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Let's analyze the given problem step-by-step.

### Problem Breakdown

1. **Utility Function**:
   The utility function for both consumers is given by:
   \[
   u(c_1, c_2) = \log(c_1) + \log(c_2)
   \]
   where \(c_1\) is the current-period consumption and \(c_2\) is the future-period consumption.

2. **Interest Rates**:
   - Deposit interest rate \(r_d = 0.10\).
   - Loan interest rate \(r_l = 0.20\).

3. **Scenario (a)**: Consumer A
   - Current-period income: \(y_1 = 200\).
   - Future-period income: \(y_2 = 0\).

4. **Scenario (b)**: Consumer B
   - Current-period income: \(y_1 = 0\).
   - Future-period income: \(y_2 = 240\).

5. **Scenario (c)**: Interest rate changes:
   - New loan interest rate: \(r_l' < 0.20\) while \(r_d = 0.10\) remains the same.

### Solution Steps

#### (a) Optimal Consumption for Consumer A

- **Budget Constraint for Consumer A**:
  Consumer A has a current income of \(y_1 = 200\) and no future income. If Consumer A saves \(s\) in the current period, the consumption in each period is:
  \[
  c_1 = y_1 - s, \quad c_2 = (1 + r_d) s.
  \]

- **Maximizing Utility**:
  Substitute the expressions for \(c_1\) and \(c_2\) into the utility function:
  \[
  u(c_1, c_2) = \log(y_1 - s) + \log((1 + r_d) s).
  \]

  The optimal \(s\) can be found by taking the derivative of \(u(c_1, c_2)\) with respect to \(s\), setting it to zero, and solving for \(s\).

#### (b) Optimal Consumption for Consumer B

- **Budget Constraint for Consumer B**:
  Consumer B has no current income (\(y_1 = 0\)) and future income \(y_2 = 240\). If Consumer B borrows \(b\) in the current period, the consumption in each period is:
  \[
  c_1 = b, \quad c_2 = y_2 - (1 + r_l) b.
  \]

- **Maximizing Utility**:
  Substitute the expressions for \(c_1\) and \(c_2\) into the utility function:
  \[
  u(c_1, c_2) = \log(b) + \log(y_2 - (1 + r_l) b).
  \]

  The optimal \(b\) can be found by taking the derivative of \(u(c_1, c_2)\) with respect to \(b\), setting it to zero, and solving for \(b\).

#### (c) Effect of Interest Rate Change

- **New Loan Interest Rate \(r_l' < 0.20\)**:
  Analyze the effect of a lower loan interest rate on both consumers. Specifically, consider how a lower \(r_l'\) affects Consumer B’s borrowing decision and overall utility, and compare it to Consumer A's situation who benefits from the fixed deposit rate \(r_d = 0.10\).

### Calculations

Let me perform the necessary calculations to find the optimal savings \(s\) for Consumer A and the optimal borrowing \(b\) for Consumer B.

#### (a) Calculating Optimal Savings \(s\) for Consumer A

1. The utility function for Consumer A is:
   \[
   u(s) = \log(200 - s) + \log(1.1s).
   \]
   
2. Take the derivative with respect to \(s\) and set it to zero:
   \[
   \frac{d}{ds} \left( \log(200 - s) + \log(1.1s) \right) = 0.
   \]

#### (b) Calculating Optimal Borrowing \(b\) for Consumer B

1. The utility function for Consumer B is:
   \[
   u(b) = \log(b) + \log(240 - 1.2b).
   \]

2. Take the derivative with respect to \(b\) and set it to zero:
   \[
   \frac{d}{db} \left( \log(b) + \log(240 -

Consider the two-period model with consumption and saving where Consumer A has current-period income y1 = 200 and future-period income y2 = 0. Consumer B has current-period income y1 = 0 and future-period income y2 = 240. Consumers receive interest on deposits at rd = 0.10 and pay on loans at rl = 0.20.

Explore how to solve a two-period consumption-saving model with different interest rates for borrowing and saving. Includes step-by-step analysis of Consumer A and B's optimal consumption decisions and the effect of changing loan interest rates.

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