Solution:

Adam's utility maximization problem is solved using the following steps:

a) Write Adam’s intertemporal budget constraint:

Let $c_1$ and $c_2$ represent consumption in periods 1 and 2, respectively. Adam’s intertemporal budget constraint equates his lifetime income and lifetime consumption, adjusted for the interest rate $r$. His income endowment in periods 1 and 2 is $m_1 = 200$ and $m_2 = 100$.

The intertemporal budget constraint is: $c_1 + \frac{c_2}{1 + r} = m_1 + \frac{m_2}{1 + r}$

Simplifying: $c_1 + \frac{c_2}{1 + r} = 200 + \frac{100}{1 + r}$

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b) Solve for Adam’s optimal consumption bundle as a function of $r$:

Adam’s utility function is: $u(c_1, c_2) = c_1 \cdot c_2$

To maximize utility, we use the method of Lagrange multipliers. Let $\lambda$ be the multiplier: $\mathcal{L}(c_1, c_2, \lambda) = c_1 c_2 + \lambda \left( m_1 + \frac{m_2}{1 + r} - c_1 - \frac{c_2}{1 + r} \right)$

The first-order conditions are:

1. $\frac{\partial \mathcal{L}}{\partial c_1} = c_2 - \lambda = 0$ $\implies$ $\lambda = c_2$
2. $\frac{\partial \mathcal{L}}{\partial c_2} = c_1 - \lambda \frac{1}{1 + r} = 0$ $\implies$ $\lambda = (1 + r)c_1$
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = m_1 + \frac{m_2}{1 + r} - c_1 - \frac{c_2}{1 + r} = 0$

Equating $\lambda$ from the first two equations: $c_2 = (1 + r)c_1$

Substitute $c_2 = (1 + r)c_1$ into the budget constraint: $c_1 + \frac{(1 + r)c_1}{1 + r} = 200 + \frac{100}{1 + r}$

Simplify: $c_1 + c_1 = 200 + \frac{100}{1 + r}$ $2c_1 = 200 + \frac{100}{1 + r}$ $c_1 = 100 + \frac{50}{1 + r}$

Using $c_2 = (1 + r)c_1$: $c_2 = (1 + r) \left( 100 + \frac{50}{1 + r} \right)$ $c_2 = 100(1 + r) + 50$ $c_2 = 100 + 100r + 50 = 150 + 100r$

Thus, the optimal consumption bundle is: $c_1 = 100 + \frac{50}{1 + r}, \quad c_2 = 150 + 100r$

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c) Find Adam’s optimal bundle when $r = 0.25$:

Substitute $r = 0.25$ into the expressions for $c_1$ and $c_2$: $c_1 = 100 + \frac{50}{1.25} = 100 + 40 = 140$ $c_2 = 150 + 100(0.25) = 150 + 25 = 175$

Adam’s optimal bundle is: $c_1 = 140, \quad c_2 = 175$

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d) Determine if Adam could be a borrower in period 1 for $r < 0.25$:

Adam is a borrower in period 1 if $c_1 > m_1 = 200$. From the optimal consumption $c_1 = 100 + \frac{50}{1 + r}$, we solve: $100 + \frac{50}{1 + r} > 200$ $\frac{50}{1 + r} > 100$ $\frac{1}{1 + r} > 2$

This inequality is not possible for any positive $r$ because $\frac{1}{1 + r} < 1$ for all $r > 0$. Thus, Adam cannot be a borrower in period 1, regardless of how low $r$ becomes.

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Summary of Results:

1. Intertemporal Budget Constraint: $c_1 + \frac{c_2}{1 + r} = 200 + \frac{100}{1 + r}$
2. Optimal Consumption: $c_1 = 100 + \frac{50}{1 + r}, \quad c_2 = 150 + 100r$
3. At $r = 0.25$: $c_1 = 140, \, c_2 = 175$
4. Borrowing at $r < 0.25$: Not possible.

Would you like further clarification on any step or additional details?

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Related Questions:

1. How would the consumption bundle change if $r = 0$?
2. What happens to Adam’s consumption in period 2 as $r \to \infty$?
3. Can Adam ever consume only in period 1?
4. How would inflation affect Adam’s budget constraint?
5. What is the marginal utility of income in periods 1 and 2 for Adam?

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Tip:

When solving intertemporal optimization problems, always confirm the nature of $r$ (positive or negative) to understand how borrowing/saving choices are affected.