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This question is driving me nuts.

Suppose, an individual lives for two periods. In each period she consumes only one good,which is rice. In period 2, she can costlessly produce 1 unit of rice, but inperiod 1 she produces nothing. However, in period 1 she can borrow rice at an interest rate $r$ > 0. That is, if she borrows $z$ units of rice in period 1, then in Period 2, she must return $z(1 + r)$ units of rice. Let $x_1$ and $x_2$ denote her consumption of rice in period 1 and period 2, respectively; $x_1$, $x_2$ ≥ 0. Her utility function is given by $$U(x_1, x_2) = x_1 + βx_2$$, where β is the discount factor, 0 < β < 1. Note that there are only two sources through which rice can be available; own production and borrowing.

Now suppose that there are N agents in the above two period economy. The agents are identical (in terms of production and utility function) except that they have different discount factors. Suppose that β follows uniform distribution in the interval [ 1/2, 1]. Assuming r ≤ 1, what would be the demand function for rice in period 1?

My approach is integrating $\beta$ from 1/2 to $\frac{1}{1+r}$, and my answer comes out to be $\frac {N(1-r)} {1+r}$, but I dont think that's right.

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  • $\begingroup$ I maximized the utility function, given the constraints on $\beta$. The MRS of U is 1/ $\beta$. The slope of budget constraint is -(1+r). The consumer would consume $c_1$ only if the slope of the indifference curve is greater than the budget slope i.e 1/ $\beta$ > 1+r, which implies beta < $\frac{1}{1+r}$. Since beta is uniformaly distributed from 1/2 to 1, the height of the function is 2. We integrate 2 from 1/2 to 1/(1+r). $\endgroup$ – dexter Jun 3 '15 at 10:35
  • $\begingroup$ The answer however, is $\frac {N* (1-r)} {(1+r)^2}$ $\endgroup$ – dexter Jun 3 '15 at 10:36
  • $\begingroup$ I don't get two things. First.: where do you use that she produces exactly 1 unit of rice in period 2? Second: You wrote "Since beta is uniformaly distributed from 1/2 to 1, the height of the function is 2.". What function? You are right that the MRS takes value 2 if $\beta = \frac{1}{2}$, but this is not constant. $\endgroup$ – Giskard Jun 3 '15 at 10:55
  • $\begingroup$ 1. I think that implies that the total production in both the time periods is 1 unit, i.e the income would be 1* price in second period. I don't really know how to use that though. 2. We're given that the $\beta$ follows a uniform distribution, so the height of the beta distribution has to be constant. Since x lies between 1/2 and 1, and integration of the function has to be 1, the height (i.e the reflection of the domain) has to be 2. $\endgroup$ – dexter Jun 3 '15 at 11:24
  • $\begingroup$ Okay, here is my suggestion. Determine demand for period 1 consumption as a function of $r$ for $\beta = 0.5$. This will help you organize your thoughts. Then do the same for $\beta$ as a parameter, so demand will be $D(r, \beta)$. Finally, to get aggregate demand take $$ \int_{\frac{1}{2}}^{\frac{1}{1+r}} D(r, \beta) \ d \beta. $$ $\endgroup$ – Giskard Jun 3 '15 at 11:35
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Due to the linearity of the utility function the utility maximization problem for individual $i$ has either one of the two corner solutions, or it is indeterminate.

The individual $i$ has to solve the following problem

$$\max_{x_{1i},x_{2i}}[ x_{1i} + \beta_i x_{2i}]$$

$$s.t. x_{2i} = 1-(1+r)x_{1i}, \;\;x_{1i} \geq 0,\;\;x_{2i} \geq 0 $$ Solvency is also imposed -i.e. the individual cannot borrow in the first period more than what he can repay in the second period (including interest).

One can deduce that

$$\text {"flatter" budget constraint:}\;\; -(1+r) > -\beta_i^{-1} \implies \{x_{1i}^* = (1+r)^{-1},\;\;x_{2i}^* = 0\} $$

$$\text {"steeper" budget constraint:}\;\; -(1+r) < -\beta_i^{-1} \implies \{x_{1i}^* = 0,\;\;x_{2i}^* = 1\}$$

.."steeper" and "flatter" with respect to the utility (linear) indiference map, and with $x_{2i}$ in the vertical axis.

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Working the implied inequalities we re-write the solution as

$$\beta_i < \frac 1{1+r} \implies \{x_{1i}^* = (1+r)^{-1},\;\;x_{2i}^* = 0\}$$

$$\beta_i > \frac 1{1+r} \implies \{x_{1i}^* = 0,\;\;x_{2i}^* = 1\}$$

In the $N$-identical individuals case, and with $\beta_i, i=1,...,N$ being a random variable we can only talk about expected demand in the first period, denote it $E(d_{1i})$ for the individual and $E(D_1) = N\cdot E(d_{1i})$ on aggregate.

We have

$$E(d_{1i}) = \frac 1{1+r} \cdot \Pr\left[\beta_i < \frac 1{1+r}\right] + 0 \cdot \Pr\left[\beta_i > \frac 1{1+r}\right] $$

The second term vanishes.

Given the assumptions on the distribution of the $\beta$'s its density function over the domain is $f(\beta_i) =2$ and so the remaining probability is (as long as $r \leq 1$)

$$\Pr\left[\beta_i \leq \frac 1{1+r}\right] = \int_{1/2}^{(1+r)^{-1}}2{\rm d}\beta_i = 2\left(\frac 1{1+r}-\frac 12\right) = \frac {1-r}{1+r}$$

and so we have

$$E(D_1) = \frac N{1+r}\cdot \frac {1-r}{1+r} = \frac {1-r}{(1+r)^2}N$$

which is decreasing in $r$.

If $r$ hits unity, expected demand will be zero -because
a) It would place a solvency ceiling to loan demand to be not higher than $1/2$ so that with interest rate $100\%$ all production of the second period would then be needed to repay the loans. This means that with $r=1$, maximum consumption (and utility) in the first period will not exceed $1/2$.
b) the discount factor being greater or equal than $1/2$, implies that by taking zero loan in the first period, one would enjoy the full unitary production of the second period, with utility higher than $1/2$. So for non-zero demand for loans in the first period, it must be the case that this demand can exceed $1/2$. When $r\geq 1$ it cannot, hence the zero expected demand.

Note: if instead of a "stochastic distribution" of the $\beta$'s we think of a "uniform allocation" of them in the $[1/2,1]$ interval, then we can think of expected demand as deterministic demand.

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