Question

help with 9-18 i am stuck and and need help with them being answered thanks!

9. Let's assume that A is fixed and define the economic impact as a function of r, that is I(r) = Graph 1-1 this function with A = $10,000. What are the domain and range of the economic impact function in the context of this problem? 10. Compute the derivatives l'(r) and 1"(r). Is the economic impact function increasing or decreasing on its domain? Is it concave up or concave down? 11. The sensitivity of the economic impact to changes in (or errors in estimating) the spending rate is important. Suppose the spending rater is changed from r tor + Ar. Use linear approximation to find the resulting approximate change Al in the economic impact in terms of A and r). 12. Suppose that with a $350 billion stimulus package the spending rate increases from r=0.3 to r=0.35. What is the approximate change in the economic impact? 13. Suppose that with a $350 billion stimulus package the spending rate increases from r= 0.8 to r=0.85. What is the approximate change in the economic impact? 14. Compare the rewalts of Steps 12 and 13. Notice that the change in the spending rate is the same (Ar=0.05) in both cases. Is the change in the economic impact greater when r is small or large? 15. Finally, it is useful to relate the percentage change in the spending rate (Arlr) to the percentage change in ΔΙ the economic impact (All). Use the general result in Step II to show that Ar changed chun 16. The interpretation of this relationship is important. The percentage change in the spending rate is related to the percentage change in the economic impact through the function ger). Graph g for 0 <r<l. 17. Explain the statement that for small spending rates (r< 1/2), the percentage change in the spending rate is greater than the percentage change in the economic impact, while for large spending rates (r> 1/2), the percentage change in the spending rate is less than the percentage change in the economic impact. 18. Explain why for countries with a large spending rate (such as the United States) errors in estimating the spending rate lead to relatively larger errors in predictions of the economic impact (compared to countries with lower spending rate)

Answer 1

9. We have to graph the function

$\small I(r)={10000\over 1-r}$

Let us take r along the horizontal axis and I(r) along the vertical axis, we get:

1000+ 8004 600+ 4004 200+ 00 -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200 1400 200+ -400+ -600+ -800+

Now, the domain of a function is the set of values where the function is valid. Now, the economic impact function given to us has (1-r) in the denominator. As division by zero is undefined, hence the function will not be defined when r = 1, thus the domain is given by

Domain I = (-0, 1) U (1,0)

Further, the range is the set of all functional values. Now, notice that the function I(r) can never be zero, as it's numerator is a constant. We can also see that the line I = 0 is a horizontal asymptote of the function, hence, the range of the function is

$\small \text{Range }I=(-\infty, 0)\cup (0,\infty)$

10.

Now, we differentiate I(r) twice with respect to r to get

$\small I'(r)={10000\over (1-r)^2}$

$\small I''(r)={20000\over (1-r)^3}$

Now, we see that the denominator on I'(r) is a square, hence no matter what the value of r is, the value of I'(r) will always be positive. Now, I'(r) gives us the instantaneous rate of change of the function I(r), thus, the economic impact function is always increasing on it's domain.

However, notice that in I'(r) the denominator is a cube, so the sign of (1 - r) will affect the sign of the function I''(r). Now, when I''(r) > 0, then I(r) is said to be concave up, and when I''(r) < 0, then I(r) is said to be concave down.

Thus, the economic impact function is concave up when r < 1 and concave down when r > 1