Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Chapter 5 The Theory of Demand Solutions to Problems 5.6 Suzie purchases two goods, food and clothing. She has the utility function U ( x , y ) = xy , where x denotes the amount of food consumed and y the amount of clothing. The marginal utilities for this utility function are MU x = y and MU y = x a) Show that the equation for her demand curve for clothing is y = I/ (2 P y ). b) Is clothing a normal good? Draw her demand curve for clothing when the level o f income is I = 200. Label this demand curve D 1 . Draw the demand curve when I = 300 and label this demand curve D 2 c) What can be said about the cross - price elasticity of demand of food with respect to the price of clothing? 5.7 K a r l ’ s p re f e re n ce s o v e r h a m b u r g er s ( H ) a n d b ee r ( B ) a r e d e s c r i b e d b y t h e u t il i ty f u n c t i o n : U ( H , B ) = m i n ( 2 H , 3 B ). H i s m o n t h l y i n c o m e i s I d o l l a r s , a n d h e o n l y bu ys t h e s e t w o goo d s o u t of h i s i n c om e D e n o t e t h e p ric e of h a m bu r g er s b y P H a n d of b ee r b y P B a) D e r i ve K a r l ’s d e m a n d c u r ve f or b ee r as a f un c t i on of t h e e xog e n o u s va ri a b l e s b ) W h i c h a ff e c ts K a rl ’ s c o n s u m p t i on o f b e e r mo r e : a o n e d o ll ar i n cre a s e i n P H or a o n e d o l l ar i n cr e a s e i n P B ? 5.11 G i n g er ’ s u t i l i t y f un c t i on i s U ( x , y ) = x 2 y , w i th a ss o c i a t e d m a r g i n al u t i li ty f un c t i o n s M U x = 2 xy a n d M U y = x 2 Sh e h as i n c o me I = 240 a n d f a ce s p rice s P x = $8 a n d P y = $2. a) D e t e r m i n e G i n g er ’s o p t i mal b a sk e t g i v e n t h e s e p rice s a n d h e r i n c o m e b ) I f t h e p r i c e o f y i n cr e a s e s to $8 a n d G i n g er ’ s i n c o m e i s un c h a n g e d , w h at m us t t h e p r i c e of x fa l l to i n o r d e r for h e r to b e e xa c t l y as w e l l o f f as b e f o r e t h e c h a n ge i n P y ? Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual 5.1 2 Ann’s utility function is U ( x , y ) = x + y , with associated marginal utility functions MU x = 1 and MU y = 1. Ann has income I = 4. a) Determine all optimal baskets given that she faces prices P x = 1 and P y = 1. b) Determine all optimal baskets given that she faces prices P x = 1 and P y = 2. c) What is demand for y when P x = 1 and P y = 1? What is demand for y when P x = 1 and P y > 1? What is demand for y when P x = 1 and P y < 1? Plot Ann’s demand for y as a function of P y d) Repeat the exercises in a), b) and c) for U ( x , y ) = 2 x + y , with associated marginal utility functions MU x = 2 and MU y = 1, and with the same level of income. 5.19 J i m’s p re f ere n ce s ov e r c oo k ie s ( x) a n d o t h e r goo d s ( y ) a r e g i v e n b y U ( x , y ) = x y w i th a ss o c i at e d m a r g i n al u t i l i t y f un c t i o n s M U x = y a n d M U y = x H i s i n c o m e i s $20. a) F i n d J i m ’ s d e m a n d s c h e d u l e f or x w h e n p r i c e of y i s P y = $1. b ) I l l us t r ate g r a ph i c a l l y t h e c h a n ge i n c o n s u m e r su r p l u s wh e n t h e p r i c e of x i n c r e a s e s f r om $1 to $ 2 5.20 Lo u ’ s p r e f e r e n ce s ov e r p izz a ( x ) a n d o t h e r goo d s ( y ) a r e g i v e n b y U ( x , y ) = x y , w i t h a ss o c i at e d m a r g i n al u t i l i t ie s M U x = y a n d M U y = x Hi s i n c o m e i s $120. a) C a l c u l ate h i s o p t i m al b a s k e t wh e n P x = 4 a n d P y = 1. b ) C a lc u l ate h i s i n c ome a n d subs t i t u t i o n e ff ec t s of a d ecre a s e i n t h e p ric e of f ood to $3. c ) C a l c u l ate t h e c o m p e ns at i n g va r i at i o n of t h e p ric e c h a n g e d ) C a lc u l ate t h e e qu i va le n t va r i a t i on of t h e p r i c e c h a n g e 5.24 T h er e a r e t w o c o nsu m er s on t h e m a r k e t : J i m a n d D o nn a. J i m ’ s u t i l i t y f un c t i o n i s U ( x , y ) = x y , w i t h a s s o c i a t e d m a r g i n al u t i l i t y f un c t i o n s M U x = y a n d M U y = x D o n n a ’ s u t i l i ty f u n c t i on i s U ( x , y ) = x 2 y , w i t h a s s o c i a t e d m a r g i n al u t i l i t y f un c t i o n s M U x = 2 xy a n d M U y = x 2 In c o m e of J i m i s I J = 100 a n d i n c ome of D o n n a i s I D = 150. a) F i n d o p t i m al b a sk e t s of J i m a n d D o nn a wh e n p ric e of y i s P y = 1 a n d p ric e of x i s P b ) O n s e p a r ate g r a ph s p l ot J i m ’s a n d D o nn a ’ s d e m a n d s c h e d u l e f or x f or a l l va l u e s o f P c ) C om pu te a n d p l o t agg r e ga t e d e m a n d w h e n J i m a n d D o nn a a r e t h e o n l y c o nsu m er s d ) Pl ot agg r e gate d e ma n d wh e n t h e r e i s o n e m o r e c o nsu m e r t h at h as i d e n t ic al u t il i t y f u n c t i on a n d i n c ome as D o nn a. Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Copyright © 2020 John Wiley & Sons, Inc. Chapter 5 - 4 5.30 C o ns i d e r N oa h ’s p re f ere n c e s for lei su r e ( L ) a n d o t h e r goo d s ( Y ) , U ( L , Y ) = √ L + √ Y T h e a ss o c i a t e d ma r g i n al u t il i t i e s a r e M U L = 1 / ( 2 √ L ) a n d M U Y = 1 / ( 2 √ Y ) Supp o s e t h at P Y = $1. I s N oa h ’ s s u pp l y of l a b or b a c kw a r d b e nd i n g? $ 6.12. Su p p o s e t h e p r o d u c t i on f u n c t i on i s g i v e n b y t h e f o l l o w i n g e qu a t i on ( wh e r e a a n d b a r e p o s i t i ve c o ns t a n t s ): Q = aL + b K W h at i s t h e m a r g i n al r a t e of t ec hn ic al sub s t i t u t i on of l a b or for c a p i t al ( M R T S L , K ) at a n y p o i n t a l o n g an i s o qu a n t? 6.9 S u p p o s e t h e p r o d u c t i on f u n c t i on for a u t omo b i l e s i s " # $ w h e r e Q i s t h e q u a n t i t y of a u tom o b i l e s p r o d u c e d p e r y e a r , L i s t h e q u a n t i t y of l a b or ! ( m a n - h o u r s ) a n d K i s t h e q u a n t i t y of c a p i t al ( m a c h i n e - h o u r s ) a) S k e t c h t h e i s o qu a n t c o r r e s p o n d i n g t o a q u a n t i t y of Q = 100? b ) W h at i s t h e g e n e r al e q u a t i on f or t h e i s o q u a n t c o r r e s p o n d i n g t o a n y l e v e l of o u t p u t Q ? c ) D o e s t h e i s o q u a n t e x h i b i t d i m i n i s h i n g m a r g i n al r ate of t e c h n i c al s u b s t i t u t i o n ? 6.10. S u p p o s e t h e p r o d u c t i on f u n c t i on i s g i v e n b y t h e e q u a t i on Q = L √ K G r a p h t h e i s o q u a n ts c o r r e s p o n d i n g t o Q = 10, Q = 20, a n d Q = 50. D o t h e s e i s o q u a n t s e x h i b i t d i m i n i s h i n g m a r g i n al r a t e of t ec hn i c al s u b s t i t u t i o n ? Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual C opyright © 20 20 John Wil ey & Sons, Inc. Chapter 6 - 3 6.14 Consider the following production functions and their associated marginal products. For each production function, determine the marginal rate of technical substitution of labor for capital, and indicate whether the isoquants for the production function exhibit diminishing marginal rate of substitution. Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual C opyright © 20 20 John Wil ey & Sons, Inc. Chapter 6 - 4 6.1 7 Let B be the number of bicycles produced from F bicycle frames and T tires. Every bicycle needs exactly two tires and one frame. a ) Draw the isoquants for bicycle production. b) Write a mathematical expression fo r the production function for bicycles. 6.1 9 What can you say about the returns to scale of the linear pro duction function Q = aK + bL , where a and b are positive constants? 6. 20 What can you say about the returns to scale of the Leontief production function Q = min( aK , bL ), where a and b are positive constants? 6.23. A f i r m’s p r o du c t i on f un c t i on i s Q = 5 L 2 / 3 K 1 / 3 w i t h M P K = ( 5 / 3 ) L 2 / 3 K − 2 / 3 a n d M P L = ( 1 0 / 3 ) L − 1 / 3 K 1 / 3 a) D o e s t h i s p r o du c t i on f un c t i on e x h i b i t c o n s t a n t , i n cre a s i n g, or d ecr e a s i n g re t u r n s t o s c a l e ? b ) W h at i s t h e m a r g i n al r ate of t ec h n ic al s ubs t i t u t i on of L f or K f or t h i s p r o du c t i on f u n c t i o n ? Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual C opyright © 20 20 John Wil ey & Sons, Inc. Chapter 6 - 6 Intuitively, in this production functio n, while you can increase the 𝐾 and 𝐿 inputs, you cannot increase the constant portion. So output cannot go up by as much as the inputs. 6.25 Consider the following production functions and their associated marginal products. For each production functio n, indicate whether (a) the marginal product of each input is diminishing, constant, or increasing in the quantity of that input; (b) the production function exhibits decreasing, constant, or increasing returns to scale. Production function MP L MP K Marginal product of labor? Marginal product of capital? Returns to scale? 𝑄 = 𝐿 + 𝐾 𝑀 𝑃 " = 1 𝑀 𝑃 # = 1 Constant in L Constant in K Constant 𝑄 = √ 𝐿𝐾 𝑀 𝑃 " = 1 2 √ 𝐾 √ 𝐿 𝑀 𝑃 # = 1 2 √ 𝐿 √ 𝐾 Diminishing in L Diminishing in K Constant 𝑄 = √ 𝐿 + √ 𝐾 𝑀 𝑃 " = 1 2 1 √ 𝐿 𝑀 𝑃 # = 1 2 1 √ 𝐾 Diminishing in L Diminishing in K Decreasing 𝑄 = 𝐿 - 𝐾 - 𝑀 𝑃 " = 3 𝐿 & 𝐾 - 𝑀 𝑃 # = 3 𝐿 - 𝐾 & Increasing in L Increasing in K Increasing 𝑄 = 𝐿𝐾 𝑀 𝑃 " = 𝐾 𝑀 𝑃 # = 𝐿 Constant in L Constant in K Increasing Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual C opyright © 20 20 John Wil ey & Sons, Inc. Chapter 6 - 1 Chapter 6 Inputs and Production Functions Solutions to Problems 6.2. A firm is required to produce 100 units of output using quantities of labor and capital ( L , K ) = (7, 6). For each of the following production functions, state whether it is possible to produce the required output with the given input combination. If it is possible, state whether the input combination is technically efficient or inefficient. a) Q = 7 L + 8 K b) Q = 20 √ KL c) Q = min(16 L , 20 K ) d) Q = 2( KL + L + 1) 6.3. For the production function Q = 6 L 2 − L 3 , fill in the following table and state how much the firm should produce so that: a) average product is maximized b) marginal pr oduct is maximized c) total product is maximized d) average product is zero C opyright © 20 20 John Wil ey & Sons, Inc. Chapter 6 - 4 c ) F rom t he T a b l e i t i s c l ea r t h a t t o t a l prod u c t i s m a x i miz e d w h e n L = 4. d) A v e r a ge P rod u c t w i l l be z e ro on l y w h e n T o t a l P r od u c t i s z e r o. T h i s h a p p e ns w h e n L = 6. 6.4. S u p p o s e t h at t h e p r o d u c t i on f u n c t i on f or D V D s i s g i v e n b y Q = K L 2 − L 3 , w h e r e Q i s t h e n u m b e r of d i s k s p r o d u c e d p e r y e a r , K i s m a c h i n e - h o u r s of c a p i t a l , a n d L i s m a n - h o u r s o f l a b o r a) S u pp o s e K = 600. F i n d t h e t ot al p r o d u c t f u n c t i on a n d g r a p h i t ov e r t h e r a n ge L = 0 to L = 500. T h e n s k e t c h t h e g r a p h s of t h e av e r age a n d ma r g i n al p r o d u c t f u n c t i o n s A t w h at l e v e l of l a b or L d o e s t h e a v e r age p r o d u c t c u r ve a p p e ar t o r e a c h i t s ma x i m u m? A t w h at l e v e l d o e s t h e m a r g i n al p r o d u c t c u r ve a p p e ar t o r e a c h i t s m ax i m u m ? b ) R e p l i c a t e t h e a n a l y s i s i n (a) for t h e c a s e i n w h i c h K = 1200. c ) W h e n e i t h e r K = 600 or K = 1200, d o e s t h e t o t al p r o d u c t f u n c t i on h ave a r e g i on of i n c r e a s i n g m a r g i n al r e t u r n s ? 6.5. A r e t h e f o l l o w i n g s t a t e m e n t s c o r r e c t or i n c o r r e c t ? a) I f av e r age p r o d u c t i s i n c r e a s i n g, ma r g i n al p r o d u c t m u s t b e l e s s t h an av e r age p r o d u c t. b ) I f ma r g i n al p r o d u c t i s n e ga t i v e , av e r age p r o d u c t m u s t b e n e ga t i v e c ) I f a v e r age p r o d u c t i s p o s i t i v e , t otal p r o d u c t m u s t b e r i s i n g. d ) I f t o t al p r o d u c t i s i n c r e a s i n g, m a r g i n al p r o d u c t m u s t a l s o b e i n c r e a s i n g. Besanko & Braeutigam – Microeconomi cs, 6 th edition Solutions Manual Chapter 7 Costs and Cost Minimization 7.1 5 Ajax, Inc. assembles gadgets. It can make each gadget either by hand or with a special gadget - making machine. Each gadget can be assembled in 15 minutes by a worker or in 5 minutes by the machine. The firm can a lso assemble some of the gadgets by hand and some with machines. Both types of work are perfect substitutes, and they are the only inputs necessary to produce t he gadgets. a) It costs the firm $30 per hour to use the machine and $10 per hour to hire a work er. The firm wants to produce 120 gadgets. What are the cost - minimizing input quantities? Illustrate your answer with a clearly labeled graph. b) What are the c ost - minimizing input quantities if it costs the firm $30 per hour to use the machine, and $10 pe r hour to hire a worker? Illustrate your answer with a graph. c) Write down the equation of the firm’s production function for the firm. Let G be the number of gadgets assembled, M the number of hour the machines are used, and L the number of hours of labo r. Besanko & Braeutigam – Microeconomi cs, 6 th edition Solutions Manual Copyright © 20 20 John Wiley & Sons, Inc. Chapter 7 - 3 7.1 6 A construction company has two types of employees: skilled and unskilled. A skilled employee can build 1 yard of a brick wall in one hour. A n unskilled employee needs twice as much time to build the same wall. The hourly wage of a skilled employee is $15. The hourly wage of an unskilled employee is $8. a) Write down a production function with labor. The inputs are the number of hours of skille d workers, L S , the number of hours worked by unskilled employees, L U , and the output is the number of yards of brick wall, Q b) The firm needs to build 100 yards of a wall. Sketch the isoquant that shows all combinations of skilled and unskilled labor tha t result in building 100 yards of the wall. c) What is the cost - minimizing way to build 100 yards of a wall? Illustrate your answer on the graph in part (b). 7.22. S u p p o s e a p r o d u c t i o n f un c t i on i s g i v e n b y Q = K + L —t h at i s , t h e i np u t s a r e p e r f e c t s u b s t i t u t e s F or t h i s p r o d u c t i on f u n c t i o n , M P L = 1 a n d M P K = 1. D r aw a g r a p h of t h e d e ma n d c u r ve f or l a b or w h e n t h e f ir m w a n t s t o p r o du c e 10 un i ts of o u t p u t a n d t h e p ri c e of c a p i tal s e r v i ce s i s $1 p e r u n i t ( Q = 10 a n d r = 1). Besanko & Braeutigam – Microeconomi cs, 6 th edition Solutions Manual Copyright © 20 20 John Wiley & Sons, Inc. Chapter 7 - 5 7.2 5 A firm has the production function Q = LK . For this production function, MP L = K and MP K = L . The firm initial ly faces input prices w = $1 and r = $1 and is required to produce Q = 100 units. Later the price of labor w goes up to $4. Find the optimal input combinations for each set of prices and use these to calculate the firm’s price elasticity of demand for labo r over this range of prices. 7.2 6 A bicycle is assembled out of a bicycle frame and two whe els. a) Write down a production function of a firm that produces bicycles out of frames and wheels. No assembly is required by th e firm, so labor is not an input in this case. Sketch the isoquant that shows all combinations of frames and wheels that result in producing 100 bicycles. b) Suppose that initially the price of a frame is $100 and the price of a wheel is $50. On the graph you drew for part (a), show the choices of frames and wheels that minimize the cost of producing 100 bicycles, and draw the iso cost line through the optimal basket. Then repeat the exercise if the price of a frame rises to $200, while the price of a wheel remains $50. 8.12. A f i r m p r o du c e s a p r o du c t w i th l a b or a n d c a p i t a l I t s p r o du c t i on f un c t i on i s d e s cri b e d b y Q = m i n ( L , K ) L e t w a n d r b e t h e p r i c e s of l a b or a n d c a p i t a l , re sp ec t i v e l y. a) F i n d t h e e qu a t i on for t h e f i r m ’s l o n g - r u n t otal c o s t c u r ve as a f un c t i on of qu a n t i t y Q a n d i n pu t p ri ce s , w a n d r b ) Fi n d t h e s o l u t i on t o t h e f ir m ’s sh o r t - r u n c o s t m i n i m iz a t i on p r o b le m wh e n c a p i t al i s f i x e d at a qu a n t i ty of 5 un i t s ( i e ., K = 5). D eri ve t h e e qu a t i on f or t h e f i r m’ s s h o r t - r u n t o t al c o s t c u r ve as a f u n c t i on of qu a n t i t y Q G r a p h t h i s c u r ve t og e t h e r w i t h t h e l o n g - r u n t o t al c o s t c u r ve for w = 1 a n d r = 1. c ) H ow d o t h e g r a p h s of t h e l o n g - r u n a n d s h o r t - r u n t o t al c o s t c u r v e s c h a n ge wh e n w = 1 a n d r = 2? d ) H ow d o t h e g r a ph s of t h e l o n g - r u n a n d s h o r t - r u n t otal c o s t c u r v e s c h a n ge w h e n w = 2 a n d r = 1? Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Chapter 7 Costs and Cost Minimization Solutions to Problems 7. 9 Supp ose the production of airframes is characterized by a Cobb – Douglas production function: Q = LK The marginal products for this production function are MP L = K and MP K = L . Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost - minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes. Use the metho dology that you have learned from cost minimization hand out. The cost - minimizing quantities of labor and capital to produce 121,000 airframes are 𝐾 = 1 , 100 and 𝐿 = 110 7. 10 The processing of payroll for the 10,000 workers in a large firm can either be done using 1 hour of computer time (denoted by K ) and no clerks or with 10 hours of clerical time (denoted by L ) and no computer time. Computers and clerks are perfect substitutes; for example, the firm could also process its payroll using 1/2 hour of computer time and 5 hours of clerical time. a) Sketch the isoquant that shows all combinations of clerical time and computer time that allows the firm to process the payroll for 10,000 workers. b) Suppose computer time costs $5 per hour and clerical time costs $7.5 0 per hour. What are the cost - minimizing choices of L and K ? What is the minimized total cost of processing the payroll? c) Suppose the price of clerical time remains at $7.50 per hour. How high would the price of an hour of computer time have to be before the firm would find it worthwhile to use only clerks to process the payroll? 7.11. A f i r m p r o d u c e s an o u t pu t w i th t h e p r o du c t i on f un c t i on Q = K L , wh e r e Q i s t h e nu m b e r of un i ts of o u t p u t p e r h o u r wh e n t h e f ir m us e s K ma c h i n e s a n d h ire s L w o r k er s e a c h h o u r T h e m a r g i n al p r o du c t s f or t h i s p r o du c t i on f u n c t i on a r e M P K = L a n d M P L = K T h e f a c tor p r i c e of K i s 4 a n d t h e f a c tor p ric e of L i s 2. T h e f i r m i s c u rre n t l y us i n g K = 16 a n d j us t e n o u gh L to p r o du c e Q = 32. H ow m u c h c o u l d t h e f ir m s ave i f i t w er e to a d j u s t K a n d L to p r o d u c e 32 u n i t s i n t h e l e a s t c o s t l y w ay p o s s i b le ? 7.21. A f i r m ’ s p r o d u c t i on f u n c t i on i s Q = m i n ( K , 2 L ) , w h e r e Q i s t h e n u m b e r of u n i t s of o u t pu t p r o d u c e d u s i n g K u n i t s of c a p i t al a n d L u n i t s of l a b o r T h e f a c t or p ri c e s a r e w = 4 ( f or l a b o r ) a n d r = 1 (f or c a p i t a l ) O n an o p t i m al c h o i c e d i ag r am w i th L on t h e h o r i z o n t al axis and K on the vertical axis, draw the isoquant for Q = 12, indicate the optimal choices of K and L on that isoquant, and calculate the total cost. Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Copyright © 2020 John Wiley & Sons, Inc. Chapter 8 - 1 Chapter 8 Cost Curves 8.5. A firm produces a product with labor and capital, and its production function is described by Q = LK . The marginal products associated with this production function are MP L = K and MP K = L . Suppose that the price of labor equals 2 and the price of capital equals 1. Derive the equations for the long - run total cost curve and the long - run average cost curve. 8. 11 A firm produces a product with labor and capital as inputs. The production function is described by Q = LK . The marginal products associated with this production function are MP L = K and MP K = L . Let w = 1 and r = 1 be the prices of labor and capital, resp ectively. a) Find the equation for the firm’s long - run total cost curve as a function of quantity Q b) Solve the firm’s short - run cost - minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive the equation for the firm’s sh ort - run total cost curve as a function of quantity Q and graph it together with the long - run total cost curve. c) How do the graphs of the long - run and short - run total cost curves change when w = 1 and r = 4? d) How do the graphs of the long - run and short - run total cost curves change when w = 4 and r = 1? Copyright © 2020 John Wiley & Sons, Inc. Chapter 8 - 5 8.1 3 A firm produces a product with labor and capital. Its production function is described by Q = L + K . The marginal products associated with this production function are MP L = 1 and MP K = 1. Let w = 1 and r = 1 be the prices of labor and capital, respectively. a) Find the equation for the firm’s long - run total cost curve as a function of quantity Q when the prices labor and capital are w = 1 and r = 1. b) Find the solution to the firm’s short - run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5), and w = 1 and r = 1. Derive the equation for the firm’s short - run total cost curve as a function of quantity Q and graph it together with the long - run total cost curve. c) How do the graphs of the short - run and long - run total cost curves change when w = 1 and r = 2? d) How do the graphs of the short - run and long - run total cost curves change when w = 2 and r = 1? d) A v e r a ge P rodu c t w il l be ze ro on l y w h e n T o ta l P rodu c t i s ze r o. T h i s h a p p e ns w h e n L = 6. 6.4. Su p p o s e t h at t h e p r o d u c t i on f u n c t i on f or D V D s i s g i v e n b y Q = K L 2 − L 3 , wh er e Q i s t h e nu m b e r of d i sk s p r o d u c e d p e r y e a r , K i s m a c h i n e - h o u r s of c a p i t a l , a n d L i s m a n - h o u r s of l a b o r a) Su p p o s e K = 600. F i n d t h e to t al p r o du c t f un c t i on a n d g r a p h i t ov e r t h e r a n ge L = 0 to L = 500. T h e n sk e t c h t h e g r a ph s of t h e av er age a n d m a r g i n al p r o du c t f un c t i o n s A t w h at l e v e l of l a b or L d o e s t h e av er age p r o d u c t c u r ve a pp e ar t o re a c h i t s max i m u m? A t wh at l e v e l d o e s t h e ma r g i n al p r o du c t c u r ve a p p e ar t o re a c h i t s m ax i m u m ? b ) R e p l i c ate t h e a n a l y s i s i n (a) for t h e c a s e i n wh ic h K = 1200. c ) W h e n ei t h e r K = 600 or K = 1200, d o e s t h e t o t al p r o d u c t f un c t i on h ave a re g i on of i n cr e a s i n g ma r g i n al r e t u r ns ? Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Copyright © 2020 John Wiley & Sons, Inc. Chapter 8 - 8 8.1 4 Consider a production function of two inputs, labor and capital, given by Q = ( √ L + √ K ) 2 . The marginal products associated with this production function are as follows: Let w = 2 and r = 1. a) Suppose the firm is required to produce Q units of output. Sh ow how the cost - minimizing quantity of labor depends on the quantity Q . Show how the cost - minimizing quantity of capital depends on the quantity Q b) Find the equation of the firm’s long - run total cost curve. To answer this problem use row 4 of Table 494A in SE Chapter I a ) Plug in (w,r) = (2, 1 ) in to row 4 of Table 494A to get 𝐿 = & ) and 𝐾 = * & ) b) 𝑐 = ! & + 8.1 5 Tricycles must be produced with 3 wheels and 1 frame for each tricycle. Let Q be the number of tricycles, W be the number of wheels, and F be the number of frames. The price of a wheel is P W and the price of a frame is P F a) What is the long - run total cost function for producing tricycles, TC ( Q,P W , P F )? b) What is the production function for tricycles, Q ( F,W )? a ) Each tricycle requires the purchase of three wheels at price P W and one frame at price P F Thus, TC ( Q , P W , P F ) = Q (3 P W + P F ). b ) Three wheels and one frame are perfect complements in production. Thus the production function is Q ( F, W ) = min{ F , (1/3) W }. Notice that ( F, W ) = (1, 3) yields Q = 1, (F, W ) = (2, 6) yields Q = 2, etc. Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Copyright © 2020 John Wiley & Sons, Inc. Chapter 9 - 1 Chapter 9 Perfectly Competitive Markets Solutions to Problems 9.6. A bicycle - repair shop charges the competitive market price of $10 per bike repaired. The firm’s short - run total cost is given by ST C ( Q ) = Q 2 / 2, and the associated marginal cost curve is SMC ( Q ) = Q a) What quantity should the firm produce if it wants to maximize its profit? b) Draw the shop’s total revenue and total cost curves, and graph the total profit function on the same diagram. Using your graph, state (approximately) the profit - maximizing quantity in each case. In answering to this question please disre gard STC (Q) and SMC (Q) notation. For the purp ose of this question , con sider STC(Q) as the cost function C(Q) you have learned in Week s 10 and 11. SMC(Q ) is the margi nal cost MC(Q)= C ′ ( Q ) a ) Since the firm is producing in a perfectly competitive market, the firm views the output price as exogenous. It should produce up to the point at which P = SMC(Q) , that is, so that 10 = Q . So it should produce 10 units of output. b) The total cost function increases in Q, and at an increasing rate. Total Profit at first increases in Q and then decreases. From the graph, it appears that Profit is maximized when Q is about 10, which we found in (a). Total Profit TR TC Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Copyright © 2020 John Wiley & Sons, Inc. Chapter 9 - 2 9.1 8 A firm in a competitive industry produces its output in two plants. Its total cost of producing Q 1 units from the first plant is TC 1 = ( Q 1 ) 2 , and the marginal cost at this plant is MC 1 = 2 Q 1 . The firm’s total cost of producing Q 2 units from the second plant is TC 2 = 2( Q 2 ) 2 ; the marginal cost at this plant is MC 2 = 4 Q 2 . The price in the market is P. What fraction of the firm’s total supply will be produced at plant 2? Given a market price P, the firm will produce from each plant so that MC = P. The profit maximizing quantity supplied at plant 1 will be 2Q 1 = P, or Q 1 = P/2. The profit maximizing quantity supplied at plant 2 will be 4Q 2 = P, or Q 2 = P/4. The quantity supplied by the whole fir m will Q Firm = Q 1 + Q 2 . Thus Q Firm = 3P/4. So 1/3 of the firm’s total production will come from plant 2. 9.2 8 The long - run total cost function for producers of mineral water is TC ( Q ) = c Q , where Q is the output of an individual firm expressed as thousands of liters per ye ar. The market demand curve is D ( P ) = a − b P . Find the long - run equilibrium price and quantity in terms of a, b, and c. Can you determine the equilibrium number of firms? If so, what is it? If not, why not? For this total cost function, 𝑀𝐶 = 𝑐 Since each firm will supply where 𝑃 = 𝑀𝐶 , in equilibrium 𝑃 = 𝑐 If in equilibrium 𝑃 = 𝑐 , 𝐷 ( 𝑃 ) = 𝑎 − 𝑏𝑐 Equilibrium market quantity is 𝑎 − 𝑏𝑐 In order to determine the number of firms we need to know the quantity that each individual firm will produc e. In this case marginal cost is constant implying perfectly elastic supply. Thus, at 𝑃 = 𝑐 a firm may produce any quantity. Therefore, the number of firms cannot be determined. 9.3 3 A price - taking firm’s supp ly curve is s ( P ) = 10 P . What is the producer surplus for this firm if the market price is $20? By how much does producer surplus change when the market price increases from $20 to $21? The solution is shown in the figure below. The producer surplus at a price of $20 is equal to the area of triangle A , or (1/2)(20)(200) = $2,000. When the price increases to $21, producer surplus increases by area B ($200) plus area C ($5), or $205. Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Chapter 10 Competitive Markets: Applications Solutions to Problems 10.1 7 Suppose the market for corn in Pulmonia is competitive. No imports and exports are possi ble. The demand curve is Q d = 10 − P d , where, Q d is the quantity demanded (in millions of bushels) when the price consumers pay is P d . The supply curve is where Q s is the quantity supplied (in millions of bushels) when the price producers receive is P s a) What are the equilibrium price and quantity? b) At the equilibrium in part (a), what is consumer surplus? producer surplus? deadweight loss? Show all of these graphically. c) Suppose the government imposes an excise tax of $2 per unit to raise governmen t revenues. What will the new equilibrium quantity be? What price will buyers pay? What price will sellers receive? d) At the equilibrium in part (c), what is consumer surplus? producer surplus? the impact on the government budget (here a positive number, the government tax receipts)? deadweight loss? Show all of these graphically. e) Suppose the government has a change of heart about the importance of corn revenues to the happiness of the Pulmonian farmers. The tax is removed, and a subsidy of $1 per unit is granted to corn producers. What will the equilibrium quantity be? What price will the buyer pay? What amount (including the subsidy) will corn farmers receive? f ) At the equilibrium in part (e), what is consumer surplus? producer surplus? What will be t he total cost to the government? deadweight loss? Show all of these graphically. g) Verify that for your answers to parts (b), (d), and (f) the following sum is always the same: consumer surplus + producer surplus + budgetary impact + deadweight loss. Why is the sum equal in all three cases? = Besanko & Braeutigam – Microeconomics, 6 th edition Solutions Manual Copyright © 2020 John Wiley & Sons, Inc. Chapter 10 - 4 10.1 9 In a perfectly competitive market, the market demand curve is Q d = 10 − P d , and the market supply curve is Q s = 1 5 P s a) Verify that the market equilibrium price and quantity in the absence of government intervention are P d = P s = 4 and Q d = Q s = 6. b) Consider two possible government interventions: (1) A price ceiling of $1 per unit; (2) a subsidy of $5 per unit paid to producers. Verify that the equilibrium market price paid by consumers under the subsidy equals $1, the same as the price ceiling. Are the quantities supplied and demanded the same under each government intervention? c) How will consumer surplus differ in t hese different government interventions? d) For which form of intervention will we expect the product to be purchased by consumers with the highest willingness to pay? e) Which government intervention results in the lower deadweight loss and why?