lecture 6: producer behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...expansion path and...
TRANSCRIPT
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Lecture 6:Producer Behavior
October 6, 2015
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Overview
Course Administration
Basics of Production
Production in the Short Run
Production in the Long Run
The Firm’s Problem: Cost Minimization
Returns to Scale
Expansion Path and Total Cost
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Course Administration
1. Return Problem Set 4
2. Collect Problem Set 5
3. Problem Set 6 is posted
4. Midterm• October 20• Review session with Monika: Sunday 10/18, 1-3 pm, Phil 217• Bring questions• I will not provide answers for last year’s midterm
5. Any questions?
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Ripped from the Headlines
As a reminder, next week
Afternoon
Finder Presenter
Giana Mandel David Meni
Evening
Finder Presenter
Alysse Henkel Julia RobbinsHoazheng Wang McKenna Saady
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
What is Production?
• Production ≡ process of producing a good or service
• Final good ≡ good bought by consumer
• Intermediate good ≡ good bought by a firm to produceanother good
• Production function ≡ mathematical relationship betweeninputs and outputs
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Simplifying Assumptions, 1 of 2
Why do we assume things? To make the problem manageable andlook carefully at a limited number of factors.
1. Firm produces a single good
2. Firm has already chosen what product it will produce
3. Firm’s goal is to minimize cost
4. Firm uses only two inputs: capital and labor
5. In the short run, the firm can change only labor. In the longrun the firm can change labor and capital
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Simplifying Assumptions, 1 of 2
Why do we assume things? To make the problem manageable andlook carefully at a limited number of factors.
1. Firm produces a single good
2. Firm has already chosen what product it will produce
3. Firm’s goal is to minimize cost
4. Firm uses only two inputs: capital and labor
5. In the short run, the firm can change only labor. In the longrun the firm can change labor and capital
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Simplifying Assumptions, 2 of 2
6. More inputs → more outputs
7. Production has diminishing marginal returns to capital andlabor
8. An infinite amount of inputs sells at fixed prices
9. The firm has no budget constraint → very well-functioningcapital market
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
The Production Function
Q = f (K , L)
• Q is output
• K is capital
• L is labor
• f () is a general function
For example, Q = K 0.5L0.5.
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
The Production Function
Q = f (K , L)
• Q is output
• K is capital
• L is labor
• f () is a general function
For example, Q = K 0.5L0.5.
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
The Production Function
Q = f (K , L)
• Q is output
• K is capital
• L is labor
• f () is a general function
For example, Q = K 0.5L0.5.
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Measuring Changes in Production
1. Marginal product of X ≡ additional output from an additionalunit of input
MPX =∆Q
∆X=
(∂Q
∂X
)
2. Average product of X
APX =Q
X
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Measuring Changes in Production
1. Marginal product of X ≡ additional output from an additionalunit of input
MPX =∆Q
∆X=
(∂Q
∂X
)
2. Average product of X
APX =Q
X
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Measuring Changes in the Short Run
• Recall that we assumed that in the short run K is fixed and Lcan change
• Suppose K = 5, and Q = f (K , L)
• Then the short run production function is
Q = f (5, L)
• Recall that we assumed diminishing marginal product of labor
• Draw short-run output as a function of labor (Q on the y axis,L on the x axis)
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Measuring Changes in the Short Run
• Recall that we assumed that in the short run K is fixed and Lcan change
• Suppose K = 5, and Q = f (K , L)
• Then the short run production function is Q = f (5, L)
• Recall that we assumed diminishing marginal product of labor
• Draw short-run output as a function of labor (Q on the y axis,L on the x axis)
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Short-Run Production Function
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
L
Q
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
L
Q
L1
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
L
Q
L1
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
L
Q
L1 L2
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
L
Q
L1 L2
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
L
Q
L1 L2
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
So what does a graph of MPL as a function of Q look like?
L
Q
L1 L2
Q1
Q2
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding the Marginal Product of Labor from theProduction Function
Q
MPL
Q1 Q2
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of diminishing marginal utility?
Consumer Producer
Diminishing marginal utility
Dimin. MPL if K constant
max U s.t. budget constraintUtility functionIndifference curvesMRSX ,Y
Price of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of diminishing marginal utility?
Consumer Producer
Diminishing marginal utility Dimin. MPL if K constantmax U s.t. budget constraintUtility functionIndifference curvesMRSX ,Y
Price of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Production in the Long Run
• In the long run, everything can change
• Diminishing returns are less of a problem, since you can addboth capital and labor
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Firm’s Problem
• Firm wants to minimize costs
• Subject to producing a given amount of output
• It could always minimize costs by shutting down, but then noone is making any money
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Firm’s Problem
• Firm wants to minimize costs
• Subject to producing a given amount of output
• It could always minimize costs by shutting down, but then noone is making any money
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of the utility function?
Consumer Producer
Diminishing marginal utility Dimin. MU of L, K constantUtility function
Production function
max U s.t. budget constraint
min C s.t. producing Q = a
Indifference curvesMRSX ,Y
Price of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of the consumer problem?
Consumer Producer
Diminishing marginal utility Dimin. MU of L, K constantUtility function Production functionmax U s.t. budget constraint
min C s.t. producing Q = a
Indifference curvesMRSX ,Y
Price of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsGiven a quantity to produce, producers minimize costs
Consumer Producer
Diminishing marginal utility Dimin. MU of L, K constantUtility function Production functionmax U s.t. budget constraint min C s.t. producing Q = aIndifference curvesMRSX ,Y
Price of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Isoquants
• “iso” ≡ same
• “quant” for quantity
• All combinations of K and L that produce some level of Q
• Properties of isoquants, for a given production function• Further from the origin → more production• Cannot intersect• Convex to the origin
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Short-Run Production Function
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Marginal Rate of Technical Substitution
• MRTSXY ≡ slope of the isoquant
• Or, the rate at which firm can trade input L for input K ,holding output constant
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Marginal Rate of Technical Substitution Along an IsoquantWhat Does the Shape of the Isoquant Tell Us About the Trade-off Between Capital and
Labor?
L
K
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Marginal Rate of Technical Substitution Along an IsoquantAnd at the Other End?
L
K
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Marginal Rate of Technical Substitution Along an IsoquantDiminishing Marginal Product in Action
L
K
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of indifference curve?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves
Isoquants
MRSX ,Y
MRTSL,K
Price of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of the marginal rate of substitution?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y
MRTSL,K
Price of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsIsoquants and MRTS
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goodsBudget Constraint
Slope of budget constraint = −PXPY
Optimality at MRSXY = PXPY
Income expansion path
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Input Substitutability and ComplementarityWhat Does it Mean for the Production Function?
• What if inputs are perfect substitutes?
• What if they are perfect complements?
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Isocost Lines
• Cost of capital is R: rental rate per period
• Cost of labor is W : wage rate per period
• For any cost C , the isocost line is C = RK + WL
• What’s the slope of the isocost line?
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Implications of Isocost Line
• Slope of isocost line is the cost consequences of trading offone unit of K for L
• What if the price of K increases? decreases?
• Labor?
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat are the relevant producer prices?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods
PL = W , PK = R
Budget Constraint
Isocost line
Slope of budget constraint = −PXPY
Slope of isocost = − PLPK
Optimality at MRSXY = PXPY
Income expansion path
![Page 46: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/46.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of the budget constraint?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods PL = W , PK = RBudget Constraint
Isocost line
Slope of budget constraint = −PXPY
Slope of isocost = − PLPK
Optimality at MRSXY = PXPY
Income expansion path
![Page 47: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/47.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of the slope of the budget constraint?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods PL = W , PK = RBudget Constraint Isocost line
Slope of budget constraint = −PXPY
Slope of isocost = − PLPK
Optimality at MRSXY = PXPY
Income expansion path
![Page 48: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/48.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsBudget constraint ≈ Isocost
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods PL = W , PK = RBudget Constraint Isocost line
Slope of budget constraint = −PXPY
Slope of isocost = − PLPK
Optimality at MRSXY = PXPY
Income expansion path
![Page 49: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/49.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding Minimum Cost
• Firm wants to produce a given output at minimum cost
• A constrained minimization problem
• Constraint is that firm produces some level of output Q• Think of this as a given: Q = a• Parallel to your income in the consumer problem, given as I
• Goal: what is the lowest cost at which it can produce thatoutput?
• Necessary but not sufficient for profit maximization – more onthis later
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Finding Minimum Cost
• Firm wants to produce a given output at minimum cost
• A constrained minimization problem
• Constraint is that firm produces some level of output Q• Think of this as a given: Q = a• Parallel to your income in the consumer problem, given as I
• Goal: what is the lowest cost at which it can produce thatoutput?
• Necessary but not sufficient for profit maximization – more onthis later
![Page 51: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/51.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Cost Minimization in PicturesHow Can You Produce Q = a at Minimum Cost?
L
K
Q= a
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Cost Minimization in PicturesFind the Slope of the Isocost Line
L
K
Q= awhat is the slope of this line?
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Cost Minimization in PicturesNot Enough Inputs to Make a
L
K
Q= a
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Cost Minimization in PicturesStill Not Enough
L
K
Q= a
![Page 55: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/55.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Cost Minimization in PicturesStill Not Enough
L
K
Q= a
![Page 56: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/56.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Cost Minimization in PicturesEnough?
L
K
Q= a
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Cost Minimization in PicturesThe Optimal Combination of K and L
L
K
Q= a
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Conditions for Cost Minimization
• Occurs where isocost is tangent to isoquant
• Occurs when
−MRTSLK = − PL
PK
−MPL
MPK= −W
R
• More intuitively,MPL
W=
MPK
R
• Marginal product per dollar is equal
![Page 59: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/59.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Conditions for Cost Minimization
• Occurs where isocost is tangent to isoquant
• Occurs when
−MRTSLK = − PL
PK
−MPL
MPK= −W
R
• More intuitively,MPL
W=
MPK
R
• Marginal product per dollar is equal
![Page 60: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/60.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer optimality condition?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods PL = W , PK = RBudget Constraint Isocost curve
Slope of budget constraint = −PXPY
Slope of isocost curve = − PLPK
Optimality at MRSXY = PXPY
MRTSLK = WR
Income expansion path
![Page 61: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/61.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsThink tangency!
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods PL = W , PK = RBudget Constraint Isocost curve
Slope of budget constraint = −PXPY
Slope of isocost curve = − PLPK
Optimality at MRSXY = PXPY
MRTSLK = WR
Income expansion path
![Page 62: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/62.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
What if Input Prices Change?
• Price of labor increases, andprice of capital decreases
• Firms adjust to use more ofthe less costly input
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
What if Input Prices Change?
• Price of labor increases, andprice of capital decreases
• Firms adjust to use more ofthe less costly input
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Minimize Costs on Your Own
A firm employs 25 workers (W = $10\hour) and 5 units of capital(R = $20\hour). At these levels, the marginal product of labor is25, and the marginal product of capital is 30.
1. Is this firm minimizing costs?
2. If not, what changes should it make?
3. How does the answer to question 2 depend on the time frameof analysis?
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Returns to Scale
Returns to Scale ≡ changes in output given changes in inputs.
Canbe
• constant → outputs increase proportionately with inputs• double inputs, double outputs
• increasing → outputs increase more than proportionately withinputs
• double inputs, more than double outputs
• decreasing → outputs increase less than proportionately withinputs
• double inputs, less than double outputs
What kind of returns to scale does the production functionQ = 10K + 8L have?In general, put in inputs, find Q. Double the inputs, find Q ′. IsQ ′ = 2Q? Q ′ > 2Q? Q ′ < 2Q?
![Page 66: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/66.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Returns to Scale
Returns to Scale ≡ changes in output given changes in inputs. Canbe
• constant → outputs increase proportionately with inputs• double inputs, double outputs
• increasing → outputs increase more than proportionately withinputs
• double inputs, more than double outputs
• decreasing → outputs increase less than proportionately withinputs
• double inputs, less than double outputs
What kind of returns to scale does the production functionQ = 10K + 8L have?In general, put in inputs, find Q. Double the inputs, find Q ′. IsQ ′ = 2Q? Q ′ > 2Q? Q ′ < 2Q?
![Page 67: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/67.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Returns to Scale
Returns to Scale ≡ changes in output given changes in inputs. Canbe
• constant → outputs increase proportionately with inputs• double inputs, double outputs
• increasing → outputs increase more than proportionately withinputs
• double inputs, more than double outputs
• decreasing → outputs increase less than proportionately withinputs
• double inputs, less than double outputs
What kind of returns to scale does the production functionQ = 10K + 8L have?In general, put in inputs, find Q. Double the inputs, find Q ′. IsQ ′ = 2Q? Q ′ > 2Q? Q ′ < 2Q?
![Page 68: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/68.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Returns to Scale
Returns to Scale ≡ changes in output given changes in inputs. Canbe
• constant → outputs increase proportionately with inputs• double inputs, double outputs
• increasing → outputs increase more than proportionately withinputs
• double inputs, more than double outputs
• decreasing → outputs increase less than proportionately withinputs
• double inputs, less than double outputs
What kind of returns to scale does the production functionQ = 10K + 8L have?In general, put in inputs, find Q. Double the inputs, find Q ′. IsQ ′ = 2Q? Q ′ > 2Q? Q ′ < 2Q?
![Page 69: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/69.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Returns to Scale
Returns to Scale ≡ changes in output given changes in inputs. Canbe
• constant → outputs increase proportionately with inputs• double inputs, double outputs
• increasing → outputs increase more than proportionately withinputs
• double inputs, more than double outputs
• decreasing → outputs increase less than proportionately withinputs
• double inputs, less than double outputs
What kind of returns to scale does the production functionQ = 10K + 8L have?
In general, put in inputs, find Q. Double the inputs, find Q ′. IsQ ′ = 2Q? Q ′ > 2Q? Q ′ < 2Q?
![Page 70: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/70.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Returns to Scale
Returns to Scale ≡ changes in output given changes in inputs. Canbe
• constant → outputs increase proportionately with inputs• double inputs, double outputs
• increasing → outputs increase more than proportionately withinputs
• double inputs, more than double outputs
• decreasing → outputs increase less than proportionately withinputs
• double inputs, less than double outputs
What kind of returns to scale does the production functionQ = 10K + 8L have?In general, put in inputs, find Q. Double the inputs, find Q ′. IsQ ′ = 2Q? Q ′ > 2Q? Q ′ < 2Q?
![Page 71: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/71.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
What Drives Returns to Scale?
• Increasing returns• Fixed costs• Learning by doing – if the firm gets bigger and better at
production by producing
• Decreasing returns• Regulation• Limited low cost/high quality inputs (violates one of our
assumptions)
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
How Does Production Change at Different Levels of Q?
• We know how to find the firm’s ideal inputs given Q
• Now we repeat this exercise for a variety of different Qs• Each optimal K and L will be where an isoquant is tangent to
an isocost line• MRTSLK will be the same at each point
• Call this optimal (L,K ) for each Q the expansion path
• And we can draw a total cost curve – with different axes
![Page 73: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/73.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveRecall Our Previous Optimum. What if the firm wants to produce b < a?
L
K
Q= a
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveRecall Our Previous Optimum. What if the firm wants to produce b < a?
L
K
Q= aQ= b
![Page 75: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/75.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveWhat is optimum (L,K)if it wants to make c < b?
L
K
Q= aQ= b
![Page 76: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/76.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveWhat is optimum (L,K)if it wants to make c < b?
L
K
Q= aQ= b
Q= c
![Page 77: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/77.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveWhat is optimum (L,K)if it wants to make c < b?
L
K
Q= aQ= b
Q= c
![Page 78: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/78.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveOptimum!
L
K
Q= aQ= b
Q= c
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveExpansion Path
L
K
Q= aQ= b
Q= c
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Drawing a Total Cost CurveTotal Cost
Q
Total Cost
Q= aQ= bQ= c
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of the income expansion path?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods PL = W , PK = RBudget Constraint Isocost curve
Slope of budget constraint = −PXPY
Slope of isocost curve = − PLPK
Optimality at MRSXY = PXPY
MRTS = WR
Income expansion path
Expansion path
![Page 82: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/82.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Parallels: Consumer and Producer ProblemsWhat is the producer parallel of the income expansion path?
Consumer Producer
Diminishing marginal utility Dimin. MP of L, K constantUtility function Production functionmax U s.t. budget constraint min cost s.t. production of QIndifference curves IsoquantsMRSX ,Y MRTSL,KPrice of consumption goods PL = W , PK = RBudget Constraint Isocost curve
Slope of budget constraint = −PXPY
Slope of isocost curve = − PLPK
Optimality at MRSXY = PXPY
MRTS = WR
Income expansion path Expansion path
![Page 83: Lecture 6: Producer Behaviorhome.gwu.edu/~lfbrooks/leahweb/teaching/pppa6085/...Expansion Path and Total Cost. AdminBasicsShort RunLong RunCost Min.ScaleTotal Cost Course Administration](https://reader033.vdocuments.mx/reader033/viewer/2022042010/5e71f7109e4ba76d0a28b622/html5/thumbnails/83.jpg)
Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Recap of Today
• Production Assumptions and Basics
• Production in the Short Run
• Production in the Long Run
• Cost Minimization Problem
• Returns to Scale
• Expansion Path and Total Cost
• Technological Change (time did not permit)
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Admin Basics Short Run Long Run Cost Min. Scale Total Cost
Next Class
• Turn in Problem Set 6
• Costs, GLS Chapter 7