<p>Question: </p>

<blockquote>
  <p>Suppose Mr X maximises his utility for two periods, his total utility is given by  log $c_1$ + $\beta$ log $c_2$ where $\beta$ $\in$ ( 0,1) and $c_1$ and $c_2$ are his consumption for period 1 and period 2 respectively. Suppose he earns a wage only in period 1 and it is given by W. He saves for the second period on which he enjoys a gross return of (1+r) where r > 0 is the net interest rate. Suppose the government implements a scheme where T $\geq$ 0 is collected from agents ( this is also from Mr. X) in the first period and returns the same amount T back in the second period. What is the optimum T for which his total utility is maximised?.</p>
</blockquote>

<p>My working: I tried putting $c_2= [(M -  c_1- T ) ( 1 + r )] + T$ and plugged in the values that helps the consumer achieve optimum utility for the given budget constraint. Using the lagrangean method of optimisation I got the following 
$$\frac{c_2}{\beta c_1} = 1+r$$ 
Solving the budget constraint using this, I got $$ c_2=\frac{\beta[ M ( 1 + r) - Tr ]}{\beta+ 1}$$Next I tried differentiating $c_2$ with respect to T so as to find the optimum T that maximises $c_2$ and in turn the utility function. This method doesn't seem to work. Could somebody correct me ?. Thank you.</p>

CC BY-SA 3.0
2017-09-19T14:37:17.383Z
Just build a budget constraint instead
123
CC BY-SA 3.0
2017-09-19T14:58:01.283Z
123, isn't the budget constraint just the same as the equation for second period consumption with all terms exception M (1 + r ) swapped to the left, what difference would that make ?
Meera Unni
CC BY-SA 3.0
2017-09-19T16:20:16.703Z
"tried maximising the budget constraint" How does one maximize a constraint...?
Giskard
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2017-09-19T17:00:39.010Z
Can you show in detail which the calculation, including the optimal c1 and c2?
luchonacho
CC BY-SA 3.0
2017-09-28T07:07:52.307Z
luchonacho I've added edits to my question, could you refer to that now. Also sorry for the very late reply.
Meera Unni
CC BY-SA 3.0
2017-09-28T07:09:38.507Z
denesp, I'm sorry that was a huge error in my wording. I've edited the question and included my working now. Can I know why my question was closed off topic as well ?
Meera Unni
CC BY-SA 3.0
2017-10-01T08:22:42.753Z
The question you are trying to answer seems weird to me. Suppose we take $T>0$ from X in period 1 but give him an extra $T$ to spend in period 2. I can't see how that can ever be better than taking zero from him in period 1. Worst case scenario is that X saves *all* of the extra $T$ we left him in period 1, giving him $T(1+r)$ to spend in period 2 (which is clearly better than having only $T$ to spend). But if X chooses not to save all of the extra $T$ then it must be because there is some other consumption plan that does even better still.
Ubiquitous
CC BY-SA 3.0
2017-10-02T11:21:51.103Z
Thanks a lot Ubiquitous, so the answer should be T = 0, it matches with the answer key that was provided. Thanks once again!
Meera Unni
<microeconomics><utility>
Intertemporal choice with no income flow in the second period
Meera Unni
# Intertemporal choice with no income flow in the second period

(Asked by: username_0 on 2017-09-19)

Question:

> Suppose Mr X maximises his utility for two periods, his total utility is given by log $c\_1$ + $\beta$ log $c\_2$ where $\beta$ $\in$ ( 0,1) and $c\_1$ and $c\_2$ are his consumption for period 1 and period 2 respectively. Suppose he earns a wage only in period 1 and it is given by W. He saves for the second period on which he enjoys a gross return of (1+r) where r > 0 is the net interest rate. Suppose the government implements a scheme where T $\geq$ 0 is collected from agents ( this is also from Mr. X) in the first period and returns the same amount T back in the second period. What is the optimum T for which his total utility is maximised?.

My working: I tried putting $c\_2= [(M - c\_1- T ) ( 1 + r )] + T$ and plugged in the values that helps the consumer achieve optimum utility for the given budget constraint. Using the lagrangean method of optimisation I got the following
$$\frac{c\_2}{\beta c\_1} = 1+r$$
Solving the budget constraint using this, I got $$ c\_2=\frac{\beta[ M ( 1 + r) - Tr ]}{\beta+ 1}$$Next I tried differentiating $c\_2$ with respect to T so as to find the optimum T that maximises $c\_2$ and in turn the utility function. This method doesn't seem to work. Could somebody correct me ?. Thank you.

**Tags:** microeconomics, utility
> **Comments:**
> **username_1**: Just build a budget constraint instead  
> **username_0**: 123, isn't the budget constraint just the same as the equation for second period consumption with all terms exception M (1 + r ) swapped to the left, what difference would that make ?  
> **username_2**: "tried maximising the budget constraint" How does one maximize a constraint...?  
> **username_3**: Can you show in detail which the calculation, including the optimal c1 and c2?  
> **username_0**: luchonacho I've added edits to my question, could you refer to that now. Also sorry for the very late reply.  
> **username_0**: denesp, I'm sorry that was a huge error in my wording. I've edited the question and included my working now. Can I know why my question was closed off topic as well ?  
> **username_4**: The question you are trying to answer seems weird to me. Suppose we take $T>0$ from X in period 1 but give him an extra $T$ to spend in period 2. I can't see how that can ever be better than taking zero from him in period 1. Worst case scenario is that X saves *all* of the extra $T$ we left him in period 1, giving him $T(1+r)$ to spend in period 2 (which is clearly better than having only $T$ to spend). But if X chooses not to save all of the extra $T$ then it must be because there is some other consumption plan that does even better still.  
> **username_0**: Thanks a lot Ubiquitous, so the answer should be T = 0, it matches with the answer key that was provided. Thanks once again!  

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## 0 answers

