<p>This is more easily seen by writing out the budget constraints for periods 1 and 2 separately, and then eliminate the saving $s$. </p>

<p>In period 1, the agent spends $\left(1+T_1^c \right)\cdot c_1$ on consumption, and saves the rest, so 
$$(1) \qquad \qquad s=y - \left(1+T_1^c \right)\cdot c_1.$$</p>

<p>In period 2, the agents lives on savings (together with interest income), i.e., consumes all the savings, so 
$$(2) \qquad \qquad \left(1+T_2^c \right)\cdot c_2 = (1+r)s,$$ where we multiplied $s$ with $1+r$ because of the interest from period 1 to period 2. </p>

<p>Now putting the two equations together, we have 
$$
\left(1+T_2^c \right)\cdot c_2 = (1+r)s= (1+r)( y - \left(1+T_1^c \right)\cdot c_1). 
$$
Divide both sides by $(1+r)$ and rearranging the terms, you get the desired result. </p>

CC BY-SA 3.0
2016-04-13T20:21:30.893Z
2016-04-13T20:21:30.893Z
Yuxin Wang
<p>An agent lives for 2 periods, consumes $c$ and saves $s$ at interest rate $r$ in the first period of life from income $y$. In the second period the agent consumes for any savings. We introduce a percentage tax $T^c$ on consumption in both periods where $0 &lt; T^c &lt; 1$. Assuming initially that $T^c_1 = T^c_2$. What is the consumer's lifetime budget constraint?</p>

<p>Answer: $$c_t (1+T^c_1)+ c_{t+1} \frac{1+T^c_2}{1+r} =y_t$$</p>

<p>I don't understand how we can calculate the budget constraint, can anyone help?</p>

CC BY-SA 3.0
2016-04-13T16:29:08.113Z
I texed the math symbols. I assume you mean $T^c_1 = T^c_t$ and $T^c_2 = T^c_{t+1}$. A hint: savings $s$ is missing in the answer. How come? $s$ is transfered from today to tomorrow, i.e. it appears in each period budget constraint and thus we may combine both constraints over $s$. So, write down the budget constraint for each period!
clueless
CC BY-SA 3.0
2016-04-13T17:44:00.553Z
thanks for sorting out the symbols! yeah s is definitely not in the answer but I don't understand why either! Would you know how to write down the budget constraints for each period?
svenja
CC BY-SA 3.0
2016-04-13T17:59:40.590Z
Well his income in period $t$ is $y_t$. Of that very income he consumes $c_t$ pays taxes $c_t T_t$ and saves $s$. What's the budget constraint in $t$? Another hint: the answers you gave is wrong as it's statet. There must be a minus somewhere. In addition it would be appreciated if you show some effort.
clueless
<macroeconomics><mathematical-economics><financial-constraints>
Lifetime Budget Constraint
svenja
# Lifetime Budget Constraint

(Asked by: username_0 on 2016-04-13)

An agent lives for 2 periods, consumes $c$ and saves $s$ at interest rate $r$ in the first period of life from income $y$. In the second period the agent consumes for any savings. We introduce a percentage tax $T^c$ on consumption in both periods where $0 < T^c < 1$. Assuming initially that $T^c\_1 = T^c\_2$. What is the consumer's lifetime budget constraint?

Answer: $$c\_t (1+T^c\_1)+ c\_{t+1} \frac{1+T^c\_2}{1+r} =y\_t$$

I don't understand how we can calculate the budget constraint, can anyone help?

**Tags:** macroeconomics, mathematical-economics, financial-constraints

> **Comments:**
> **username_1**: I texed the math symbols. I assume you mean $T^c_1 = T^c_t$ and $T^c_2 = T^c_{t+1}$. A hint: savings $s$ is missing in the answer. How come? $s$ is transfered from today to tomorrow, i.e. it appears in each period budget constraint and thus we may combine both constraints over $s$. So, write down the budget constraint for each period!  
> **username_0**: thanks for sorting out the symbols! yeah s is definitely not in the answer but I don't understand why either! Would you know how to write down the budget constraints for each period?  
> **username_1**: Well his income in period $t$ is $y_t$. Of that very income he consumes $c_t$ pays taxes $c_t T_t$ and saves $s$. What's the budget constraint in $t$? Another hint: the answers you gave is wrong as it's statet. There must be a minus somewhere. In addition it would be appreciated if you show some effort.  

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## 1 answers

### Answer by username_2 on 2016-04-13. Score: 3

This is more easily seen by writing out the budget constraints for periods 1 and 2 separately, and then eliminate the saving $s$.

In period 1, the agent spends $\left(1+T\_1^c \right)\cdot c\_1$ on consumption, and saves the rest, so
$$(1) \qquad \qquad s=y - \left(1+T\_1^c \right)\cdot c\_1.$$

In period 2, the agents lives on savings (together with interest income), i.e., consumes all the savings, so
$$(2) \qquad \qquad \left(1+T\_2^c \right)\cdot c\_2 = (1+r)s,$$ where we multiplied $s$ with $1+r$ because of the interest from period 1 to period 2.

Now putting the two equations together, we have
$$
\left(1+T\_2^c \right)\cdot c\_2 = (1+r)s= (1+r)( y - \left(1+T\_1^c \right)\cdot c\_1).
$$
Divide both sides by $(1+r)$ and rearranging the terms, you get the desired result.


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