Math for Econ II, Written Assignment 8 (30 points) Due Friday, December 4th Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you will not be given any credit. Write your name and recitation section number. Staple your homework if you have multiple pages! 1. Assume an economics model where we have three industries producing turkey, cranberry sauce, and pumpkin pie, with the following consumption matrix 0 0.1 0.4 12 C = 0.5 0.4 0.2 , and nal demand d = 6 , 0.1 0.2 0.3 8 where the components of d are in millions of dollars. (a) (2 pts) What does the entry c12 = 0.1 mean? (b) (2 pts) Compute the output levels to meet the total demand. Note: You can use a calculator to do the computations for problem 2, but you must write every step of your work on paper. Do not skip any steps!!! (c) (2 pts) Suppose that we increase the nal demand for pumpkin pie from 8 to 9. What is the corresponding increase of the output level of cranberry sauce? (Hint: the answer isn't just an entry in the consumption matrix. Use Cramer's rule). 2. (2 pts) Find the present and future values of a continuous income stream of $1500 per year over the next 20 years, assuming an annual interest rate of 3% compounded continuously. 3. (4 pts) Determine whether or not the following improper integrals converge, either by integrating directly or comparing them with a known integral: (a) 1 (b) 1 ln x dx. x 1 dx (x2 + 4)3/2 4. (The power of exponential growth) Alice promises to pay Bob $1000 today, $2000 in one year, $3000 in two years, and so on indenitely. Assume that the interest rate is 2%, compounded continuously. (a) (4 pts) Express the total present value of Alice's promise to Bob as an indenite integral, and evaluate it. (b) (2 pts) In approximately what year will Alice make the largest payment to Bob, in terms of present value? 5. (12 pts) We dene the error function erf(x) by 2 erf x = x 2 ex dx. 0 We can prove that lim erf x = 1, which means that x+ e 0 x2 b dx = lim b x2 e 0 dx = lim erf b = , 2 b 2 2 ex dx = 2 2 ex dx = 0 (a) (4 pts) Find the indenite integral 2 x2 ex dx as a function involving erf(x). (Hint: you will need to use integration by parts). . (b) (2 pts) Use this formula to nd the improper integral 2 x2 ex dx. (c) (6 pts) In statistics, the Gaussian density function with mean and variance is dened by f (x) = Use the above calculations to prove that f (x) dx = 1, i. ii. xf (x) dx = , (x )2 f (x) dx = 2 . iii. 1 exp[(x )2 /2 2 ]. 2