Problem 1. Costs for Sedate Sedan, Inc. (adapted from Nicholson). Sedate Sedan, Inc. uses robots and workers to assemble automobiles. There is a diminishing rate of technical substitution between robots and labour: robots do well at tasks that are purely repetitive, while labour does better at jobs that are less boring and require more thought. The various combinations of hours of robot use (r) and hours of labour use (1) needed to assemble sedans are described by the Cobb-Douglas production function: q= f (r,l) = ²/√√² 50 Note that marginal products for this production function are given by: MP, 1 √r 100 Vi = MP₁ Please make sure you know how to calculate them. This production function exhibits diminishing marginal products: the marginal product of robots decreases when more robots are used; the marginal product of labour decreases when more labour is used. 1 √ī 100 √r The company can rent all the robots it wants for v = 40 dollars per hour and hire all the workers it wants at w = 40 dollars per hour. It hires you to find its cost functions. a) Returns to scale (tick) d) MC (tick) d) AC (tick) and a) First, show that the production function exhibits constant return to scale (CRS): what happens to the output (number of sedans) when inputs (robot use and labour) are doubled? b) What are the cost-minimising robot and labour inputs for Sedate Sedan, Inc.? They are called the factor demands for robots r(q) and labour l(q). Hint: use two conditions (the isoquant and tangency) to calculate optimal inputs. Illustrate on a diagram with an isoquant and isocosts. Put robots on a horizontal axis. CRS c) Given your answer to (b), find the long-run cost function CLR (q) - how much it costs to produce q sedans. d) How do long-run cost, marginal cost, and average cost depend on output? IRS DRS constant increasing decreasing increasing decreasing constant (as a function of q) (as a function of q)
Problem 1. Costs for Sedate Sedan, Inc. (adapted from Nicholson). Sedate Sedan, Inc. uses robots and workers to assemble automobiles. There is a diminishing rate of technical substitution between robots and labour: robots do well at tasks that are purely repetitive, while labour does better at jobs that are less boring and require more thought. The various combinations of hours of robot use (r) and hours of labour use (1) needed to assemble sedans are described by the Cobb-Douglas production function: q= f (r,l) = ²/√√² 50 Note that marginal products for this production function are given by: MP, 1 √r 100 Vi = MP₁ Please make sure you know how to calculate them. This production function exhibits diminishing marginal products: the marginal product of robots decreases when more robots are used; the marginal product of labour decreases when more labour is used. 1 √ī 100 √r The company can rent all the robots it wants for v = 40 dollars per hour and hire all the workers it wants at w = 40 dollars per hour. It hires you to find its cost functions. a) Returns to scale (tick) d) MC (tick) d) AC (tick) and a) First, show that the production function exhibits constant return to scale (CRS): what happens to the output (number of sedans) when inputs (robot use and labour) are doubled? b) What are the cost-minimising robot and labour inputs for Sedate Sedan, Inc.? They are called the factor demands for robots r(q) and labour l(q). Hint: use two conditions (the isoquant and tangency) to calculate optimal inputs. Illustrate on a diagram with an isoquant and isocosts. Put robots on a horizontal axis. CRS c) Given your answer to (b), find the long-run cost function CLR (q) - how much it costs to produce q sedans. d) How do long-run cost, marginal cost, and average cost depend on output? IRS DRS constant increasing decreasing increasing decreasing constant (as a function of q) (as a function of q)
Chapter1: Making Economics Decisions
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Transcribed Image Text:Problem 1. Costs for Sedate Sedan, Inc. (adapted from Nicholson).
Sedate Sedan, Inc. uses robots and workers to assemble automobiles. There is a diminishing
rate of technical substitution between robots and labour: robots do well at tasks that are purely
repetitive, while labour does better at jobs that are less boring and require more thought. The
various combinations of hours of robot use (r) and hours of labour use (1) needed to assemble
sedans are described by the Cobb-Douglas production function:
q = f(r,l) = ²√√√²
50
Note that marginal products for this production function are given by: MP:
=
1 √r
MP₁
Please make sure you know how to calculate them. This production function
100 Vi
exhibits diminishing marginal products: the marginal product of robots decreases when more
robots are used; the marginal product of labour decreases when more labour is used.
The company can rent all the robots it wants for v = 40 dollars per hour and hire all the workers
it wants at w = 40 dollars per hour. It hires you to find its cost functions.
a) First, show that the production function exhibits constant return to scale (CRS): what
happens to the output (number of sedans) when inputs (robot use and labour) are doubled?
a) Returns to scale (tick)
d) MC (tick)
d) AC (tick)
1 √ī
100 √r
b) What are the cost-minimising robot and labour inputs for Sedate Sedan, Inc.? They are
called the factor demands for robots r(q) and labour l(q). Hint: use two conditions (the
isoquant and tangency) to calculate optimal inputs. Illustrate on a diagram with an
isoquant and isocosts. Put robots on a horizontal axis.
and
c) Given your answer to (b), find the long-run cost function CLR (q) - how much it costs to
produce q sedans.
d) How do long-run cost, marginal cost, and average cost depend on output?
IRS
DRS
increasing
decreasing
increasing decreasing
CRS
constant
constant
(as a function of q)
(as a function of q)
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