Cost Minimization Accountants vs Economists Economists think of
![Cost Minimization Cost Minimization](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-1.jpg)
![Accountants vs. Economists • Economists think of costs differently from accountants. – Economics Cost Accountants vs. Economists • Economists think of costs differently from accountants. – Economics Cost](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-2.jpg)
![Fixed Cost and Sunk Cost • Fixed vs. Sunk cost – “fixed” means its Fixed Cost and Sunk Cost • Fixed vs. Sunk cost – “fixed” means its](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-3.jpg)
![Short Run – Long Run • Key distinction in terms of optimizing behavior if Short Run – Long Run • Key distinction in terms of optimizing behavior if](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-4.jpg)
![To Distinguish Short Run from Long Run • We need two inputs, one always To Distinguish Short Run from Long Run • We need two inputs, one always](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-5.jpg)
![To be clear • L is assumed always variable • K is always fixed To be clear • L is assumed always variable • K is always fixed](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-6.jpg)
![Notation • K, capital; v, rental rate of capital • L, Labor; w, wage Notation • K, capital; v, rental rate of capital • L, Labor; w, wage](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-7.jpg)
![Cost • Clearly, C can be stated as: C = v. K + w. Cost • Clearly, C can be stated as: C = v. K + w.](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-8.jpg)
![q Graphically q 3 q 2 K q 1 q = f(K, L) q q Graphically q 3 q 2 K q 1 q = f(K, L) q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-9.jpg)
![Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2, Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-10.jpg)
![Short Run, K constant q Let’s look at just one slice. q 2 At Short Run, K constant q Let’s look at just one slice. q 2 At](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-11.jpg)
![Production to Cost Flip the axis q. L Lq Production to Cost Flip the axis q. L Lq](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-12.jpg)
![Production to Cost L L = L(q, K) q Production to Cost L L = L(q, K) q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-13.jpg)
![Production to Cost $/time Multiply L by w to get labor cost SC = Production to Cost $/time Multiply L by w to get labor cost SC =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-14.jpg)
![Production to Cost $/time And we can figure a few things SC = w· Production to Cost $/time And we can figure a few things SC = w·](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-15.jpg)
![Production to Cost $ SMC SAC AVC AFC q Production to Cost $ SMC SAC AVC AFC q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-16.jpg)
![Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2, Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-17.jpg)
![Production to Cost $ SC = w· L 2(q)+r. K 2 SC = w· Production to Cost $ SC = w· L 2(q)+r. K 2 SC = w·](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-18.jpg)
![C $ C SC = w· L 2(q)+r. K 2 SC = w· L C $ C SC = w· L 2(q)+r. K 2 SC = w· L](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-19.jpg)
![SAC $ Min value at higher q with more K. SAC = C 1/q SAC $ Min value at higher q with more K. SAC = C 1/q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-20.jpg)
![MC C $ MC is the slope of the C curve at any q MC C $ MC is the slope of the C curve at any q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-21.jpg)
![Short Run – Long Run • Firms ALWAYS produce in a short run situation Short Run – Long Run • Firms ALWAYS produce in a short run situation](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-22.jpg)
![Different Shapes for the long run C Curve C $ There are four different Different Shapes for the long run C Curve C $ There are four different](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-23.jpg)
![C -- CRS C $ CRS, as K and L are scaled upwards, C C -- CRS C $ CRS, as K and L are scaled upwards, C](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-24.jpg)
![C -- CRS $ Min value at higher q with more K. ATC = C -- CRS $ Min value at higher q with more K. ATC =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-25.jpg)
![C -- IRS C $ IRS, as K and L are scaled upwards, C C -- IRS C $ IRS, as K and L are scaled upwards, C](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-26.jpg)
![C -- IRS $ AC MC q CRS, as K and L are scaled C -- IRS $ AC MC q CRS, as K and L are scaled](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-27.jpg)
![C -- DRS $ DRS, as K and L are scaled upwards, TC rises C -- DRS $ DRS, as K and L are scaled upwards, TC rises](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-28.jpg)
![LTC -- DRS $ Note, it is not the low point of each ATC LTC -- DRS $ Note, it is not the low point of each ATC](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-29.jpg)
![C – IRS, CRS, DRS $ C IRS, then CRS, then DRS q C – IRS, CRS, DRS $ C IRS, then CRS, then DRS q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-30.jpg)
![C – IRS, CRS, DRS $ So we have a U-shaped AC curve for C – IRS, CRS, DRS $ So we have a U-shaped AC curve for](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-31.jpg)
![Firm Decisions • So which short run curve should we be on? – That Firm Decisions • So which short run curve should we be on? – That](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-32.jpg)
![Profit Maximization • Economic profits ( ) are equal to = total revenue - Profit Maximization • Economic profits ( ) are equal to = total revenue -](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-33.jpg)
![Profit Maximization • Solving to maximize profit means jointly choosing q = q* along Profit Maximization • Solving to maximize profit means jointly choosing q = q* along](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-34.jpg)
![Cost Minimization • That firms minimize cost is a weaker hypothesis of firm behavior Cost Minimization • That firms minimize cost is a weaker hypothesis of firm behavior](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-35.jpg)
![New Direction in Graphing • In all the graphs above, we have illustrated the New Direction in Graphing • In all the graphs above, we have illustrated the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-36.jpg)
![Intuitively K Isoquant, all combinations of factors that yield the same output. Slope is Intuitively K Isoquant, all combinations of factors that yield the same output. Slope is](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-37.jpg)
![Cost-Minimizing Input Choices • When K and L change a small amount • Along Cost-Minimizing Input Choices • When K and L change a small amount • Along](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-38.jpg)
![Cost-Minimizing Input Choices • TRS is the change in K needed to replace one Cost-Minimizing Input Choices • TRS is the change in K needed to replace one](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-39.jpg)
![Intuitively K w=20, v=10 w/v=2 TRS=. 25 • Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=. 25 • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-40.jpg)
![Intuitively K w=20, v=10 w/v=2 TRS=1 • Firing one L has MB of $20 Intuitively K w=20, v=10 w/v=2 TRS=1 • Firing one L has MB of $20](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-41.jpg)
![Intuitively K w=20, v=10 w/v=2 TRS=1. 5 • Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=1. 5 • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-42.jpg)
![Intuitively K w=20, v=10 w/v=2 TRS=2 L • Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=2 L • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-43.jpg)
![Intuitively K w=20, v=10 w/v=2 TRS=2. 5 • Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=2. 5 • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-44.jpg)
![Plan 1. Figure out the quantity of K and L that minimize total cost, Plan 1. Figure out the quantity of K and L that minimize total cost,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-45.jpg)
![Cost-Minimizing Input Choices • We seek to minimize total costs given q =q 0 Cost-Minimizing Input Choices • We seek to minimize total costs given q =q 0](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-46.jpg)
![Cost-Minimizing Input Choices • Dividing the first two conditions we get • The cost-minimizing Cost-Minimizing Input Choices • Dividing the first two conditions we get • The cost-minimizing](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-47.jpg)
![Cost-Minimizing Input Choices • The inverse of this equation is also of interest • Cost-Minimizing Input Choices • The inverse of this equation is also of interest •](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-48.jpg)
![Cost-Minimizing Input Choices • SOC to ensure costs are RISING, along the isoquant, away Cost-Minimizing Input Choices • SOC to ensure costs are RISING, along the isoquant, away](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-49.jpg)
![Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC satisfied as moving Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC satisfied as moving](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-50.jpg)
![SOC fail, Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC not SOC fail, Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC not](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-51.jpg)
![Conditional Factor Demand (aka Constant Output) • Solve the FOC to derive – K Conditional Factor Demand (aka Constant Output) • Solve the FOC to derive – K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-52.jpg)
![The Firm’s Expansion Path The expansion path is the locus of cost-minimizing tangencies K The Firm’s Expansion Path The expansion path is the locus of cost-minimizing tangencies K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-53.jpg)
![The Firm’s Expansion Path • The cost-minimizing combinations of K and L for every The Firm’s Expansion Path • The cost-minimizing combinations of K and L for every](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-54.jpg)
![The Firm’s Expansion Path • The expansion path does not have to be a The Firm’s Expansion Path • The expansion path does not have to be a](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-55.jpg)
![Cost Functions • Long Run Cost • Short Run Cost – The factor demand Cost Functions • Long Run Cost • Short Run Cost – The factor demand](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-56.jpg)
![Short Run Cost Functions • With only two inputs, once K is fixed, the Short Run Cost Functions • With only two inputs, once K is fixed, the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-57.jpg)
![Short Run & Long Run • Pick a level of output, q 1 and Short Run & Long Run • Pick a level of output, q 1 and](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-58.jpg)
![Short Long Relationship $ MC SMC ATC AC AVC q Short Long Relationship $ MC SMC ATC AC AVC q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-59.jpg)
![SMC vs. LMC $ When q falls in SR, cost falls by less than SMC vs. LMC $ When q falls in SR, cost falls by less than](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-60.jpg)
![Short Long Relationship (Easier to see if assume CRS) $ SMC actually lower from Short Long Relationship (Easier to see if assume CRS) $ SMC actually lower from](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-61.jpg)
![Fixed Proportions Factor Demand Cost Functions • Suppose we have a fixed proportions technology Fixed Proportions Factor Demand Cost Functions • Suppose we have a fixed proportions technology](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-62.jpg)
![Cobb-Douglas Cost Minimization • Suppose that the production function is Cobb. Douglas: q = Cobb-Douglas Cost Minimization • Suppose that the production function is Cobb. Douglas: q =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-63.jpg)
![Cobb-Douglas Cost Minimization • Dividing the first equation by the second gives us • Cobb-Douglas Cost Minimization • Dividing the first equation by the second gives us •](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-64.jpg)
![Cobb-Douglas Cost Minimization • Expansion path equation: – the expansion path is a straight Cobb-Douglas Cost Minimization • Expansion path equation: – the expansion path is a straight](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-65.jpg)
![Cobb-Douglas Input Demand • Using the remaining FOC (production function), solve for the input Cobb-Douglas Input Demand • Using the remaining FOC (production function), solve for the input](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-66.jpg)
![Cobb-Douglas Cost • Now we can derive total costs as – Where – MC Cobb-Douglas Cost • Now we can derive total costs as – Where – MC](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-67.jpg)
![CES Cost Minimization • Suppose that the production function is CES: q = (K CES Cost Minimization • Suppose that the production function is CES: q = (K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-68.jpg)
![CES Cost Minimization • Dividing the first FOC equation by the second gives us CES Cost Minimization • Dividing the first FOC equation by the second gives us](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-69.jpg)
![CES Expansion Path • Expansion path • Capital Labor ratio CES Expansion Path • Expansion path • Capital Labor ratio](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-70.jpg)
![CES Input Demand • Using the remaining FOC (production function), solve for the input CES Input Demand • Using the remaining FOC (production function), solve for the input](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-71.jpg)
![CES Cost Functions • To derive the total cost, we would use the inputs CES Cost Functions • To derive the total cost, we would use the inputs](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-72.jpg)
![Law of Demand? • With Cobb-Douglass, CES etc. we can take the partial derivative Law of Demand? • With Cobb-Douglass, CES etc. we can take the partial derivative](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-73.jpg)
![Comparative Statics • Setting up the Lagrangian: ℒ = w. L + v. K Comparative Statics • Setting up the Lagrangian: ℒ = w. L + v. K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-74.jpg)
![Comparative Statics • Plug solutions into FOC w - λ(w, v, q)·f. L(Lq(w, v, Comparative Statics • Plug solutions into FOC w - λ(w, v, q)·f. L(Lq(w, v,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-75.jpg)
![Comparative Statics • Differentiate these w. r. t. w To get: Notationally, replace, Lq(w, Comparative Statics • Differentiate these w. r. t. w To get: Notationally, replace, Lq(w,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-76.jpg)
![Matrix Notation and Cramer’s Rule Matrix Notation and Cramer’s Rule](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-77.jpg)
![Matrix Notation and Cramer’s Rule Matrix Notation and Cramer’s Rule](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-78.jpg)
![Matrix Notation and Cramer’s Rule • With a higher wage, the amount of L Matrix Notation and Cramer’s Rule • With a higher wage, the amount of L](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-79.jpg)
![Comparative Statics • Differentiate these w. r. t. q To get: Comparative Statics • Differentiate these w. r. t. q To get:](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-80.jpg)
![Matrix Notation and Cramer’s Rule If labor is an inferior input, we COULD minimize Matrix Notation and Cramer’s Rule If labor is an inferior input, we COULD minimize](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-81.jpg)
![Matrix Notation and Cramer’s Rule If there is IRS, although total cost is still Matrix Notation and Cramer’s Rule If there is IRS, although total cost is still](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-82.jpg)
![Shephard’s Lemma (Again) • Remember how we used Shephard’s Lemma to derive the compensated Shephard’s Lemma (Again) • Remember how we used Shephard’s Lemma to derive the compensated](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-83.jpg)
![Marginal Cost Function And also: That is, λ*= λ(v, w, q) tells us the Marginal Cost Function And also: That is, λ*= λ(v, w, q) tells us the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-84.jpg)
![Fixed Proportions Shephard’s Lemma Results • Suppose we have a fixed proportions technology • Fixed Proportions Shephard’s Lemma Results • Suppose we have a fixed proportions technology •](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-85.jpg)
![Cobb-Douglas Shephard’s Lemma Result • Suppose we have a Cobb-Douglas technology • The cost Cobb-Douglas Shephard’s Lemma Result • Suppose we have a Cobb-Douglas technology • The cost](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-86.jpg)
![CES Shephard’s Lemma Result • Suppose we have a CES technology • The cost CES Shephard’s Lemma Result • Suppose we have a CES technology • The cost](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-87.jpg)
![Properties of Cost Functions • Homogeneity – cost functions are all homogeneous of degree Properties of Cost Functions • Homogeneity – cost functions are all homogeneous of degree](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-88.jpg)
![Properties of AC and MC functions • Both AC and MC are HD 1 Properties of AC and MC functions • Both AC and MC are HD 1](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-89.jpg)
![Properties of Cost Functions • The total cost function is non-decreasing in q, v, Properties of Cost Functions • The total cost function is non-decreasing in q, v,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-90.jpg)
![Concavity of Cost Function At w 1, the firm’s costs are C(v, w 1, Concavity of Cost Function At w 1, the firm’s costs are C(v, w 1,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-91.jpg)
![Measuring Input Substitution • A change in the price of an input will cause Measuring Input Substitution • A change in the price of an input will cause](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-92.jpg)
![Input Substitution • In the Production lecture: • But more usefully, because RTS = Input Substitution • In the Production lecture: • But more usefully, because RTS =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-93.jpg)
![Input Substitution • This alternative definition of the elasticity of substitution – in the Input Substitution • This alternative definition of the elasticity of substitution – in the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-94.jpg)
![Size of Shifts in Costs Curves • The increase in cost caused by a Size of Shifts in Costs Curves • The increase in cost caused by a](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-95.jpg)
![Appendix, Envelope Derivation • The marginal cost function (MC) is found by computing the Appendix, Envelope Derivation • The marginal cost function (MC) is found by computing the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-96.jpg)
![Marginal Cost Function and • Back to the cost function Marginal Cost Function and • Back to the cost function](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-97.jpg)
![Marginal Cost Function So as long as cost are being minimized, this is true Marginal Cost Function So as long as cost are being minimized, this is true](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-98.jpg)
![Marginal Cost Function and • And combine: Marginal Cost Function and • And combine:](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-99.jpg)
- Slides: 99
![Cost Minimization Cost Minimization](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-1.jpg)
Cost Minimization
![Accountants vs Economists Economists think of costs differently from accountants Economics Cost Accountants vs. Economists • Economists think of costs differently from accountants. – Economics Cost](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-2.jpg)
Accountants vs. Economists • Economists think of costs differently from accountants. – Economics Cost = Explicit Costs + Implicit Costs – Accounting Cost = Explicit Costs – Opportunity Cost is synonymous with Cost – Measuring implicit opportunity costs is difficult • generally, these are resources “owned” by the firm • Includes the value of the entrepreneur’s time • Includes accounting profit at next best alternative use of firm’s resources – or, accounting profit from selling all owned resources and investing the proceeds in the best possible alternative investment.
![Fixed Cost and Sunk Cost Fixed vs Sunk cost fixed means its Fixed Cost and Sunk Cost • Fixed vs. Sunk cost – “fixed” means its](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-3.jpg)
Fixed Cost and Sunk Cost • Fixed vs. Sunk cost – “fixed” means its cost does not vary with output – “sunk” means its cost cannot be avoided • The security guard outside the parking lot is a fixed cost, but if they can be fired at a moment’s notice, the cost is not sunk • A long term lease for renting a building is fixed and sunk – Varian calls a fixed cost that is not sunk a “quasifixed cost” – Sunk, but not fixed? No, because if it varies, it can be avoided.
![Short Run Long Run Key distinction in terms of optimizing behavior if Short Run – Long Run • Key distinction in terms of optimizing behavior if](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-4.jpg)
Short Run – Long Run • Key distinction in terms of optimizing behavior if everything can be controlled vs. not everything can be controlled. • Short run, the quantities of some inputs used in production are fixed and some are variable. – Short run, firms can shut down (q=0), but cannot exit the industry • Long run, the quantity of all inputs used are variable. – Long run, firms can enter or exit an industry.
![To Distinguish Short Run from Long Run We need two inputs one always To Distinguish Short Run from Long Run • We need two inputs, one always](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-5.jpg)
To Distinguish Short Run from Long Run • We need two inputs, one always variable, and one that can only be adjusted periodically, but is then re-fixed. • To accomplish that, we assume two inputs • Homogeneous labor (L), measured in labor per time • Homogeneous capital (K), measured in machine per time • Entrepreneurial costs assumed to be zero (or included in fixed costs) • Inputs are hired in perfectly competitive markets, so price of L, w, and price of K, v, are not a function of L and K
![To be clear L is assumed always variable K is always fixed To be clear • L is assumed always variable • K is always fixed](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-6.jpg)
To be clear • L is assumed always variable • K is always fixed when the firm is in production, but can periodically be adjusted to a new fixed amount (so in the long run it can be varied to a new fixed amount). • That is, production always takes place in a short run situation.
![Notation K capital v rental rate of capital L Labor w wage Notation • K, capital; v, rental rate of capital • L, Labor; w, wage](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-7.jpg)
Notation • K, capital; v, rental rate of capital • L, Labor; w, wage rate 5340 Nomenclature (Typical Intermediate Micro Abbreviations in Parentheses) Short Run Long Run SC (TC) C (LRTC) SAC (ATC) AC (LRAC) Variable Cost VC - Average Variable Cost AVC - Fixed Cost FC - Average Fixed Cost AFC - SMC (MC) MC (LRMC) Total Cost Average Total Cost Marginal Cost
![Cost Clearly C can be stated as C v K w Cost • Clearly, C can be stated as: C = v. K + w.](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-8.jpg)
Cost • Clearly, C can be stated as: C = v. K + w. L • But this is meaningless, we need cost as a function of output, not K and L. • To get that, back to the production function
![q Graphically q 3 q 2 K q 1 q fK L q q Graphically q 3 q 2 K q 1 q = f(K, L) q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-9.jpg)
q Graphically q 3 q 2 K q 1 q = f(K, L) q 0 An entire range of input combinations can be used to produce every level of output. q 0 q 2 q 3 q 1 L
![Short Run K constant q fKK 3 L q q fKK 2 Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-10.jpg)
Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2, L) q = f(K=K 1, L) q 3 q 2 q 1 Possible to produce the same level of output at different combinations of K and L. Three different slices through the production function at different levels of K. L
![Short Run K constant q Lets look at just one slice q 2 At Short Run, K constant q Let’s look at just one slice. q 2 At](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-11.jpg)
Short Run, K constant q Let’s look at just one slice. q 2 At some fixed level of K, for every q, we know how much L it will take. L 2 L
![Production to Cost Flip the axis q L Lq Production to Cost Flip the axis q. L Lq](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-12.jpg)
Production to Cost Flip the axis q. L Lq
![Production to Cost L L Lq K q Production to Cost L L = L(q, K) q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-13.jpg)
Production to Cost L L = L(q, K) q
![Production to Cost time Multiply L by w to get labor cost SC Production to Cost $/time Multiply L by w to get labor cost SC =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-14.jpg)
Production to Cost $/time Multiply L by w to get labor cost SC = w·L(q)+v. K Add in FC, SC=SVC+FC VC = w· L(q) FC q
![Production to Cost time And we can figure a few things SC w Production to Cost $/time And we can figure a few things SC = w·](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-15.jpg)
Production to Cost $/time And we can figure a few things SC = w· L(q)+r. K VC = w· L(q) FC q
![Production to Cost SMC SAC AVC AFC q Production to Cost $ SMC SAC AVC AFC q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-16.jpg)
Production to Cost $ SMC SAC AVC AFC q
![Short Run K constant q fKK 3 L q q fKK 2 Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-17.jpg)
Short Run, K constant q = f(K=K 3, L) q q = f(K=K 2, L) Inflection point, q 3 Inflection point, q 2 q = f(K=K 1, L) With more K, you can produce more with a given L and the inflection point moves Inflection towards more L and q. point, q 1 L What does that mean for cost curves?
![Production to Cost SC w L 2qr K 2 SC w Production to Cost $ SC = w· L 2(q)+r. K 2 SC = w·](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-18.jpg)
Production to Cost $ SC = w· L 2(q)+r. K 2 SC = w· L 3(q)+r. K 3 Inflection Points FC SC = w· L 1(q)+r. K 1 q Fixed cost is higher with more K, but the inflection point is further to the right, with a slower build-up of crowding.
![C C SC w L 2qr K 2 SC w L C $ C SC = w· L 2(q)+r. K 2 SC = w· L](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-19.jpg)
C $ C SC = w· L 2(q)+r. K 2 SC = w· L 3(q)+r. K 3 SC = w· L 1(q)+r. K 1 q C is the lowest point on any of the SC curves for any q
![SAC Min value at higher q with more K SAC C 1q SAC $ Min value at higher q with more K. SAC = C 1/q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-20.jpg)
SAC $ Min value at higher q with more K. SAC = C 1/q SAC = C 2/q SAC = C 3/q LAC q AC is the lowest point on any of the SR ATC curves for any q
![MC C MC is the slope of the C curve at any q MC C $ MC is the slope of the C curve at any q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-21.jpg)
MC C $ MC is the slope of the C curve at any q q
![Short Run Long Run Firms ALWAYS produce in a short run situation Short Run – Long Run • Firms ALWAYS produce in a short run situation](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-22.jpg)
Short Run – Long Run • Firms ALWAYS produce in a short run situation with at least one fixed and one variable input. • Being in the long run simply means the firm has adjusted to an optimal level of capital to minimize the cost of producing any chosen (profit maximizing) level of output.
![Different Shapes for the long run C Curve C There are four different Different Shapes for the long run C Curve C $ There are four different](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-23.jpg)
Different Shapes for the long run C Curve C $ There are four different potential shapes q
![C CRS C CRS as K and L are scaled upwards C C -- CRS C $ CRS, as K and L are scaled upwards, C](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-24.jpg)
C -- CRS C $ CRS, as K and L are scaled upwards, C rises at the same rate. That is, MC is constant. q
![C CRS Min value at higher q with more K ATC C -- CRS $ Min value at higher q with more K. ATC =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-25.jpg)
C -- CRS $ Min value at higher q with more K. ATC = TC 1/q ATC = TC 2/q ATC = TC 3/q AC=MC q CRS, as K and L are scaled upwards, LAC rises at the same rate. That is, LAC is constant.
![C IRS C IRS as K and L are scaled upwards C C -- IRS C $ IRS, as K and L are scaled upwards, C](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-26.jpg)
C -- IRS C $ IRS, as K and L are scaled upwards, C rises more slowly That is, MC is decreasing. q
![C IRS AC MC q CRS as K and L are scaled C -- IRS $ AC MC q CRS, as K and L are scaled](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-27.jpg)
C -- IRS $ AC MC q CRS, as K and L are scaled upwards, AC rises at a slower rate. AC is decreasing.
![C DRS DRS as K and L are scaled upwards TC rises C -- DRS $ DRS, as K and L are scaled upwards, TC rises](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-28.jpg)
C -- DRS $ DRS, as K and L are scaled upwards, TC rises faster That is, MC is increasing. C q
![LTC DRS Note it is not the low point of each ATC LTC -- DRS $ Note, it is not the low point of each ATC](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-29.jpg)
LTC -- DRS $ Note, it is not the low point of each ATC curve, but the simply the lowest point on any ATC. MC AC q CRS, as K and L are scaled upwards, AC rises at a faster rate. AC is increasing.
![C IRS CRS DRS C IRS then CRS then DRS q C – IRS, CRS, DRS $ C IRS, then CRS, then DRS q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-30.jpg)
C – IRS, CRS, DRS $ C IRS, then CRS, then DRS q
![C IRS CRS DRS So we have a Ushaped AC curve for C – IRS, CRS, DRS $ So we have a U-shaped AC curve for](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-31.jpg)
C – IRS, CRS, DRS $ So we have a U-shaped AC curve for a completely different reason than the SAC curve is U-shaped. MC AC q
![Firm Decisions So which short run curve should we be on That Firm Decisions • So which short run curve should we be on? – That](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-32.jpg)
Firm Decisions • So which short run curve should we be on? – That is, how much K (and then L) do we want to hire (with K fixed in the SR)? • Two choices – Profit Maximization: Firms maximize profit by choosing q*, K and L to minimize cost all at once. – Cost minimization: Firms choose a q*, then choose K and L to minimize the cost of producing q*.
![Profit Maximization Economic profits are equal to total revenue Profit Maximization • Economic profits ( ) are equal to = total revenue -](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-33.jpg)
Profit Maximization • Economic profits ( ) are equal to = total revenue - total cost • Total costs for the firm are given by C = w. L + v. K • Total revenue for the firm is given by total revenue = R = p·q = p·f(K, L) • Economic profits ( ) are equal to = p·f(K, L) - w. L - v. K
![Profit Maximization Solving to maximize profit means jointly choosing q q along Profit Maximization • Solving to maximize profit means jointly choosing q = q* along](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-34.jpg)
Profit Maximization • Solving to maximize profit means jointly choosing q = q* along with K* and L*. – Yields profit maximizing factor (input) demand functions which provide L* and K* that minimizes the cost of producing q*. – When r, w, and p change, L*, K*, and q* all change. – L*=L(w, v, p); K*=K(w, v, p); q*=q(w, v, p) – These functions allow for a change in q* (the isoquant) when prices change.
![Cost Minimization That firms minimize cost is a weaker hypothesis of firm behavior Cost Minimization • That firms minimize cost is a weaker hypothesis of firm behavior](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-35.jpg)
Cost Minimization • That firms minimize cost is a weaker hypothesis of firm behavior than profit maximization. – Yields quantity constant (quantity contingent) factor (input) demand functions which provide L* and K* that minimize the cost of producing q 0. – When r and w change, L* and K* do change, but q does not, stay on one isoquant. • Why bother? It is how we get the cost functions and curves (i. e. what we use in principles and intermediate micro)
![New Direction in Graphing In all the graphs above we have illustrated the New Direction in Graphing • In all the graphs above, we have illustrated the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-36.jpg)
New Direction in Graphing • In all the graphs above, we have illustrated the long run as a series of short run curves and traced out the envelope. • Good for intuition, but not terribly tied to the math of the optimization • Let’s switch to the isoquant graph.
![Intuitively K Isoquant all combinations of factors that yield the same output Slope is Intuitively K Isoquant, all combinations of factors that yield the same output. Slope is](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-37.jpg)
Intuitively K Isoquant, all combinations of factors that yield the same output. Slope is -d. K/d. L Isocost (total cost), all combinations of factors that yield the same total cost of production. Slope is -w/v L
![CostMinimizing Input Choices When K and L change a small amount Along Cost-Minimizing Input Choices • When K and L change a small amount • Along](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-38.jpg)
Cost-Minimizing Input Choices • When K and L change a small amount • Along an Isoquant,
![CostMinimizing Input Choices TRS is the change in K needed to replace one Cost-Minimizing Input Choices • TRS is the change in K needed to replace one](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-39.jpg)
Cost-Minimizing Input Choices • TRS is the change in K needed to replace one L while maintaining output. • Minimum cost occurs where the TRS is equal to w/v – the rate at which K can be traded for L in the production process = the rate at which they can be traded in the marketplace
![Intuitively K w20 v10 wv2 TRS 25 Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=. 25 • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-40.jpg)
Intuitively K w=20, v=10 w/v=2 TRS=. 25 • Firing one L has MB of $20 and MC of $2. 5 • Lowers cost by $17. 50 L
![Intuitively K w20 v10 wv2 TRS1 Firing one L has MB of 20 Intuitively K w=20, v=10 w/v=2 TRS=1 • Firing one L has MB of $20](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-41.jpg)
Intuitively K w=20, v=10 w/v=2 TRS=1 • Firing one L has MB of $20 and MC of $10 • Lowers cost by $10 L
![Intuitively K w20 v10 wv2 TRS1 5 Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=1. 5 • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-42.jpg)
Intuitively K w=20, v=10 w/v=2 TRS=1. 5 • Firing one L has MB of $20 and MC of $15 • Lowers cost by $5 L
![Intuitively K w20 v10 wv2 TRS2 L Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=2 L • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-43.jpg)
Intuitively K w=20, v=10 w/v=2 TRS=2 L • Firing one L has MB of $20 and MC of $20 • Lowers cost by $0 • Further reductions in L require an increase in cost as TRS >w/r.
![Intuitively K w20 v10 wv2 TRS2 5 Firing one L has MB of Intuitively K w=20, v=10 w/v=2 TRS=2. 5 • Firing one L has MB of](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-44.jpg)
Intuitively K w=20, v=10 w/v=2 TRS=2. 5 • Firing one L has MB of $20 and MC of $25 • Raises cost by $5 L
![Plan 1 Figure out the quantity of K and L that minimize total cost Plan 1. Figure out the quantity of K and L that minimize total cost,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-45.jpg)
Plan 1. Figure out the quantity of K and L that minimize total cost, holding q constant. 2. Use the resulting factor demand curves to derive the cost functions.
![CostMinimizing Input Choices We seek to minimize total costs given q q 0 Cost-Minimizing Input Choices • We seek to minimize total costs given q =q 0](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-46.jpg)
Cost-Minimizing Input Choices • We seek to minimize total costs given q =q 0 q = f(K, L) = q 0 • Setting up the Lagrangian: FOCs are
![CostMinimizing Input Choices Dividing the first two conditions we get The costminimizing Cost-Minimizing Input Choices • Dividing the first two conditions we get • The cost-minimizing](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-47.jpg)
Cost-Minimizing Input Choices • Dividing the first two conditions we get • The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices • But also • Which tells us that for the last unit of all inputs hired should provide the same bang-for-thebuck.
![CostMinimizing Input Choices The inverse of this equation is also of interest Cost-Minimizing Input Choices • The inverse of this equation is also of interest •](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-48.jpg)
Cost-Minimizing Input Choices • The inverse of this equation is also of interest • The value of the Lagrange multiplier is the extra costs that would be incurred by increasing the output constraint slightly by hiring enough L or K to increase output by 1. • That is λ = MC of increasing production by one unit.
![CostMinimizing Input Choices SOC to ensure costs are RISING along the isoquant away Cost-Minimizing Input Choices • SOC to ensure costs are RISING, along the isoquant, away](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-49.jpg)
Cost-Minimizing Input Choices • SOC to ensure costs are RISING, along the isoquant, away from the tangency: • Bordered Hessian The part in brackets is the same condition required for strict quasi concavity of the production function
![Intuitively w20 r10 wr 2 K Total Cost RTS2 SOC satisfied as moving Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC satisfied as moving](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-50.jpg)
Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC satisfied as moving along the isoquant means increasing TC K* L* L • Cost minimized for q 0 when L=L* and K=K *
![SOC fail Intuitively w20 r10 wr 2 K Total Cost RTS2 SOC not SOC fail, Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC not](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-51.jpg)
SOC fail, Intuitively w=20, r=10 w/r = 2 K Total Cost RTS=2 SOC not satisfied as moving along the isoquant means decreasing TC K* L* L • Cost maximized for q 0 when L=L* and K=K *
![Conditional Factor Demand aka Constant Output Solve the FOC to derive K Conditional Factor Demand (aka Constant Output) • Solve the FOC to derive – K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-52.jpg)
Conditional Factor Demand (aka Constant Output) • Solve the FOC to derive – K * =Kq (w, v, q 0) – L * =Lq (w, v, q 0) • Earlier, we considered an individual’s expenditureminimization problem – to develop the compensated demand for a good • In the present case, cost minimization leads to a demand for capital and labor to produce a constant quantity of output.
![The Firms Expansion Path The expansion path is the locus of costminimizing tangencies K The Firm’s Expansion Path The expansion path is the locus of cost-minimizing tangencies K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-53.jpg)
The Firm’s Expansion Path The expansion path is the locus of cost-minimizing tangencies K K=KE(w, v, L) The curve shows how inputs increase as output increases K* K* K* L* L
![The Firms Expansion Path The costminimizing combinations of K and L for every The Firm’s Expansion Path • The cost-minimizing combinations of K and L for every](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-54.jpg)
The Firm’s Expansion Path • The cost-minimizing combinations of K and L for every level of output (according to the quantity constant factor demand functions) • If input prices remain constant for all amounts of K and L, locus of cost-minimizing choices is the expansion path • Solve the minimization condition for K to get the expansion path
![The Firms Expansion Path The expansion path does not have to be a The Firm’s Expansion Path • The expansion path does not have to be a](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-55.jpg)
The Firm’s Expansion Path • The expansion path does not have to be a straight line – the use of some inputs may increase faster than others as output expands • depends on the shape of the isoquants • The expansion path does not have to be upward sloping – if the use of an input falls as output expands, that input is an inferior input
![Cost Functions Long Run Cost Short Run Cost The factor demand Cost Functions • Long Run Cost • Short Run Cost – The factor demand](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-56.jpg)
Cost Functions • Long Run Cost • Short Run Cost – The factor demand curves we have just derived are not appropriate as they each assume L and K are variable.
![Short Run Cost Functions With only two inputs once K is fixed the Short Run Cost Functions • With only two inputs, once K is fixed, the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-57.jpg)
Short Run Cost Functions • With only two inputs, once K is fixed, the production function dictates how much L is needed for each q. I am just using the “s” superscript here to denote Short Run
![Short Run Long Run Pick a level of output q 1 and Short Run & Long Run • Pick a level of output, q 1 and](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-58.jpg)
Short Run & Long Run • Pick a level of output, q 1 and hold w and v constant C q 1 q
![Short Long Relationship MC SMC ATC AC AVC q Short Long Relationship $ MC SMC ATC AC AVC q](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-59.jpg)
Short Long Relationship $ MC SMC ATC AC AVC q
![SMC vs LMC When q falls in SR cost falls by less than SMC vs. LMC $ When q falls in SR, cost falls by less than](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-60.jpg)
SMC vs. LMC $ When q falls in SR, cost falls by less than in the LR; Means from q 1 to q 2 the SMC is lower as C starts higher at q 1. When q rises in SR, cost rises by more than in the LR q 2 q 1 q 3 q
![Short Long Relationship Easier to see if assume CRS SMC actually lower from Short Long Relationship (Easier to see if assume CRS) $ SMC actually lower from](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-61.jpg)
Short Long Relationship (Easier to see if assume CRS) $ SMC actually lower from q 1 to q 2 in the short run as total cost is higher to start SMC (K=K 1) ATC (K=K 1) MC=AC AVC (K=K 1) q 1 q 2 q 3 q
![Fixed Proportions Factor Demand Cost Functions Suppose we have a fixed proportions technology Fixed Proportions Factor Demand Cost Functions • Suppose we have a fixed proportions technology](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-62.jpg)
Fixed Proportions Factor Demand Cost Functions • Suppose we have a fixed proportions technology such that q = f(K, L) = min(a. K, b. L) • To minimize cost, production will occur at the vertex of the L-shaped isoquants where q = a. K = b. L (any extra K or L only drives up cost) • Expansion path: • Factor Demand Curves Kq=q/a and Lq=q/b • Cost function: C(w, v, q) = v. K + w. L = v(q/a) + w(q/b)
![CobbDouglas Cost Minimization Suppose that the production function is Cobb Douglas q Cobb-Douglas Cost Minimization • Suppose that the production function is Cobb. Douglas: q =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-63.jpg)
Cobb-Douglas Cost Minimization • Suppose that the production function is Cobb. Douglas: q = K L • The Lagrangian expression for cost minimization of producing q 0 is ℒ = v. K + w. L + (q 0 - K L ) • The FOCs for a minimum are ℒK = v - K -1 L = 0 ℒL = w - K L -1 = 0 ℒ = q 0 - K L = 0
![CobbDouglas Cost Minimization Dividing the first equation by the second gives us Cobb-Douglas Cost Minimization • Dividing the first equation by the second gives us •](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-64.jpg)
Cobb-Douglas Cost Minimization • Dividing the first equation by the second gives us • This production function is homothetic – the TRS depends only on the ratio of the two inputs
![CobbDouglas Cost Minimization Expansion path equation the expansion path is a straight Cobb-Douglas Cost Minimization • Expansion path equation: – the expansion path is a straight](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-65.jpg)
Cobb-Douglas Cost Minimization • Expansion path equation: – the expansion path is a straight line – The K/L ratio is a function of w and v.
![CobbDouglas Input Demand Using the remaining FOC production function solve for the input Cobb-Douglas Input Demand • Using the remaining FOC (production function), solve for the input](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-66.jpg)
Cobb-Douglas Input Demand • Using the remaining FOC (production function), solve for the input demand equations K* = Kq (v, w, q) L* = Lq (v, w, q)
![CobbDouglas Cost Now we can derive total costs as Where MC Cobb-Douglas Cost • Now we can derive total costs as – Where – MC](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-67.jpg)
Cobb-Douglas Cost • Now we can derive total costs as – Where – MC – AC
![CES Cost Minimization Suppose that the production function is CES q K CES Cost Minimization • Suppose that the production function is CES: q = (K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-68.jpg)
CES Cost Minimization • Suppose that the production function is CES: q = (K +L ) / • The Lagrangian expression for cost minimization of producing q 0 is ℒ = v. K + w. L + [q 0 - (K +L ) / ] • The FOCs for a minimum are ℒ L = w - ( / )(K +L )( - )/ ( )L -1 = 0 ℒ K = v - ( / )(K +L )( - )/ ( )K -1 = 0 ℒ = q 0 - (K + L ) / = 0
![CES Cost Minimization Dividing the first FOC equation by the second gives us CES Cost Minimization • Dividing the first FOC equation by the second gives us](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-69.jpg)
CES Cost Minimization • Dividing the first FOC equation by the second gives us • This production function is also homothetic
![CES Expansion Path Expansion path Capital Labor ratio CES Expansion Path • Expansion path • Capital Labor ratio](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-70.jpg)
CES Expansion Path • Expansion path • Capital Labor ratio
![CES Input Demand Using the remaining FOC production function solve for the input CES Input Demand • Using the remaining FOC (production function), solve for the input](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-71.jpg)
CES Input Demand • Using the remaining FOC (production function), solve for the input demand equations K* = Kq (v, w, q) L* = Lq (v, w, q)
![CES Cost Functions To derive the total cost we would use the inputs CES Cost Functions • To derive the total cost, we would use the inputs](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-72.jpg)
CES Cost Functions • To derive the total cost, we would use the inputs demand functions and get
![Law of Demand With CobbDouglass CES etc we can take the partial derivative Law of Demand? • With Cobb-Douglass, CES etc. we can take the partial derivative](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-73.jpg)
Law of Demand? • With Cobb-Douglass, CES etc. we can take the partial derivative w. r. t. price of the inputs and see how quantity demanded responds. • But if we don’t know the functional form, how do we know how demand for L and K will respond to changes in w and v? • Comparative Statics
![Comparative Statics Setting up the Lagrangian ℒ w L v K Comparative Statics • Setting up the Lagrangian: ℒ = w. L + v. K](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-74.jpg)
Comparative Statics • Setting up the Lagrangian: ℒ = w. L + v. K + [q 0 - f(K, L)] • FOCs are ℒL = w - ·f. L = 0 ℒK = v - ·f. K = 0 ℒλ = q 0 - f(K, L) = 0 • Solve for L* = Lq(w, v, q) K* = Kq (w, v, q) λ* = λq (w, v, q)
![Comparative Statics Plug solutions into FOC w λw v qf LLqw v Comparative Statics • Plug solutions into FOC w - λ(w, v, q)·f. L(Lq(w, v,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-75.jpg)
Comparative Statics • Plug solutions into FOC w - λ(w, v, q)·f. L(Lq(w, v, q), Kq (w, v, q)) ≡ 0 v - λ(w, v, q)·f. K(Lq (w, v, q), Kq (w, v, q)) ≡ 0 q 0 - f(Kq (w, v, q), Lq (w, v, q)) ≡ 0 • These are identities because the solutions (FOC) are substituted into the equations from which they were solved. • Whatever prices and output may be, the firm will instantly adjust K and L to minimize cost of that level of output.
![Comparative Statics Differentiate these w r t w To get Notationally replace Lqw Comparative Statics • Differentiate these w. r. t. w To get: Notationally, replace, Lq(w,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-76.jpg)
Comparative Statics • Differentiate these w. r. t. w To get: Notationally, replace, Lq(w, v, q) with L*, etc.
![Matrix Notation and Cramers Rule Matrix Notation and Cramer’s Rule](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-77.jpg)
Matrix Notation and Cramer’s Rule
![Matrix Notation and Cramers Rule Matrix Notation and Cramer’s Rule](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-78.jpg)
Matrix Notation and Cramer’s Rule
![Matrix Notation and Cramers Rule With a higher wage the amount of L Matrix Notation and Cramer’s Rule • With a higher wage, the amount of L](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-79.jpg)
Matrix Notation and Cramer’s Rule • With a higher wage, the amount of L used will definitely fall. • It is possible that K becomes more productive with less L being used (if f. KL< 0). • If this effect is large enough, the higher productivity of capital can more than compensate for the higher cost of labor and MC can fall. • Unlikely in the real world.
![Comparative Statics Differentiate these w r t q To get Comparative Statics • Differentiate these w. r. t. q To get:](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-80.jpg)
Comparative Statics • Differentiate these w. r. t. q To get:
![Matrix Notation and Cramers Rule If labor is an inferior input we COULD minimize Matrix Notation and Cramer’s Rule If labor is an inferior input, we COULD minimize](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-81.jpg)
Matrix Notation and Cramer’s Rule If labor is an inferior input, we COULD minimize cost with more K and less L. I. e. , a “backward bending” expansion path.
![Matrix Notation and Cramers Rule If there is IRS although total cost is still Matrix Notation and Cramer’s Rule If there is IRS, although total cost is still](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-82.jpg)
Matrix Notation and Cramer’s Rule If there is IRS, although total cost is still rising, MC may well be falling along the expansion path.
![Shephards Lemma Again Remember how we used Shephards Lemma to derive the compensated Shephard’s Lemma (Again) • Remember how we used Shephard’s Lemma to derive the compensated](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-83.jpg)
Shephard’s Lemma (Again) • Remember how we used Shephard’s Lemma to derive the compensated demand curves from the expenditure function? • We can do it again from the cost function
![Marginal Cost Function And also That is λ λv w q tells us the Marginal Cost Function And also: That is, λ*= λ(v, w, q) tells us the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-84.jpg)
Marginal Cost Function And also: That is, λ*= λ(v, w, q) tells us the MC of increasing production along the expansion path
![Fixed Proportions Shephards Lemma Results Suppose we have a fixed proportions technology Fixed Proportions Shephard’s Lemma Results • Suppose we have a fixed proportions technology •](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-85.jpg)
Fixed Proportions Shephard’s Lemma Results • Suppose we have a fixed proportions technology • The cost function is • For this cost function, output constant demand functions are quite simple:
![CobbDouglas Shephards Lemma Result Suppose we have a CobbDouglas technology The cost Cobb-Douglas Shephard’s Lemma Result • Suppose we have a Cobb-Douglas technology • The cost](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-86.jpg)
Cobb-Douglas Shephard’s Lemma Result • Suppose we have a Cobb-Douglas technology • The cost function is • where • then the output constant factor demand functions are:
![CES Shephards Lemma Result Suppose we have a CES technology The cost CES Shephard’s Lemma Result • Suppose we have a CES technology • The cost](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-87.jpg)
CES Shephard’s Lemma Result • Suppose we have a CES technology • The cost function is • The quantity constant demand functions for capital and labor are
![Properties of Cost Functions Homogeneity cost functions are all homogeneous of degree Properties of Cost Functions • Homogeneity – cost functions are all homogeneous of degree](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-88.jpg)
Properties of Cost Functions • Homogeneity – cost functions are all homogeneous of degree one in the input prices • a doubling of all input prices will double cost. • As derivatives of HD 1 functions are HD 0, the contingent demand functions must be HD 0, that is, a doubling of w and v will not affect the cost minimizing input mix. • inflation will shift the cost curves up and will not change Kc, Lc
![Properties of AC and MC functions Both AC and MC are HD 1 Properties of AC and MC functions • Both AC and MC are HD 1](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-89.jpg)
Properties of AC and MC functions • Both AC and MC are HD 1 meaning a doubling of input prices means a doubling of MC and AC – True enough, C=C(v, w, q) is HD 1 and – However, C is HD 1 in input prices, and MC is the derivative w. r. t. q. So the derivative of an HD 1 function being HD 0 does not apply here.
![Properties of Cost Functions The total cost function is nondecreasing in q v Properties of Cost Functions • The total cost function is non-decreasing in q, v,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-90.jpg)
Properties of Cost Functions • The total cost function is non-decreasing in q, v, and w. • Fixed factor production has a linear cost function w. r. t. w and v • To the extent that one factor can be substituted for another, the function will be concave to input prices.
![Concavity of Cost Function At w 1 the firms costs are Cv w 1 Concavity of Cost Function At w 1, the firm’s costs are C(v, w 1,](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-91.jpg)
Concavity of Cost Function At w 1, the firm’s costs are C(v, w 1, q 1) If the firm continues to buy the same input mix as w changes, its cost function would be Cpseudo C=w. L 1+v 1 K 1 C C*=C(v, w, q 1) Since the firm’s input mix will likely change, actual costs will be less than C such as C*=C(v, w, q 1) C(v, w 1, q 1) w 1 w
![Measuring Input Substitution A change in the price of an input will cause Measuring Input Substitution • A change in the price of an input will cause](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-92.jpg)
Measuring Input Substitution • A change in the price of an input will cause the firm to alter its input mix • The change in K/L in response to a change in w/v, while holding q constant is
![Input Substitution In the Production lecture But more usefully because RTS Input Substitution • In the Production lecture: • But more usefully, because RTS =](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-93.jpg)
Input Substitution • In the Production lecture: • But more usefully, because RTS = w/v at cost minimum
![Input Substitution This alternative definition of the elasticity of substitution in the Input Substitution • This alternative definition of the elasticity of substitution – in the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-94.jpg)
Input Substitution • This alternative definition of the elasticity of substitution – in the two-input case, s must be nonnegative – large values of s indicate that firms change their input mix significantly if input prices change – s = 0 for fixed factor production
![Size of Shifts in Costs Curves The increase in cost caused by a Size of Shifts in Costs Curves • The increase in cost caused by a](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-95.jpg)
Size of Shifts in Costs Curves • The increase in cost caused by a change in the price of an input will be largely influenced by • the relative significance of the input in the production process • the ability of firms to substitute another input for the one that has risen in price – easy substitution means little change in costs.
![Appendix Envelope Derivation The marginal cost function MC is found by computing the Appendix, Envelope Derivation • The marginal cost function (MC) is found by computing the](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-96.jpg)
Appendix, Envelope Derivation • The marginal cost function (MC) is found by computing the change in total costs for a change in output produced
![Marginal Cost Function and Back to the cost function Marginal Cost Function and • Back to the cost function](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-97.jpg)
Marginal Cost Function and • Back to the cost function
![Marginal Cost Function So as long as cost are being minimized this is true Marginal Cost Function So as long as cost are being minimized, this is true](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-98.jpg)
Marginal Cost Function So as long as cost are being minimized, this is true
![Marginal Cost Function and And combine Marginal Cost Function and • And combine:](https://slidetodoc.com/presentation_image_h2/b53a8541a9a22ceb6cdcae8ded4963f3/image-99.jpg)
Marginal Cost Function and • And combine:
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