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Intermediate Microeconomics W3211

Lecture 15: Perfect Competition 5 Introduction The Short Run and the Long Run

Columbia University, Spring 2016 Mark Dean: [email protected]

The Story So Far…. Today

• We have now modeled the perfectly competitive firm in • Think more about the behavior of the firm in the short and some detail the long run

• Set up the firm’s problem

• Discussed how to split the problem into two • Cost minimization • Profit maximization

• Solved both parts

• Thought a bit about how firm behavior will change as prices change

The Long-Run and the Short-Runs

 We now introduce the distinction between long run and short run

 The long-run is the circumstance in which a firm is unrestricted in its choice of all input levels.  In the long run a firm can choose how many workers to hire and how many machines to use The Short and the Long Run  Technology in the Short and the Long Run The short-run is a circumstance in which a firm is restricted in some way in its choice of at least one input level.  They have already purchased machines, and can now only decide how many workers to hire  Notice that there are many possible short runs

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The Long-Run and the Short-Runs The Long-Run and the Short-Runs

 Notice, there are other possible causes of the firm being in  What do short-run restrictions imply for a firm’s technology? a ‘short run’ situation  Suppose the short-run restriction is fixing the level of input 2.  i.e. being unable to change one of its inputs:  Input 2 is thus a fixed input in the short-run. Input 1 remains  temporarily being unable to install, or remove, machinery variable.  being required by law to meet affirmative action quotas  having to meet domestic content regulations.

The Long-Run and the Short-Runs The Long-Run and the Short-Runs yxx 1/3 1/3 1 2 y  x1/3101/3 is the long-run production 1 function (both x and x are variable). 1/3 1/3 1 2 y  x1 5 The short-run production function when y  x1/321/3 x2  1 is 1 1/ 3 1/ 3 1/ 3 y 1/3 1/3 y  x1 1  x1 . y  x1 1

The short-run production function when

x2  10 is 1/3 1/3 x y  x1 10 . 1 Four short-run production functions

Short-Run & Long-Run Total Costs

 In the long-run a firm can vary all of its input levels.

 Consider a firm that cannot change its input 2 level from x2’ units.

 How does the short-run total cost of producing y output units compare to the long-run total cost of producing y units of The Short and the Long Run output? Cost Minimization in the Short and the Long Run

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13 Short-Run & Long-Run Total Costs Short-Run & Long-Run Total Costs

 The short-run cost-min. problem is the long-run problem

subject to the extra constraint that x2 = x2’.  The long-run cost-minimization problem is  How does this affect costs? min p1x1  p2 x2 x1 ,x2 0  If the long-run choice for x was x ’ then the extra constraint x subject to 2 2 2 f (x1,x2 )  y. = x2’ is not really a constraint at all  Long-run and short-run total costs of producing y output units are  The short-run cost-minimization problem is the same.

 But, if the long-run choice for x  x ’ then the extra constraint  2 2 min p1x1  p2 x2 x = x ’ prevents the firm from achieving its long-run x1 0 2 2 subject to f (x ,x )  y. production cost 1 2  Short-run total cost exceed the long-run total cost of producing y output units.

Short-Run & Long-Run Total Costs Short-Run & Long-Run Total Costs

y y In the long-run when the firm x Consider three output levels. x 2 2 is free to choose both x and y y 1 x2, the least-costly input y y bundles are ...

x1 x1

Short-Run & Long-Run Total Costs Short-Run & Long-Run Total Costs

Long-run costs are: y y cy()  wx11  wx 22 x2 x2 Long-run Long-run cy()  wx  wx y output y output 11 22 expansion expansion cy()  wx11 wx 22 y path y path

x2 x2 x2 x2 x2 x2

x1 x1 x1 x1 x1 x1 x1 x1

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Short-Run & Long-Run Total Costs Short-Run & Long-Run Total Costs

Long-run costs are: y Short-run  Now suppose the firm becomes subject to the short-run cy()  wx  wx x output 11 22 constraint that x = x “ 2 2 2 cy() wx wx y expansion  11 22 path cy()  wx11 wx 22

 Denote by c (y) the corresponding short run cost function s y

x2 x2 x2

x1 x1 x1 x1

Short-Run & Long-Run Total Costs Short-Run & Long-Run Total Costs

Long-run costs are: Long-run costs are: y Short-run y Short-run cy()  wx11  wx 22 cy()  wx11  wx 22 x2 output x2 output cy() wx wx cy() wx wx y expansion  11 22 y expansion  11 22 path cy()  wx11 wx 22 path cy()  wx11 wx 22 y y Short-run costs are: cys ()  cy () x2 x2 x2 x2 x2 x2

x1 x1 x1 x1 x1 x1 x1 x1

Short-Run & Long-Run Total Costs Short-Run & Long-Run Total Costs

Long-run costs are: Long-run costs are: y Short-run y Short-run cy()  wx11  wx 22 cy()  wx11  wx 22 x2 output x2 output cy() wx wx cy() wx wx y expansion  11 22 y expansion  11 22 path cy()  wx11 wx 22 path cy()  wx11 wx 22 y Short-run costs are: y Short-run costs are: cys ()  cy () cys ()  cy () x x 2 cys ()  cy () 2 cys ()  cy () x2 x2 x2 x2

x1 x1 x1 x1 x1 x1 x1 x1

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Short-Run & Long-Run Total Costs Short-Run & Long-Run Total Costs

Long-run costs are: y Short-run cy()  wx11  wx 22  Short-run total cost exceeds long-run total cost except for the x2 output cy()  wx  wx output level where the short-run input level restriction is the long- y expansion 11 22 run input level choice. path cy()  wx11 wx 22  This says that the long-run total cost curve always has one point y Short-run costs are: in common with any particular short-run total cost curve. cys ()  cy () x 2 cys ()  cy () x 2 cys ()()  cy x2

x1 x1 x1 x1

Short-Run & Long-Run Total Costs

A short-run total cost curve always has $ one point in common with the long-run total cost curve, and is elsewhere higher than the long-run total cost curve.

cs(y)

c(y) The Short and the Long Run F  Cost Curves in the Short and the Long Run wx22

y y y y

Types of Cost Curves Types of Cost Curves

 We are now going to think a little bit more about the cost  We now have lots of different types of costs curves of a firm  Total vs Fixed vs Variable

 In order to do so, we are going to differentiate between two  Long run vs Short run different types of cost  Costs vs Average Costs vs Marginal Costs  Fixed Costs: these do not change regardless of how much the firm produces  How are these cost curves related to each other?  Variable Costs: these do change depending on how much the firm produces

 Typically, in the long run, all costs are variable  If the firm produces no output it uses no input

 In the short run, the firm may have some fixed costs  If they are ‘forced’ to use a certain amount of one input, then they have to pay for that input regardless of how much they produce

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Fixed, Variable & Total Cost Fixed, Variable & Total Cost Functions Functions

 What do these various cost curves look like?  F is the total cost to a firm of its short-run fixed inputs. F, the  Fixed ’ ’ firm s fixed cost, does not vary with the firm s output level.  Variable  Total  cv(y) is the total cost to a firm of its variable inputs when producing y output units. cv(y) is the firm’s variable cost function.

 cv(y) depends upon the levels of the fixed inputs.

 c(y) is the total cost of all inputs, fixed and variable, when producing y output units. c(y) is the firm’s total cost function;

cy() F cv (). y

$ $

cv(y)

F

y y

$ $ c(y)

cv(y) cv(y) cy() F cv () y

F

F F

y y

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Av. Fixed, Av. Variable & Av. Total Av. Fixed, Av. Variable & Av. Total Cost Curves Cost Curves

 What about average costs? (Remember )  What does an average fixed cost curve look like? F For y > 0, the firm’s average total cost function is AFC() y  y F cy() AC() y v  AFC(y) is a rectangular hyperbola so its graph looks like ... y y AFCy() AVCy ().

$/output unit Av. Fixed, Av. Variable & Av. Total Cost Curves

 What about average variable costs?

 Well, as we have seen, this will depend on whether the firm has increasing, decreasing, or constant returns to scale AFC(y)  0 as y   In the short run, at least some of the inputs are fixed

 We therefore typically assume that there will be diminishing returns to scale (at least eventually)  If we fix the number of computers, at some point we will have decreasing returns to scale if we keep adding economists AFC(y)

0 y

Av. Fixed, Av. Variable & Av. Total $/output unit Cost Curves

 Think of a Cobb Douglas Production function

 If 1then the firm will exhibit decreasing returns to scale in the short run

 i.e. if is fixed at  )

 This is true even if the firm exhibits increasing returns to scale in AVC(y) the long run

 i.e 1

0 y

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$/output unit $/output unit

AFC(y) = ATC(y) - AVC(y)

ATC(y)

AVC(y) AFC AVC(y)

AFC(y) AFC(y)

0 y 0 y

$/output unit $/output unit Two things to notice: Two things to notice: 1. Since AFC(y)  0 as y , ATC(y)  AVC(y) as y 

ATC(y) ATC(y)

AFC AVC(y) AFC AVC(y)

AFC(y) AFC(y)

0 y 0 y

$/output unit Two things to notice: Marginal Cost Function 1. Since AFC(y)  0 as y , ATC(y)  AVC(y) as y  2. since short-run AVC(y) must  What about marginal cost? eventually increase, ATC(y) must eventually increase in a short-run  Well, marginal fixed cost is zero

 So marginal costs are just equal to marginal variable costs ATC(y)  cy() MC() y  v . AFC AVC(y)  y

AFC(y)

0 y

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Relationship Between Marginal and Marginal and Variable Cost Total Cost $/output unit y cyv ()   MCzdz ()  Since MC(y) is the derivative of cv(y), cv(y) must be the integral of MC(y). 0  Fundamental Theorem of Calculus MC(y)  c ()y MC() y  v  y Area is the variable y cost of making y’ units c (y)  MC(z)dz  c (0) v  v 0 0 y y y c(y)   MC(z)dz  F 0

Marginal & Average Cost Functions Marginal & Average Cost Functions cy() Since AVC() y  v , y  How is marginal cost related to average variable cost?  AVC() y yMCy ()1 cy ()  v .  y y2

Marginal & Average Cost Functions Av. Fixed, Av. Variable & Av. Total cy() Since AVC() y  v , Cost Curves y  What does this look like in practice?  AVC() y yMCy ()1 cyv ()  .  To make things more interesting, let’s think about a production  y y2 function which has both increasing and decreasing returns to scale Therefore,

 AVC() y   cy()  0 as MC() y  v  AVC(). y  y   y

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A Non-Concave Production Function Av. Fixed, Av. Variable & Av. Total Cost Curves

 What does this look like in practice? Production Function is  To make things more interesting, let’s think about a production Convex function which has both increasing and decreasing returns to y* scale

Production  Implies Marginal Cost first decreases then increases Function is Concave

*

$/output unit $/output unit  AVC()y MC() y AVC () y 0  y

MC(y) MC(y)

AVC(y) AVC(y)

y y

$/output unit $/output unit  AVC() y  AVC()y MC() y AVC () y 0 MC() y AVC () y 0  y  y

MC(y) MC(y)

AVC(y) AVC(y)

y y

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$/output unit Marginal & Average Cost Functions  AVC()y c()y MC() y AVC () y 0 Similarly, since ATC() y  ,  y y The short-run MC curve intersects  ATC() y yMCy ()1 cy () the short-run AVC curve from  . below at the AVC curve’s MC(y)  y 2 minimum. y

AVC(y)

y

Marginal & Average Cost Functions $/output unit c()y  ATC() y   Similarly, since ATC() y  ,  0 as MC() y ATC () y y  y    ATC() y yMCy ()1 cy ()  .  y y2 MC(y)

 ATC() y   cy() ATC(y)  0 as MC() y   ATC(). y  y   y

y

Marginal & Average Cost Functions $/output unit

 The short-run MC curve intersects the short-run AVC curve from below at the AVC curve’s minimum.

 And, similarly, the short-run MC curve intersects the short-run MC(y) ATC curve from below at the ATC curve’s minimum. ATC(y)

 This is also intuitive AVC(y)  If marginal cost is below average, the average must be going down  If marginal cost is above average, the average must be going up

y

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Short-Run & Long-Run Total Cost $

Curves cs(y;x2) F = w2x2

 Remember, a firm has a different short-run total cost curve for each possible short-run circumstance.  By ‘circumstance’ we mean ‘level of the fixed input’

 Suppose the firm can be in one of just three short-runs;

x2 = x2 or x2 = x2 x2 < x2 < x2. or x2 = x2.

F 0 y

$ $

cs(y;x2) cs(y;x2) F = w2x2 F = w2x2

F = w2x2 F = w2x2 A larger amount of the fixed cs(y;x2) cs(y;x2) input increases the firm’s fixed cost.

F F F F 0 y 0 y

$ $

cs(y;x2) cs(y;x2) F = w2x2 F = w2x2

F = w2x2 F = w2x2 F = w x  A larger amount of the fixed 2 2 cs(y;x2) cs(y;x2) input increases the firm’s fixed cost.

cs(y;x2) Why does F a larger amount of the fixed input reduce the slope of the firm’s F total cost curve? F F F 0 y 0 y

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Short-Run & Long-Run Total Cost $ For 0  y  y, choose x = ? Curves 2 cs(y;x2)

 The firm has three short-run total cost curves.

 In the long-run the firm is free to choose amongst these three cs(y;x2) since it is free to select x2 equal to any of x2, x2, or x2.

 How does the firm make this choice?

cs(y;x2) F

F F 0 y y y

$ $ For 0  y  y, choose x = x . For 0  y  y, choose x = x . 2 2 cs(y;x2) 2 2 cs(y;x2)

For yy  y, choose x2 = ?

cs(y;x2) cs(y;x2)

cs(y;x2) cs(y;x2) F F

F F F F 0 y y y 0 y y y

$ $ For 0  y  y, choose x = x . For 0  y  y, choose x = x . 2 2 cs(y;x2) 2 2 cs(y;x2)

For yy  y, choose x2 = x2. For yy  y, choose x2 = x2.

For y  y, choose x2 = ? cs(y;x2) cs(y;x2)

cs(y;x2) cs(y;x2) F F

F F F F 0 y y y 0 y y y

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$ $ For 0  y  y, choose x = x . For 0  y  y, choose x = x . 2 2 cs(y;x2) 2 2 cs(y;x2)

For yy  y, choose x2 = x2. For yy  y, choose x2 = x2.

For y  y, choose x2 = x2. For y  y, choose x2 = x2. cs(y;x2) cs(y;x2)

c (y;x ) c (y;x ) s 2 s 2 c(y), the F F firm’s long- run total F F cost curve. F F 0 y y y 0 y y y

Short-Run & Long-Run Total Cost Short-Run & Long-Run Total Cost Curves Curves

 The firm’s long-run total cost curve consists of the lowest parts of  If input 2 is available in continuous amounts then there is an the short-run total cost curves. The long-run total cost curve is infinity of short-run total cost curves but the long-run total cost the lower envelope of the short-run total cost curves. curve is still the lower envelope of all of the short-run total cost curves.

$ Short-Run & Long-Run Average Total

cs(y;x2) Cost Curves

 For any output level y, the long-run total cost curve always gives the lowest possible total production cost. cs(y;x2)  Therefore, the long-run av. total cost curve must always give the lowest possible av. total production cost.

cs(y;x2) c(y)  The long-run av. total cost curve must be the lower envelope of all of the firm’s short-run av. total cost curves. F

F F 0 y

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Short-Run & Long-Run Average Total $/output unit Cost Curves

 E.g. suppose again that the firm can be in one of just three short-runs; ACs(y;x2)

x2 = x2 or x2 = x2 (x2 < x2 < x2) or x2 = x2 ACs(y;x2) then the firm’s three short-run average total cost curves are ...

ACs(y;x2)

y

Short-Run & Long-Run Average Total $/output unit

Cost Curves ACs(y;x2)

 The firm’s long-run average total cost curve is the lower envelope of the short-run average total cost curves ...

ACs(y;x2)

ACs(y;x2)

The long-run av. total cost curve is the lower envelope AC(y) of the short-run av. total cost curves.

y

Short-Run & Long-Run Marginal Cost Short-Run & Long-Run Marginal Cost Curves Curves

 Q: Is the long-run marginal cost curve the lower envelope of the  Q: Is the long-run marginal cost curve the lower envelope of the firm’s short-run marginal cost curves? firm’s short-run marginal cost curves?

 A: No.

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Short-Run & Long-Run Marginal Cost $/output unit AC (y;x ) Curves s 2 AC (y;x )  The firm’s three short-run average total cost curves are ... s 2

ACs(y;x2)

y

$/output unit MCs(y;x2) MCs(y;x2) $/output unit MCs(y;x2) MCs(y;x2)

ACs(y;x2) ACs(y;x2)

ACs(y;x2) ACs(y;x2)

ACs(y;x2) ACs(y;x2)

AC(y)

MCs(y;x2) MCs(y;x2)

y y y y

+ $/output unit MC (y;x ) MC (y;x ) Short-Run & Long-Run Marginal Cost s 2 s 2 AC (y;x ) Curves s 2

ACs(y;x2)  Below y’, will choose ′, despite the fact it has higher marginal cost than either ′′ or ′′′ ACs(y;x2)  Because it gives lower total cost

AC(y)

MCs(y;x2)

y y y

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$/output unit MC (y;x ) MC (y;x ) s 2 s 2 Short-Run & Long-Run Marginal Cost AC (y;x ) s 2 Curves AC (y;x ) s 2  For any output level y > 0, the long-run marginal cost of AC (y;x ) production is the marginal cost of production for the short-run s 2 chosen by the firm.

MCs(y;x2)

MC(y), the long-run marginal cost curve.

y

$/output unit MC (y;x ) MC (y;x ) s 2 s 2 Short-Run & Long-Run Marginal Cost

ACs(y;x2)

AC (y;x ) s 2  For any output level y > 0, the long-run marginal cost is the marginal cost for the short-run chosen by the firm. ACs(y;x2)  This is always true, no matter how many and which short-run circumstances exist for the firm.

MCs(y;x2)

MC(y), the long-run marginal cost curve.

y

Short-Run & Long-Run Marginal Cost Short-Run & Long-Run Marginal Cost $/output unit SRACs  For any output level y > 0, the long-run marginal cost is the marginal cost for the short-run chosen by the firm. AC(y)  So for the continuous case, where x2 can be fixed at any value of zero or more, the relationship between the long-run marginal cost and all of the short-run marginal costs is ...

y

17 4/3/2016

Short-Run & Long-Run Marginal Cost Short-Run & Long-Run Marginal Cost $/output unit $/output unit MC(y) SRMCs SRMCs

AC(y) AC(y)

y y For each y > 0, the long-run MC equals the MC for the short-run chosen by the firm.

Summary

• Today we focused on the difference between the short and the long run • How technology changes • How costs change

Summary

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