<p>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.  </p>

<p>The individual $i$ has to solve the following problem</p>

<p>$$\max_{x_{1i},x_{2i}}[ x_{1i} + \beta_i x_{2i}]$$</p>

<p>$$s.t. x_{2i} = 1-(1+r)x_{1i}, \;\;x_{1i} \geq 0,\;\;x_{2i} \geq 0 $$
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).</p>

<p>One can deduce that</p>

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

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

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

<p><img src="https://i.sstatic.net/uBRC0.png" alt="enter image description here"></p>

<p>Working the implied inequalities we re-write the solution as</p>

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

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

<p>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.</p>

<p>We have</p>

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

<p>The second term vanishes.</p>

<p>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 $r \leq 1$)</p>

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

<p>and so we have</p>

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

<p>which is decreasing in $r$.</p>

<p>If $r$ hits unity, <em>expected</em> demand will be zero -because<br>
<strong>a)</strong> 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 $1/2$.<br>
<strong>b)</strong> 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 <em>must</em> be the case that this demand can exceed $1/2$. When $r\geq 1$ it cannot, hence the zero expected demand. </p>

<p><strong>Note:</strong> 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.</p>

CC BY-SA 3.0
2015-06-03T13:47:22.800Z
2015-06-04T20:30:15.747Z
2015-06-04T20:30:15.747Z
Alecos Papadopoulos
<p>This question is driving me nuts. </p>

<p>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 &lt; β &lt; 1. Note that there are only two sources through which rice can be available; own production and borrowing.</p>

<p>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?</p>

<p>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. </p>

CC BY-SA 3.0
2015-06-03T10:35:46.700Z
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).
dexter
CC BY-SA 3.0
2015-06-03T10:36:54.200Z
The answer however, is $\frac {N* (1-r)} {(1+r)^2}$
dexter
CC BY-SA 3.0
2015-06-03T10:55:16.463Z
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.
Giskard
CC BY-SA 3.0
2015-06-03T11:24:51.173Z
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.
dexter
CC BY-SA 3.0
2015-06-03T11:35:31.240Z
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
$$
\int_{\frac{1}{2}}^{\frac{1}{1+r}} D(r, \beta) \ d \beta.
$$
Giskard
CC BY-SA 3.0
2015-06-03T12:17:43.437Z
@denesp I again believe this is already an answer, not a comment :)
FooBar
<microeconomics><consumer-theory><decision-theory>
Intertemporal consumption question
dexter
# Intertemporal consumption question

(Asked by: username_0 on 2015-06-03)

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.

**Tags:** microeconomics, consumer-theory, decision-theory
> **Comments:**
> **username_0**: 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).  
> **username_0**: The answer however, is $\frac {N* (1-r)} {(1+r)^2}$  
> **username_1**: 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.  
> **username_0**: 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.  
> **username_1**: 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 $$ \int_{\frac{1}{2}}^{\frac{1}{1+r}} D(r, \beta) \ d \beta. $$  
> **username_2**: @denesp I again believe this is already an answer, not a comment :)  

---

## 1 answers

### Answer by username_3 on 2015-06-03. Score: 3 | Accepted answer

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.

![enter image description here](https://i.sstatic.net/uBRC0.png)

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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