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Unformatted text preview: Econ 300, Problem Set 1 Professor Cramton 2.1.2. Determine which of the following relationships represent functions. Assume that the interval is the set of real numbers unless otherwise indicated. (a) y = 5 x (b) y ≤ x (c) y = a + √ x ;(0 , ∞ ) (d) y =x 2 (e) y 2 = x (f) y = 1 x3 ;(0 , ∞ ) (g) y 2 = x 4 ;(0 , ∞ ) 2.1.8. Which functions are continuous over the given intervals? (a) y = 8 + 1 x7 ;(0 , ∞ ) (b) y = 4ax 7 ;(∞ , ∞ ) (c) y =3 + 1 x +7 ;(0 , ∞ ) (d) y =  2 x4  ;[3 , 3] 2.2.8. Show that the average rate of change of a strictly increasing function is positive and that the average rate of change of a strictly decreasing function is negative. 2.2.10. Sketch the function y = 8 + 10 xx 2 over the domain [0,7]. (a) Assume that x A = 1 ,x B = 4, and λ = 0 . 4. Using the formula x = λx A + (1λ ) x B , determine the value of x . What is the value of f ( x )? (b) Calculate the value of y , which is the value of the secant line at x , using the formula y = λf ( x A ) + (1λ ) f ( x B ). (c) Prove that the above function is strictly concave by demonstrating that f ( x ) > y . 1 2.3.2. Condense the following expressions. (a) x a · x b · x c ÷ x d (b) x 1 / 2 x 3 / 2 ÷ x 1 / 3 (c) (( x 1 / 3 ) 8 ) 1 / 2 · x 2 ÷ x 3 / 4 (d) x 2 y 3 · x 3 ÷ xy 2 · x2 y 2.3.12. Consider a function that relates tax revenues R , in billions of dollars, to the average tax rate t such that R = 350 t500 t 2 (a) What tax rate(s) is consistent with raising tax revenues equal to $60 billion? (b) What tax rate(s) is consistent with raising tax revenues equal to $61.25 billion? 2 Econ 300, Problem Set 2 Professor Cramton 3.1.6. Assume a ﬁrm’s net proﬁts are $50 million in 2000 and are expected to grow at a steady rate of 6% per year through the end of the decade. How much would you expect the ﬁrm to earn in 2001? In 2003? Now assume that the ﬁrm’s proﬁts have been growing at 6% since 1997. If a negative value of n can be interpreted as the number of time periods before period t , how much did the company earn in 1998? Graph the path of income growth between 1998 and 2003 and explain why the curve gets steeper over time. 3.2.8. While you are a senior in high school, your parents decide to invest in the bond market to help pay for your college tuition. They purchase a bond that will pay $15,000 in one year plus an interest coupon payment of 7% of the bond’s value when the interest rate on comparable assets is also 7%. What is the present value of this bond? If interest rates fall to 5% after the bond purchase, what is the present value of that same investment? What is the present value if interest rates rise to 9.5%? 3.3.4. On a secretive search through your grandfather’s basement you uncover an old, dustcovered stamp collection. You take the collection to a stamp dealer who tells you that it currently is worth $1,000, but in ten years it will triple its value....
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 Spring '11
 williams
 Derivative, Probability theory

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