<p>Every lagrangian with two goods has the same two steps basically. First, find the optimal relationship between the two goods (often through derivatives). Second, plug that relationship back into the budget constraint to solve for each good.</p>

<p>This problem is no different and you have essentially only done step 1. So you just need to do step 2.</p>

<p>The idea is that now you have 2 unknowns (<span class="math-container">$c$</span>,<span class="math-container">$c'$</span>), so you need 2 equations. One equation is the relationship you found between consumption today and tomorrow from the first order conditions. The second equation is the the budget constraint (or equation 3 as you called it). Your equation 3 has still not been used toward solving for consumption. </p>

<p>Plugging in the equation for <span class="math-container">$c'$</span> that you found into your equation 3 you get:</p>

<p><span class="math-container">$c + \frac{c(1+r)}{1+r} =y+\frac{y'}{1+r}$</span></p>

<p>So <span class="math-container">$c= 0.5 *(y+\frac{y'}{1+r})$</span></p>

<p>Now plug in this equation back into the relationship you found through the lagrangian, which was:</p>

<p><span class="math-container">$c' = (1+r) c$</span>.</p>

<p>Then you'll be done. </p>

CC BY-SA 4.0
2019-04-24T07:55:03.700Z
2019-04-24T07:55:03.700Z
BB King
<p>This is the problem I've been given.
Assume the consumers’preference over current and future consumption is given by <span class="math-container">$U(c,c')=c^\frac{1}{2}*c'^\frac{1}{2}$</span>.<br>
Further assume there is no government.<br>
1) solve for the optimal decisions: <span class="math-container">$c,c′$</span> and <span class="math-container">$s$</span> ? </p>

<p>This is as far as I have come</p>

<p><span class="math-container">$\max U(c,c')=c^\frac{1}{2}c'^\frac{1}{2}$</span><br>
s.t. <span class="math-container">$c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$</span><br>
<span class="math-container">$c, c',s \ge 0 $</span> </p>

<p><span class="math-container">$L=c^\frac{1}{2}*c'^\frac{1}{2}+λ(c+\frac{c'}{1+r}-y-\frac{y'}{1+r})$</span> </p>

<p>(1)derv of <span class="math-container">$c$</span>: <span class="math-container">$\frac{1}{2}c^{-\frac{1}{2}}*c'^\frac{1}{2}-λ=0 $</span><br>
(2)derv of <span class="math-container">$c'$</span>:<span class="math-container">$\frac{1}{2}c^\frac{1}{2} c'^{-\frac{1}{2}}-λ\frac{1}{1+r}=0$</span><br>
(3)(<span class="math-container">$λ$</span>):<span class="math-container">$c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$</span></p>

<p>Divide equation 1 by 2 </p>

<p><span class="math-container">$\frac{ \frac{1}{2}c^{-\frac{1}{2}}c'^\frac{1}{2}} {\frac{1}{2}c^\frac{1}{2}c'^{-\frac{1}{2}}}= \frac{λ}{\frac{λ}{1+r}}$</span>  = <span class="math-container">$\frac{c'}{c}=1+r$</span> </p>

<p>I would appreciate if someone could give some help on how to continue so I can find the optimal decisions for <span class="math-container">$c, c', s$</span> </p>

CC BY-SA 4.0
2019-04-23T21:39:47.230Z
I recommend you edit your text to make your equations understandable! (i.e. write them in equation form) As it is it is very difficult to read
Ali
CC BY-SA 4.0
2019-04-24T03:57:49.343Z
and please show us your efforts to answer the question
emeryville
<macroeconomics>
Macroeconomics two period model
Dogge
# Macroeconomics two period model

(Asked by: username_0 on 2019-04-23)

This is the problem I've been given.
Assume the consumers’preference over current and future consumption is given by $U(c,c')=c^\frac{1}{2}\*c'^\frac{1}{2}$.  
Further assume there is no government.  
1) solve for the optimal decisions: $c,c′$ and $s$ ?

This is as far as I have come

$\max U(c,c')=c^\frac{1}{2}c'^\frac{1}{2}$  
s.t. $c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$  
$c, c',s \ge 0 $

$L=c^\frac{1}{2}\*c'^\frac{1}{2}+λ(c+\frac{c'}{1+r}-y-\frac{y'}{1+r})$

(1)derv of $c$: $\frac{1}{2}c^{-\frac{1}{2}}\*c'^\frac{1}{2}-λ=0 $  
(2)derv of $c'$:$\frac{1}{2}c^\frac{1}{2} c'^{-\frac{1}{2}}-λ\frac{1}{1+r}=0$  
(3)($λ$):$c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$

Divide equation 1 by 2

$\frac{ \frac{1}{2}c^{-\frac{1}{2}}c'^\frac{1}{2}} {\frac{1}{2}c^\frac{1}{2}c'^{-\frac{1}{2}}}= \frac{λ}{\frac{λ}{1+r}}$ = $\frac{c'}{c}=1+r$

I would appreciate if someone could give some help on how to continue so I can find the optimal decisions for $c, c', s$

**Tags:** macroeconomics
> **Comments:**
> **username_1**: I recommend you edit your text to make your equations understandable! (i.e. write them in equation form) As it is it is very difficult to read  
> **username_2**: and please show us your efforts to answer the question  

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## 1 answers

### Answer by username_3 on 2019-04-24. Score: 1

Every lagrangian with two goods has the same two steps basically. First, find the optimal relationship between the two goods (often through derivatives). Second, plug that relationship back into the budget constraint to solve for each good.

This problem is no different and you have essentially only done step 1. So you just need to do step 2.

The idea is that now you have 2 unknowns ($c$,$c'$), so you need 2 equations. One equation is the relationship you found between consumption today and tomorrow from the first order conditions. The second equation is the the budget constraint (or equation 3 as you called it). Your equation 3 has still not been used toward solving for consumption.

Plugging in the equation for $c'$ that you found into your equation 3 you get:

$c + \frac{c(1+r)}{1+r} =y+\frac{y'}{1+r}$

So $c= 0.5 \*(y+\frac{y'}{1+r})$

Now plug in this equation back into the relationship you found through the lagrangian, which was:

$c' = (1+r) c$.

Then you'll be done.


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