Intermediate Microeconomics Profit Maximization and Cost Minimization Tin Cheuk (Tommy) Leung CUHK

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Profit Maximization

Profit = Revenue - Cost profit maximizing: marginal revenue = marginal cost

Short run vs long run fixed and variable factors

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Short Run Profit Maximization Fixed factor: capital Variable factor: labor Cobb-Douglas production function Profit maximization problem becomes: max pAK α L1−α − wL − rK L

Profit maximizing condition: pA(1 − α)(K /L)α = w |{z} | {z } marginal cost marginal revenue

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Isoprofit Lines

Denote profit as π and output as y : π = py − wL − rK Isoprofit line: y=

w π + rK L+ p p |{z} | {z } intercept slope

Profit maximizing condition: tangent point of isoprofit line and production function

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Short Run Comparative Statics

w↓ marginal cost ↓⇒ L ↑ slope of isoprofit line ↓⇒ L ↑

p↓ marginal revenue ↓⇒ L ↓ slope of isoprofit line ↑⇒ L ↓

r↓ marginal revenue and marginal cost unchanged ⇒ L unchanged slope of isoprofit line unchanged ⇒ L unchanged

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Exercise

√ Production function: f (x) = 4 x Price per unit of output: $100 Price per unit of input: $50 1 2

3

Profit maximizing input and output? Profit maximizing input and output with $20 per-unit tax on output and $10 per-unit subsidy on input? Profit maximizing input and output with 50% profit tax?

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Long Run Profit Maximization

Profit maximization with cost constraint max f (K , L) s.t. C¯ = wL + rK K ,L

First Order Condition implies: TRS = ⇒

Tin Cheuk (Tommy) Leung (CUHK)

MPL w = MPK r

MPL MPK = w r

Intermediate Microeconomics

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Long Run Profit Maximization

Profit maximization with fixed output min C = wL + rK s.t. f (K , L) = y¯ K ,L

First Order Condition implies: TRS = ⇒

Tin Cheuk (Tommy) Leung (CUHK)

MPL w = MPK r

MPL MPK = w r

Intermediate Microeconomics

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Long Run Profit Maximization

General maximization without constraints max π = pf (K , L) − wL − rK K ,L

First Order Condition implies: TRS = ⇒

Tin Cheuk (Tommy) Leung (CUHK)

MPL w = MPK r

MPL MPK = w r

Intermediate Microeconomics

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Elasticity of Scale

Elasticity of scale: Eσ =

σ %4output df (σL, σK ) = %4scale dσ f (σL, σK )

Eσ > 1: increasing returns to scale Eσ = 1: constant returns to scale Eσ > 1: decreasing returns to scale

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Profit Maximization and Constant Returns to Scale

Zero profit in the long run can double profit otherwise

Cannot pin down output level without further info can increase/decrease scale without altering profit

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Revealed Profitability

Use firm’s choices to study firm’s behavior derive supply and factor demand trace out production technology

Make use of two things: 1 2

feasible production plan (y , L, K ) under prices (p, w , r ) firm are profit maximizing

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Weak Axiom of Profit Maximization (WARM)

If a firm faces (p i , w i , r i ) and chooses (y i , Li , K i ) WARM holds if p t y t − w t Lt − r t K t ≥ p t y s − w t Ls − r t K s and p s y s − w s Ls − r s K s ≥ p s y t − w s Lt − r s K t

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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From WARM to Firm’s Behavior

From WARM we can derive (p t − p s )(y t − y s ) − (w t − w s )(Lt − Ls ) − (r t − r s )(K t − K s ) ≥ 0 ⇒ 4p4y − 4w 4L − 4r 4K ≥ 0 Two properties of firm’s behavior 1 2

product supply curve upward sloping factor demand curve downward sloping

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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From WARM to Production Technology

Suppose there’s only one input - labor π t = p t y t − w t Lt Isoprofit line becomes wt πt y = t L+ t p p Points above isoprofit line not feasible from WARM Can trace out a quasi-concave production function

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Cost Minimization

Profit maximization with fixed output min C = wL + rK s.t. f (K , L) = y¯ K ,L

First Order Condition implies: TRS = ⇒

Tin Cheuk (Tommy) Leung (CUHK)

MPL w = MPK r

MPL MPK = w r

Intermediate Microeconomics

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From Cost Minimization to Cost Curve

Perfect Complements f (l, k) min{l, k} Cost wy + ry = (w + r )y

Tin Cheuk (Tommy) Leung (CUHK)

Perfect Substitutes l +k min{w , k}y

Intermediate Microeconomics

Cobb-Douglas l ak b a b 1 Kw a+b r a+b y a+b

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Cost Curve with Cobb-Douglas Production

L = mink,l wl + rk − µ[l a k b − y ] −b

b

b

1

l(w , r , y ) = ( ba ) a+b w a+b r a+b y a+b a

a

−a

1

k(w , r , y ) = ( ba )− a+b w a+b r a+b y a+b c(w , r , y ) = wl(w , r , y ) + rk(w , r , y ) a b 1 a b a −a = [( ) a+b + ( ) a+b ]w a+b r a+b y a+b b b

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Cost and Returns to Scale

Constant Returns to Scale Decreasing Returns to Scale Increasing Returns to Scale

Tin Cheuk (Tommy) Leung (CUHK)

Total Cost Increase linearly in y Increase > linearly in y Increase < linearly in y

Intermediate Microeconomics

Average Cost constant in y ↑ in y ↓ in y

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Short Run Cost Functions

Total Cost: C (y ) = y 3 + 1 Variable Cost (VC) Fixed Cost (FC) Average Variable Cost (AVC) Average Fixed Cost (AFC) Average Cost (AC) Marginal Cost (MC)

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Properties of Short Run Cost Functions

U shape AC curve AFC ↓ at the beginning AVC ↑ later

MC cuts the minimum of AC/AVC from below if MCAC ⇒ AC ↑

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Long Run Cost Functions

Short run total cost: cs (y , k) Optimal capital level for a given output level: k(y ) Long run total cost: c(y ) = cs (y , k(y )) Long run cost ≤ short run cost ¯ c(y ) ≤ cs (y , k) ∵ k freely adjustable in long run long run AC curve = envelope of short run AC curves

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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