Suppose that Congress enacts a one-time-only 10% tax rebate that is expected to infuse Sy billion, 5 < y < 7,into the economy: If every person and every corpo...

Suppose that Congress enacts a one-time-only $10 \%$ tax rebate that is expected to infuse \$y billion, $5 \leq y \leq 7$, into the economy. If every person and every corporation is expected to spend a proportion $x, 0.6 \leq x \leq 0.8,$ of each dollar received, then, by the multiplier principle in economics, the total amount of spending $S$ (in billions of dollars) generated by this tax rebate is given by $$ S(x, y)=\frac{y}{1-x} $$ What is the average total amount of spending for the indicated ranges of the values of $x$ and $y$ ? Set up a double integral and evaluate it.

In the problem, they total additional spending in millions. Dollar created by such a tax code is given by this, and so we have to find as infinitely that equals 100 into 0.8, the word by one minus 0.8 that equals eight 100 divided by zero point two. Or we can have this 800 to 1 by two. So this cancer is this four 100 times. Therefore we have a space infinity as 400 million darla's. So this is the answer.

We will use this formula here to solve this problem. So, um or as how much additional spending will be generated by a 10 billion tax rebate? 60% of rolling on this matter. So because 60% of only commencement, that is our common ratio on because any time we have received money, you're spending 60% so layer of spending would be 60% of the previous layer. So our is 0.6. Now, we need to find a one. So what is the first layer spending? Well, the first layer of spending is going to be 60% of whatever rebate got. So what? Training a one is going to be 60% of 10 billion. Uh, but that would be 10 billion times zero for six, which is? I think so. So now we have all we need to find, um, the definite some. So this is going to be equal to six union over one minus 0.6. So that is legal team. 15 1,000,000,000

All right So far live for an infinite geometric serious. His is this Now was important. No, it is that we can't just plug in 10 billion for a because it's not this tent. The 10 billion isn't the first term. The first term is actually a 0.6 times 10 billion, which, um, gives us six billion. So that's our first Her for the four million and plug in is six billion over one minus R, which is 0.6. This ends up giving US 15 billion. That's how much additional spending worker.

So here in this problem, we are given that the government pumps an extra $1 billion dollars into the economy So they pumped in $1 billion dollars into the economy. Now we have to assume that each businessman and individual saves 25% of its income and spends the rest on the initial one billion. That means 75% is being re spent. Now the amount of money that is spent during the government's pumping can be written as an infinite geometric series. And what we are given, what we are asked is that what is the total increase in spending due to the government action? So let me first right the geometric series. So initially it was $1 billion dollars Then re spent is 75% of one billion. So 75% means 0.75 of $1 billion. No, again 75% of previous value. So it would be 0.75 and it's square of $1 billion. And it will goes on like this. Now basically we have to find the sum. So some we can use the formula a divide by one minus our. Now here is the first time we have one billion so this is $1 billion dollars And the common ratio are we have 0.75 So you can plug the village. Here is one Divide by one -R is 0.75. On simplification will get the answer to be four billion. The further amount the total increase is $4 billion. So this is a required answer. I hope you have understood the problem. Thank you.