Econ Review Set 3 - Answers

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1 Econ 4808 Review Set 3 - Answers Outline: 1. Limits, continuity & derivatives. 2. Economic applications of derivatives. Unconstrained optimization. Elasticities. 2.1 Revenue and pro t functions 2.2 Productions functions and marginal products. 2.3 Cost functions. LIMITS, CONTINUITY & DERIVATIVES 1. Find the limit of q (3v+5) (v+2) as v! 0, as v! 5 3 as v! 2. (3v + 5) lim (3v + 5) lim v!0 (v + 2) v!0 (3 (0) + 5) lim (v + 2) (0 + 2) (3v + 5) lim v! 5 3 v!0 (v + 2) (3v + 5) (3 ( 2) + 5) lim 1 v! 2 (v + 2) ( 2 + 2) 0 undefined 2. Assume c(x) a + bx 3 where a; b > 0. Derive c 0 (x) using the de nition of a derivative. Show all of your work. First use the de nition of a derivative to nd c 0 (x) when x x 0 : c 0 c (x 0 + t) c (x 0 ) (x 0 ) lim t!0 t ha + b (x 0 + t) 3i a + bx 3 0 lim t!0 t ha + b (x 0 + t) 3i a + bx 3 0 lim t!0 t where (x 0 + t) 3 x x2 0 t + 3x 0t 2 + t 3. Plugging this into the de nition of the derivative, one obtains, after simplifying 1

2 a + b x x 2 0 t + 3x 0t 2 + t 3 a + bx 3 0 c 0 (x 0 ) lim t!0 t b 3x 2 0 lim t + 3x 0t 2 + t 3 t!0 t limb 3x x 0 t + t 2 t!0 b3x 2 0 Since this is true for any x, c 0 (x 0 ) 3bx Assume f f (x). A) Write out the de nition of df(x) at x x 0 in words. B) Write out the de nition of df(x) at x x 0 in functional notation. C) Draw an example graph that would show the relationship of df(x) to f (x). A) In words, the derivative df(x) at x x 0 is the instantaneous rate of change of f (x) at x x 0, or said more loosely, it is the change in f (x) for a small change in x. B) In functional notation, df(x) at x x 0 is df(x) lim f(x 0 +t) f(x 0 ) t!0 t. C) Graphically, df(x) at x x 0 is the slope of f (x) at x x 0, or said another way, it is the slope of the tangent line to the graph of f (x) at x x Find the derivative of f (x) 4. f 0 (x) 0 5. Find the derivative of f (m) e 3m2. Using the e x rule and the chain rule: f 0 (m) e 3m2 3m 2 0 e 3m2 (6m) 6. Find the derivative of f (t) (ln t) 4 + t 1, where is a parameter (or a constant). Using the product rule: f 0 (t) (ln t) t 1 + (ln t) 4 + t t 1 + (ln t) t 2 t 2

3 7. Find the derivative of f (w) w 3 w 5w 4 + w 2. Using the product rule: f 0 (w) w 3 w 0 5w 4 + w 2 + w 3 w 5w 4 + w 2 0 3w 2 1 5w 4 + w 2 + w 3 w 20w 3 + 2w 8. Find the derivative of f (v) (constants). Using the quotient rule: f 0 (v) You can now simplify: h (v + m) :5i 0 v 2 :5 (v+m), where and m are exogenous parameters v 2 h (v + m) :5i v 2 0 (v 2 ) 2 h( :5) (v + m) 1:5 (v + m) 0i h v 2 (v + m) :5i (2v) h i h ( :5) (v + m) 1:5 () v 2 (v + m) :5i (2v) v 4 v 4 f 0 (v) :5v 2 (v + m) 1:5 2v (v + m) :5 v 4 :5v (v + m) 1:5 2 (v + m) :5 v 3 9. Find the derivative of f (x) (1 + 3 ln x) 2. Using the chain rule f 0 (x) 2 (1 + 3 ln x) (1 + 3 ln x) (1 + 3 ln x) 3 x 6 (1 + 3 ln x) x 10. Find the derivative of f (x) + x 3x :5 e x + ln (3x), where and are parameters (or constants). 3

4 f 0 (x) (:5) 3x 1:5 e x + 1 3x (3x)0 1:5x 1:5 e x + 1 x 11. Find the derivative of w (x) a bx 2 + cx 2 using the chain rule. Don t worry about algebraically simplifying your answer. w 0 (x) 2a bx 2 + cx bx 2 + cx 0 2a bx 2 + cx (2bx + c) ECONOMIC APPLICATIONS OF DERIVATIVES. UNCONSTRAINED OPTIMIZATION. ELASTICITIES. REVENUE AND PROFIT FUNCTIONS 12. Assume that you own a movie theater that operates at zero costs. Therefore, your total pro ts,, from the sale of movie ticket is pq, where p is the price per ticket and q is the number of tickets sold. Assume the demand function for your tickets is q 4 :6p, where is the probability that it will rain and 0 < < 1. A) Show, using a partial derivation, whether pro ts will always increase when you raise your price. B) What will happen to pro ts if p $6 and :5 and you then increase your price a little? A) The rm s pro t function in terms of price is and the marginal pro t function is (p) p (4 :6p) 4p :6p 2 0 (p) 4 1:2p > 0 if p < 4 1:2 So, pro ts will increase with price, p, only if p < 4 1:2. B) When p $6 and :5, the marginal pro t function is 0 (6) 4 1:2 (:5) (6) 0:4 > 0 4

5 and so if you increase the price a little, you will increase pro ts. 13. Assume the following theory of supply and demand for orange- avored popsicles Q d a bp Q s cp Q d Q s where Q d and Q s are the quantity demanded and the quantity supplied for popsicles, respectively. P is the price of a popsicle, a is the amount of orange used to produce each popsicle, b is an increasing function of the number of substitutes for orange- avored popsicles, and c is the number of individuals working in the orange growing industry. Assume that the equilibrium price is positive. A) Determine how much equilibrium total revenue in the orange-popsicle industry will decrease, or increase, if the orange content of the popsicles is increased by an incremental amount. B) Determine in percentage terms how much equilibrium total revenue in the orangepopsicle industry will decrease, or increase, if the orange content of the popsicles is increased by one percent. C) Determine in percentage terms how much equilibrium total revenue in the orangepopsicle industry will decrease, or increase, if the number of people in the orange growing industry increases by one percent. Show and explain all of your work. A) Determine the equilibrium price and quantity. In equilibrium Therefore, equilibrium quantity is a bp cp P a b + c Q a bp a ba b + c a (b + c) ba b + c ac b + c 5

6 So, equilibrium total revenue is T R P Q a 2 c (b + c) 2 The issue is what happens to this amount if a increases by an incremental amount. determine this, take the partial derivative of T R with respect to a R a 2 c (b+c) 2ac (b + c) 2 So, if a increases by an incremental amount, then total revenue will increase by 2ac (b+c) 2. B) The issue is what happens to T R in percentage terms if a increases by 1%. Recall the formula Note that %T R (ln T R (ln a) a ln (T R 2 c ) ln (b + c) 2 ln a 2 c ln (b + c) 2 ln a 2 + ln c 2 ln (b + c) 2 ln a + ln c 2 ln (b + c) Then %T R (ln T R (ln (2 ln a + ln c 2 ln (b + (ln a) 2 That is, if orange content of popsicles increases by 1%, total revenue will increase by 2%. T R C) The issue is what happens to T R in percentage terms if c increases by 1%. Since a2 c (b+c) 2 and c appears in the additive term, I will use the elasticity formula 6

7 The partial derivative is %T R c T R a 2 c (b+c) a 2 c 0 (b + c) 2 h(b + c) 2i 2 a 2 c h (b + c) 2i 0 a2 (b + c) 2 a 2 c 2 (b + c) (b + c) 0 (b + c) 4 a2 (b + c) 2 a 2 c 2 (b + c) (b + c) 4 After simplifying a2 (b + c) 2a 2 c (b + c) 3 %T R c T R a2 (b + c) 2a 2 c (b + c) 3 c a 2 c (b+c) 2 Simplifying %T R %c a2 (b + c) 2a 2 c (b + c) 2 (b + c) 3 a 2 (b + c) 2c b + c b c b + c That is, if the numberof people in the orange growing industry increases by 1%, total revenue will change by b %. c b+c 14. Assume that the Gomer Corporation sells product x at the price p and that the demand function for its output is x x (p) 3p :5 7

8 A) What is the corporation s total revenue function as a function of p? As part of your answer de ne, in words, total revenue as a function of p. B) What is total, marginal and average revenue as a function of p if p 16. C) Now assume that the rm has complete control over the price (i.e., the rm is a monopolist). What price should the rm charge if its intent is to maximize the total amount of revenue it receives? Explain your answer. A) Total revenue as a function of p is T R (p) p 3p :5 3p :5 Total revenue as a function of p is the total amount of money that the corporation takes in from the sale of its product as a function of the price it charges. B) Marginal and average revenue are If p 16, dt R (p) MR (p) 1:5p :5 dp AR (p) T R (p) x (p) 3p :5 p T R (16) 3 (16) :5 12 MR (16) 1:5 (16) :5 1 1:5 :375 4 AR (16) 3 (16) :5 1 3 :75 4 Note that average revenue as a function p is quantity demanded, x (p). C) If the corporation wants to maximize its total revenue, it should raise its price as long as marginal revenue as a function of price is positive. increases total revenue, do it, if the intent is to maximize total revenue. MR (p) dt R (p) dp 1:5p :5 > 0 if p > 0 That is, if increasing the price That is marginal revenue as a function of price is always positive so the rm should charge in nity. 15. Assume that McDonald s sell burgers, b, at the price p b > 0, and the demand function for its output is b b (p b ) 3p :5 b 8

9 A) What is McDonald s total revenue from burgers as a function of the price of its burgers? As part of your answer de ne, in words, total revenue as a function of p b. B) Now assume that McDonald s has complete control over the price it charges for its burgers. What price should the rm charge if its intent is to maximize the total amount of revenue it receives from the sale of burgers? Explain your answer in words and or graphs. A) Total revenue as a function of p b is T R (p b ) p b 3p :5 b 3p :5 b Total revenue as a function of p b is the total amount of money that McDonald s takes in from the sale of its burgers as a function of the price it charges for them. B) We could use the marginal revenue function to gure out what price McDonald s should charge to maximize its revenues from the sale of burgers. MR (p b ) dt R (p b) d 3p:5 1:5p :5 dp b dp b > 0 if p b > 0 b That is marginal revenue as a function of price is always positive so the rm should charge in nity. Obviously, demand function for burgers at McDonald s isn t really given by b b (p) 3p :5 b. One can better see what is going on by graphing the demand function and the total revenue function. Total revenue, for this demand function, is always increasing in the price of burgers. b TR(p) p 16. Assume that the Gomer Corporation sells product x at the price p and that the demand function for its output is 9

10 x x (p) 3p :5 where > 0. A) What is the corporation s total revenue function? B) Write out the total revenue function in log form. C) Determine the elasticity of total revenue with respect to price. D) How does this total revenue elasticity, with respect to price, change as the price rises? Explain. E) Assume that the rm has complete control over the price (i.e., the rm is a monopolist). What price should the rm charge if its intent is to maximize the total amount of revenue it receives? Explain your answer. A) Total revenue as a function of p is T R (p) p 3p :5 3p 1:5 B) The total revenue function in log form is C) ln T R (p) ln 3 + 1:5 ln p %T R %p d ln T R d ln p 1:5 D) The elasticity of total revenue with respect to price is constant. It does not change with price, p. In particular, a 1% increase in price causes a 1.5% increase in T R. E) Because T R always increase if we increase price, you should charge in nity. 17. Assume that the Gomer Corporation sells product x at the price p and that the demand function for its output is x x (p) p where > 0. A) What is the corporation s total revenue function? B) Write out the total revenue function in log form. C) Determine the elasticity of total revenue with respect to price. How does this total revenue elasticity, with respect to price, change as the price rises? Explain. D) Assume that the rm has complete control over the price (i.e., the rm is a monopolist). What price should the rm charge if its intent is to maximize the total amount of revenue it receives? Explain your answer. 10

11 A) Total revenue as a function of p is T R (p) p p p 1 B) The total revenue function in log form is ln T R (p) ln + (1 ) ln p C) %T R %p d ln T R d ln p (1 ) The elasticity of total revenue with respect to price is constant. Further, it is positive if (1 ) > 0, that is, if < 1, and negative if > 1. D) If < 1, charge in nity, because total revenues continuously increase as price increases. If > 1, charge zero, because total revenues continuously increase as price decreases (approaches zero). PRODUCTION FUNCTIONS AND MARGINAL PRODUCTS 18. The short-run production function y f(l) identi es the maximum number output, y, that can be produced as a function of the amount of labor, L, used. A) Describe, in words, the marginal product of labor, MP L, when L L 0. B) Now assume the short-run production function y f(l) 4L 2. Using the basic de nition of a derivative, nd MP L (L 0 ). Given your answer, what is the MP L (L 0 ) when L 0 4? What does your answer mean? C) Do you think that a real world production function could have the marginal product function implied by the mathematical function y f(l) 4L 2. Yes or no and explain. A) The marginal product of labor evaluated at L L 0 is how much maximum output increases when the amount of labor used is marginally increased from L 0. B) 11

12 MP L (L 0 ) df (L 0) dl When L 0 4, then MP L (L 0 ) 32. will cause maximum output to increase by 32 units. f (L 0 + t) f (L 0 ) lim t!0 t (1) 4 (L 0 + t) 2 4 (L 0 ) 2 lim (2) t!0 t 4 L 2 0 lim 0t + t 2 4 (L 0 ) 2 (3) t!0 t 4 2L 0 t + t 2 lim (4) t!0 t lim4 (2L 0 + t) t!0 (5) 8L 0 (6) Said loosely, when L 4 increasing labor by 1 unit C) No, I do not think this production function could exist in the real world. This production function has the marginal product of labor forever increasing d(mpl (L)) dl 8 > A) De ne, in words, the short-run production function x f(l). B) De ne in words (without using the word derivative) the marginal product of labor function MP L (L). C) Now assume x f(l) 4L :5. Given this, what is the marginal product of labor when L 1? D) Given x f(l) 4L :5, show (using a derivative) what happens to the marginal product of labor when the quantity of labor employed increases. Assume L is positive. A) The short-run production function, x f(l), identi es the maximum output as a function of the number of units of labor employed. B) The marginal product of labor, MP L (L) df(l) dl, identi es how much maximum output changes if the amount of labor employed is marginally increased. C) MP L (L) df (L) dl d 4L:5 2L :5 dl MP L (1) 2 (1) :5 2 D) To determine what happens to the marginal product of labor when labor is increased take the derivative of MP L (l) with respect to L: 12

13 d (MP L (L)) d 2L :5 L 1:5 < 0 if L > 0 dl dl That is, given this particular production function, the marginal product of labor is always decreasing. 20. A) De ne, in words, the production function Y f(l; K), where K > 0 and L > 0. B) De ne, in words, the partial derivative of the marginal product of labor with respect to the level of capital. C) Identify the class of production functions that have the property that the marginal product of labor is always positive but always declining. D) What mathematical condition might one impose on the general form of the function Y f(l; K) that would be necessary, but not su cient, for Y f(l; K) to ful ll the property in part C. E) Give an example for a production function that ful lls the property in part C. As part of your answer convince me that your example ful lls the property. A) The production function identi es the amount of output Y that can be produced by employing L units of labor and K units of capital. B) The partial derivative! of the marginal product of labor with respect to the level of @K identi es the e ect on labor productivity (MP L ) of employing additional amount of capital. C) Functions Y f (L; K) such > 0 f(l;k) < 0. 2 D) A condition that is necessary but not su cient > 0. A di erent necessary condition ) < 0. The two conditions together are necessary and ) > 0 and < 0 is: E) A speci c production function that ful Y KL where 0 < < (KL ) KL 1 > dy KL 1 ( 1) KL 2 < 0 So, Y KL is one such function, where 1. 13

14 21. Assume that the Snerd Corporation s technology for producing widgets can be described by the following production function x f (K; L) K + (1 ) L 1 where > 0, 0 < < 1 and 1 < 6 0. This production function is called the constant elasticity of substitution (CES) production function. The Cobb-Douglas is a special case of the CES. A) Find the marginal product of labor function. Simplify the function as much as you can. (Show your work.) B) Given the CES technology, is the marginal product of labor always positive? Why? C) Does the marginal product of labor always decrease as the amount of labor used increases? Don t answer this question directly but rather explain in words, how you would go about addressing this question. (K; L) MP L 1 (1 ) ( ) L 1 K + (1 ) L 1 1 (1 ) L 1 K + (1 ) L 1 1 B) Yes, since > 0, (1 ) > 0, L 1 > 0 and [K + (1 ) L ] C) Take the partial derivative of MP L with respect to labor f (K; L) > 0. If it is positive (negative), it means that the marginal product of labor always increases (decreases) as the amount of labor used increases. 22. Assume a production function x f (L; K) where x is units of output, L is units of labor and K is units of capital. A) De ne in words this production function. B) Now assume that it is the short-run such that K is xed, K, so that we can write the production function x f L; K. Also assume that x f L; K 1 6 L3 + 2KL 2. We all remember from principles of microeconomics the "law of diminishing marginal productivity", which says that if one keeps increasing the amount of one of the inputs, holding the other inputs constant, at some point the marginal product of the input being varied will start to decrease and once is starts to decrease it will continue to decrease. Is this production function consistent with this "law"? Explain your answer and show all of your work. 14

15 A) The production function identi es max output as a function of how much labor and capital are used in production. B) We are concerned with showing that after some level of labor the marginal product of labor starts and continues to decline, so we need to look at the derivative of the marginal production of labor with respect to labor. First, nd the marginal product of labor function, which is the derivative of the production function with respect to labor: MP L L; 1 6 L3 + 2KL L2 + 4KL Note that the marginal product of labor is positive if 1 2 L2 + 4KL > 0, that is, if L < 8K. Then, nd the derivative of the marginal production of labor with respect to L (L) L + 4K This is positive (and so marginal product of labor is increasing) if L < 4K but negative (marginal product of labor is decreasing) if L > 4K, showing that for this production function the marginal product of labor, MP L (L) 1 2 L2 + 4KL, declines forever once L > 4K. The following graphs production assuming K L3 + 2L 2 x Graph of x f L; K 1 6 L3 + 2KL 2 L 15

16 MPL L 5 10 Graph of MP L (L) 1 2 L2 + 4KL MPL' L 4 Graph L(L) L + 4K 23. Assume a production function x f (L; K) where x is units of output, L is units of labor and K is units of capital. A) De ne in words this production function. B) Now assume that it is the short-run production function such that K is xed, K, so that we can write the production function x f L; K. Prove that it is possible for the marginal product of labor to always be positive and declining. Explain your answer and show all of your work. A) The production function identi es the amount of output x that can be produced by employing L units of labor and K units of capital. B) All that is required is > 0 f(l;k) < 0. 2 For the Cobb-Douglas function f (L; K) AL K 1, with A > 0 and 0 < < 1, 16

17 @f (L; K) AL 1 K 1 > 2 f (L; K) 2 A ( 1) L 2 K 1 < 0 because ( 1) < 0 COST FUNCTIONS 24. Assume the Gomer Corporation produces product x and that its cost function is: c(x) ax 3 + bx 2 + A) What units is c (x) expressed in? B) What is the rm s marginal cost function? C) What is its average cost function? D) Now assume a 1, b 10, and d 50. Determine, using a derivative, whether average cost is increasing or decreasing at the output level x 4. E) Continue to assume a 1, b 10, and d 50. Determine, using a derivative, whether marginal cost is increasing or decreasing at the output level x 4. A) Monetary units. B) Marginal cost as a function of the output level is how much total minimum costs of production increase if output is marginally increased from its current level. In functional notation, MC (x) c 0 (x) 3ax 2 + 2bx + d is C) Average cost is total minimum cost divided by the number of units produced. That AC (x) c(x) x ax2 + bx + d To determine whether AC (x) is increasing or decreasing at a particular point nd dac(x) : If x 4, a 1, b decreasing. dac (x) 2ax + b 10, and d 50, then dac(x) 2 < 0, so at x 4 average cost is D) To determine whether M C (x) is increasing or decreasing at a particular point nd dmc(x) : 17

18 If x 4, a 1, b increasing. dmc (x) 6ax + 2b 10, and d 50, then dmc(x) 4 > 0, so at x 4 marginal cost is 25. Assume the cost function c c (x; w; r) x 0 1 w + 2 r + 3 w :5 r :5 where x is output, w is the wage rate, r is the rental price of capital, and i > 0, where i 0; 1; 2; 3. A) Determine what happens to the marginal cost of producing an additional unit of output, when the wage rate increases. Does it always increase, always decrease? B) If it always increases (or decreases), does it always increase (decrease) at an increasing rate. C) Does the marginal cost of production always increase as output increases? A) First determine the marginal cost of production Then check c 0 (x; w; 0 x w + 2 r + 3 w :5 @x 0 x :5 3 w :5 r :5 > That is, the marginal cost of production always increases when w increases. B) To determine whether it is increasing at an increasing rate need @w 0 x 0 1 :25 3 w 1:5 r :5 < The marginal cost of production increases as w increases, but at a decreasing rate. C) Need @x 0 ( 0 1) x w + 2 r + 3 w :5 r This derivative can be positive, negative, or zero depending on whether 0 is less than 1, greater than 1, or equal to 1. always increase as output increases. So, the answer is the marginal cost of production does not 18

19 27. Consider the cost function c (x; w; r) x wr 2, where x is output, w is the wage rate, r is the rental price of capital, and > 0. Determine the elasticity of cost with respect to output. One way to nd the elasticity, take the natural logarithm of the cost function: and then nd ln c ln x + ln w + 2 ln r %c (x; w; r) ln ln x which is the elasticity of cost with respect to output. Whenever output increases by 1%, minimum cost of production increases by %. Alternatively, you can nd the elasticity the more di cult way, %c (x; w; r) x wr 2 x wr 2 x 1 wr 2 x x wr 2 x wr 2 x wr The Gomer Corporation produces gubers and you are its production manager. Your cost function is c (x; w; r) x 0 h 1 w + 2 r i where 0 > 1, 1 ; 2 > 0, and 0 < < 1. Your job is to determine how much your conditional demand for labor will change, in percentage terms, if the price of labor, w, increases by one percent, everything else constant. Explain, in words, how to solve this problem. Determine the answer. Show all of your work and identify, in words each of the critical steps in you mathematical derivation. labor First, use Shepard s lemma to derive the conditional demand function (x; w; l c (x; w; r) x 0 1 w 1 19

20 Then, determine the percentage change in l c given a one percent change in w. way to proceed is to convert the conditional demand function into logarithmic form Therefore ln l c ln x 0 1 w 1 0 ln x + ln + ln 1 + ( 1) ln w %l c (ln l (ln w) 1 This elasticity is a constant independent of either input price. The easiest 29. Assume the cost function c c (x; w; r) x 0 1 w + 2 r + 3 w :5 r :5 where x is output, w is the wage rate, r is the rental price of capital, and i > 0 8i. A) Determine whether this rm s conditional demand for capital services always increases when the price of labor increases. B) Determine, in percentage terms, what happens to the conditional demand for capital services when output is increased by one percent. A) By Shepard s Lemma: kc d kc (x; w; r) (x; w; r) x :5 3 w :5 r This is the conditional demand function for capital services. Note that it depends on x, w, and r. What happens to this demand as the price of labor d (x; w; c (x; w; r) x 0 :25 3 w :5 r :5 If the price of labor increases, the conditional demand for capital services will always increase. B) Take the log of the conditional demand function for capital services So, ln k d c ln x 0 + ln 2 + :5 3 w :5 r :5 0 ln x + ln 2 + :5 3 w :5 r :5 %k d c ln kd ln x 0 20

21 That is, a 1% desired increase in output (everything else constant), will always cause the production manager to purchase 0 % more capital services. 30. De ne, in words, the production function x f(k; L). De ne, in words, the cost function c c(x; w; r). De ne, in words, the conditional demand function l l(x; w; r). 31. De ne the short-run as a situation where the rm is required to use K units of capital. Given this, de ne, both in words and in functional notation, the rm s short-run cost function. 21