly shapes credit market outcomes. Symmetric information allows lenders to optimize profits with tailored contracts, while asymmetric information necessitates pooling strategies at the expense of potential adverse selection. Risk preferences further influence individual activity choices and utility outcomes. Graphical models and equations elucidate the thresholds, incentives, and profits associated with each scenario, emphasizing the importance of information and risk attitudes in financial market functioning.

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