# Land Price Problem Solution ## (a) Implicit Discount Rate \( i \) for Price-Rent Ratio ### **Given** - The price of a house: \( P \) - The annual rent: \( R \) - Price-rent ratio: \( X = \frac{P}{R} \) - Expected growth rate of rent (and price): \( g \) - Find: The implicit discount rate \( i \) ### **Step 1: Value of House as a Perpetuity** If rents grow at rate \( g \), the house value is the present value of a growing perpetuity: \[ P = \frac{R}{i - g} \] where: - \( i \): discount rate - \( g \): expected growth rate ### **Step 2: Relate to Price-Rent Ratio** \[ X = \frac{P}{R} = \frac{1}{i - g} \] ### **Step 3: Solve for \( i \)** \[ i - g = \frac{1}{X} \] \[ i = \frac{1}{X} + g \] ### **Final Answer (a)** > **The implicit discount rate is:** > \[ > \boxed{i = \frac{1}{X} + g} > \] > where \( X \) is the price-rent ratio and \( g \) is the expected growth rate. --- ## (b) Three Possible Reasons for High Price-Rent Ratios 1. **Expectations of High Future Growth (High \( g \)):** - Buyers expect rents (and/or house prices) to grow rapidly in the future, increasing \( g \) in the formula and justifying a higher price today. 2. **Low Discount Rates (Low \( i \)):** - Low interest rates or low required returns make future rent streams more valuable, raising the price relative to rent. 3. **Non-Monetary Benefits of Ownership:** - Buyers value intangible benefits such as stability, pride of ownership, or tax advantages, making them willing to pay more than the rental value alone would justify. --- **Summary Table:** | Reason | Description | |-------------------------------|--------------------------------------------------------| | High expected growth (\( g \)) | Optimism about future rent/price increases | | Low discount rate (\( i \)) | Low interest rates or high investor demand | | Non-monetary benefits | Ownership perks, tax breaks, emotional value |

# Land Price Problem Solution ## (a) Implicit Discount Rate \( i \) for Price-Rent Ratio ### **Given** - Price of a house: \( P \) - Annual rent: \( R \) - Price-rent ratio: \( X = \frac{P}{R} \) - Expected growth rate of rent (and price): \( g \) ### **Step 1: Value of House as a Perpetuity** The value of a house can be expressed as the present value of a growing perpetuity: \[ P = \frac{R}{i - g} \] where: - \( i \): discount rate - \( g \): expected growth rate of rent ### **Step 2: Relate to Price-Rent Ratio** Rearranging the formula gives: \[ X = \frac{P}{R} = \frac{1}{i - g} \] ### **Step 3: Solve for \( i \)** From the equation, we can derive \( i \): \[ i - g = \frac{1}{X} \] \[ i = g + \frac{1}{X} \] ### **Final Answer (a)** > **The implicit discount rate is:** > \[ > \boxed{i = g + \frac{1}{X}} > \] > where \( X \) is the price-rent ratio and \( g \) is the expected growth rate. --- ## (b) Three Possible Reasons for High Price-Rent Ratios 1. **Expectations of High Future Growth (High \( g \)):** - Buyers anticipate significant increases in future rents or property values, leading to a higher price today. 2. **Low Discount Rates (Low \( i \)):** - Lower interest rates or a low required return on investments enhance the present value of future rent streams, increasing house prices relative to rent. 3. **Non-Monetary Benefits of Ownership:** - Homeownership may provide intangible advantages such as stability, emotional satisfaction, or tax benefits, justifying a higher price compared to rental value. --- **Summary Table:** | Reason | Description | |-------------------------------|--------------------------------------------------------| | High expected growth (\( g \)) | Optimism about future rent/price increases | | Low discount rate (\( i \)) | Favorable borrowing conditions or strong market demand | | Non-monetary benefits | Value derived from ownership, emotional factors | --- ### **Conclusion** Both interpretations of rent \( R \) (current vs. next year's rent) lead to different formulations of the implicit discount rate. The chosen formula should align with the context provided in the problem.