Impatience and interest rate

Question

I'm having difficulties solving the following problem in economics. I come from a mathematical background, and it's hard for me to get some of the terms:

Consider a two-period economy with a single good and two types of agents ($i=a,b$). They have the same utility function $$u(x_0,x_1)=\ln x_0+\beta \ln x_1 $$ where $x_0,x_1$ are consumption levels at $t=0$ and $t=1$. Agent $a$ has resources only at date $0$: $\omega^a=(\omega_0,0)$ while agent $b$ has resources only at date $1$: $\omega^b=(0,\omega_1)$. The agents can borrow and lend on a competitive market for loans, where $r$ denotes the interest rate.

1. Compute the equilibrium interest rate $\bar r$, the equilibrium consumption $\bar{x}^a$ and $\bar{x}^b$ of the two agents and the quantity they borrow or lend at equilibrium.
2. Given $u(\cdot,\cdot)$ the rate of impatience of an agent at $\bf{x}$ (any $\bf{x}$) is defined as $$r(\mathbf{x})=\frac{\partial u/\partial x_0-\partial u/ \partial{x_1}}{\partial u/\partial x_1} $$ Notice that this can be re-arranged; explain what $r(\bf{x})$ represents in terms of the compensation the agent requires at date $1$ to give up consumption today.
3. Compute the rate of impatience $r(\mathbf{x})$ using $u(\cdot,\cdot)$. Explain intuitively how $r(\bf{x})$ changes with the parameter $\beta$ and the allocations $x_0,x_1$.
4. What is the relation between $r^i(\bar{\mathbf{x}})$, $i=a,b$ (at equilibrium) and the equilibrium interest rate $\bar{r}$.
5. What happens to $\bar{r}$ when the parameters $\beta,\omega_0,\omega_1$ change? Using questions 3 and 4 give some intuitions for these comparative statics.

My attempt:

I could do question 3 because it's basically just maths:

We found the partial derivatives $$\partial u/ \partial x_0=\frac{1}{x_0}, $$ $$\partial u/\partial x_1=\frac{\beta}{x_1}.$$

From there it's easy to see that $$r(\mathbf{x})= \frac{x_1}{\beta x_0}-1. $$

The way $r$ changes with $\beta,x_0,x_1$ is seen by differentiating.

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If I would understand how to find the quantities in part 1, I think I could handle parts 4,5.

I would appreciate if anyone could shed some light on how to find the required terms from part 1, and also any other kind of help.

Thank you!

Answer 1

The first order condition from the maximization problem is

$$ \frac{1}{1 + r} \frac{1}{x_0} = \beta \frac{1}{x_1} $$

for both agents. (In a calculus class, this is the parallel gradients condition from Lagrange multipliers.) This is the agent trading off marginal utilities between today and tomorrow. For example, if you have two goods $A$ and $B$ in your basket at the store, you would substitute one for another until the marginal utilities are equal. Here the two goods are consumption today and tomorrow.

In this case, the marginal utility of tomorrow, $\frac{1}{x_1}$, gets discounted by $\beta$, which is a measure of the agent's time preference/degree of impatience. But in present value terms, tomorrow's consumption is cheaper than today by a factor of $1 + r$, $r$ being the interest rate.

Plug this into the agents budget constraints gives you demand functions

$$ x^a_0(w^a, r), x^a_1(w^a, r) \;\; \mbox{for agent a} $$

and

$$ x^b_0(w^b, r), x^b_1(w^b, r) \;\; \mbox{for agent b}. $$

Market clearing means the two markets, one for each period, clear:

$$ x^a_0(w^a, r) + x^b_0(w^b, r) = w_0, $$

$$ x^a_1(w^a, r) + x^b_1(w^b, r) = w_1. $$

Solving for $r$ should give $\frac{\beta w_0 - w_1}{w_1}$. Both equations give the same $r$---if all but one market clears so must the remaining one. This is called Walras's law in microeconomics.