TUTORIAL 5 - WEDNESDAY - JUNE 4, 2014 - 4PM - 5PM ...
TUTORIAL 5 - WEDNESDAY - JUNE 4, 2014 - 4PM - 5PM
QUESTION 6.2 Frisbees are produced according to the production functions q = 2K + L where q = Output of Frisbees per hour, K = Capital input per hour, and L = Labour input per hour. (a) If K = 10, how much L is needed to produced 100 Frisbees per hour? (b) If K = 25, how much L is needed to produced 100 Frisbees per hour? (c) Graph the q = 100 isoquant. Indicate the points on that isoquant defined in part (a) and part (b). What is the RTS along the isoquant? Explain why the RTS is the same at every point on the isoquant. (d) Graph the q = 50 and q = 200 isoquants for this production function also. Describe the shape of the entire isoquant map. (e) Suppose technical progress resulted in the production function for Frisbees becoming q = 3K + 1.5L. Answer part (a) through part (d) for this new production function and discuss how it compares to q = 2K + L. QUESTION 6.3 Digging clams by hand in Sunset Bay requires only labour input. The total number of clams obtained per hour (q) is given by 100(L)(1/2) where L is labour input per hour. (a) Graph the relationship between q and L. (b) What is the average productivity of labour (output per unit of labour input) in Sunset Bay? Graph this relationship and show that output per unit of labour input diminishes for increases in labour input. (c) The marginal productivity of labour in Sunset Bay is given by MPL = 50/(L)(1/2). Graph this relationship and show that labour's marginal productivity is less than average productivity for all values of L. Explain why this is so.
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QUESTION 6.5 Grapes must be harvested by hand. This production function is characterized by fixed proportions – each worker must have one pair of stem clippers to produce any output. A skilled worker with clippers can harvest 50 pounds of grapes per hour. (a) Sketch the grape production isoquants for q = 500, q = 1,000, and q = 1,500 and indicate where on these isoquants firms are likely to operate. (b) Suppose a vineyard owner currently has 20 clippers. If the owner wishes to utilize fully these clippers, how many workers should be hired? What should grape output be? (c) Do you think the choices described in part (b) are necessarily profitmaximizing? Why might the owner hire fewer workers than indicated in this part? (d) Ambidextrous harvesters can use two clippers – one in each hand – to produce 75 pounds of grapes per hour. Draw an isoquant map (for q = 500, 1,000, and 1,500) for ambidextrous harvesters. Describe in general terms the considerations that would enter into an owner's decisions to hire such harvesters. QUESTION 6.7 The production function q = KL where 0 < , < 1 is called a CobbDouglas production function. This function is widely used in economic research. Using the function, show the following: (a) The production function q = 10(KL)(1/2) is a special case of the CobbDouglas production function. (b) If = 1, a doubling of K and L will double q. (c) If < 1, a doubling of K and L will less than double q. (d) If > 1, a doubling of K and L will more than double q. (e) Using the results from part (b) through part (d), what can you say about the returns to scale exhibited by the Cobb-Douglas function?
Page 3 QUESTION 6.8 For the Cobb-Douglas production function in QUESTION 6.7, it can be shown (using calculus) that MPK = KL MPL = KL If the Cobb—Douglas exhibits constant returns to scale ( = 1), show that (a) Both marginal productivities are diminishing. (b) The RTS for this function is given by RTS = k/L (c) The function exhibits a diminishing RTS. QUESTION 6.9 The production function for puffed rice is given by q = 100(KL)(1/2) where q is the number of boxes produced per hour, K is the number of puffing guns used each hour, and L is the number of workers hired each hour. (a) Calculate the q = 1,000 isoquant for this production function and show it on a graph. (b) If K = 10, how many workers are required to produce q = 1,000? What is the average productivity of puffed-rice workers? (c) Suppose technical progress shifts the production function q = 100(KL)(1/2 to q = 200(KL)(1/2. Answer parts (a) and (b) for this new situation.
QUESTION 6.2 Frisbees are produced according to the production functions q = 2K + L where q = Output of Frisbees per hour, K = Capital input per hour, and L = Labour input per hour. (a) If K = 10, how much L is needed to produced 100 Frisbees per hour? (b) If K = 25, how much L is needed to produced 100 Frisbees per hour? (c) Graph the q = 100 isoquant. Indicate the points on that isoquant defined in part (a) and part (b). What is the RTS along the isoquant? Explain why the RTS is the same at every point on the isoquant. (d) Graph the q = 50 and q = 200 isoquants for this production function also. Describe the shape of the entire isoquant map. (e) Suppose technical progress resulted in the production function for Frisbees becoming q = 3K + 1.5L. Answer part (a) through part (d) for this new production function and discuss how it compares to q = 2K + L. QUESTION 6.3 Digging clams by hand in Sunset Bay requires only labour input. The total number of clams obtained per hour (q) is given by 100(L)(1/2) where L is labour input per hour. (a) Graph the relationship between q and L. (b) What is the average productivity of labour (output per unit of labour input) in Sunset Bay? Graph this relationship and show that output per unit of labour input diminishes for increases in labour input. (c) The marginal productivity of labour in Sunset Bay is given by MPL = 50/(L)(1/2). Graph this relationship and show that labour's marginal productivity is less than average productivity for all values of L. Explain why this is so.
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QUESTION 6.5 Grapes must be harvested by hand. This production function is characterized by fixed proportions – each worker must have one pair of stem clippers to produce any output. A skilled worker with clippers can harvest 50 pounds of grapes per hour. (a) Sketch the grape production isoquants for q = 500, q = 1,000, and q = 1,500 and indicate where on these isoquants firms are likely to operate. (b) Suppose a vineyard owner currently has 20 clippers. If the owner wishes to utilize fully these clippers, how many workers should be hired? What should grape output be? (c) Do you think the choices described in part (b) are necessarily profitmaximizing? Why might the owner hire fewer workers than indicated in this part? (d) Ambidextrous harvesters can use two clippers – one in each hand – to produce 75 pounds of grapes per hour. Draw an isoquant map (for q = 500, 1,000, and 1,500) for ambidextrous harvesters. Describe in general terms the considerations that would enter into an owner's decisions to hire such harvesters. QUESTION 6.7 The production function q = KL where 0 < , < 1 is called a CobbDouglas production function. This function is widely used in economic research. Using the function, show the following: (a) The production function q = 10(KL)(1/2) is a special case of the CobbDouglas production function. (b) If = 1, a doubling of K and L will double q. (c) If < 1, a doubling of K and L will less than double q. (d) If > 1, a doubling of K and L will more than double q. (e) Using the results from part (b) through part (d), what can you say about the returns to scale exhibited by the Cobb-Douglas function?
Page 3 QUESTION 6.8 For the Cobb-Douglas production function in QUESTION 6.7, it can be shown (using calculus) that MPK = KL MPL = KL If the Cobb—Douglas exhibits constant returns to scale ( = 1), show that (a) Both marginal productivities are diminishing. (b) The RTS for this function is given by RTS = k/L (c) The function exhibits a diminishing RTS. QUESTION 6.9 The production function for puffed rice is given by q = 100(KL)(1/2) where q is the number of boxes produced per hour, K is the number of puffing guns used each hour, and L is the number of workers hired each hour. (a) Calculate the q = 1,000 isoquant for this production function and show it on a graph. (b) If K = 10, how many workers are required to produce q = 1,000? What is the average productivity of puffed-rice workers? (c) Suppose technical progress shifts the production function q = 100(KL)(1/2 to q = 200(KL)(1/2. Answer parts (a) and (b) for this new situation.
