8.1. Prot maximization, cost minimization and function cost. December 12, The production function has decreasing returns to scale.
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1 Prot maximization, cost minimization and function cost December 12, f(λl, λk) (λl) 1 / (λk) 1 /2 λ 1 / λ 1 /2 L 1 / }{{} λ 3 / f(k, L) < λf(k, L) f(k,l) The production function has decreasing returns to scale. 2. Min {L,K} CT wl + rk s.a. L 1 / Y We use the substitution method (you can also use Lagrangian method). L ( ) Y K 1 2 Y K 2 Then we introduce it in the objective function an we have a maximization problem without restrictions: Min {L,K} CT w ( Y K 2) + rk Then: CT K 0 2wY K 3 + r 0 1
2 Conditional demand of capital is: r 2wY 1 K 3 ( 2wY K(w, r, Y ) r To nd the conditional demand repeat the process clearing the restriction K as a function of L: K )1/3 ( ) 2 Y L 1 / Y 2 L 1 /2 We introduce it in the objective function an we have a maximization problem without restrictions: Min {L,K} T C wl + ry 2 L 1 /2 Then: T C L 0 w 1 2 ry 2 L 3 /2 0 Conditional demand of labour is: w ry 2 2L 3 /2 ( ry 2 L(w, r, Y ) 2w )2/3 3. T C wl + rk Substituting the conditional demands found previously: ( ry 2 T C w 2w )2/3 ( 2wY + r r )1/3 2
3 T C 2 2 /3 w 1 w 2 /3 }{{} r 2 /3 Y /3 + r 1 r 1 /3 }{{} 2 1 /3 w 1 /3 Y /3 w 1 /3 r 2 /3 Out common factor: ( ) T C r 2 /3 Y /3 w 1 / /3 21 /3 Medium cost: Marginal cost: } {{ } /3 MeC T C Y ( ) MeC Y 1 /3 w 1 /3 r 2 / /3 MgC T C Y ( ) MgC 22 3 Y 1/3 w 1 /3 r 2 / /3 2 /3 Y 1 /3 w 1 /3 r 2 / T RS (L,K) MgP L MgP K F/ L F/ K Calculating TRS for f(l, K) 3L 1 /3 K 1 /3 1 3 T RS 3L 2 /3 K 1 /3 2/ L1 /3 K L K 1 /3 2 /3 L 1 /3 K 2 /3 K L The TRS is the slope of the isoquant and measures the rate which you need to replace one factor production by another to keep production steady. In this case, we must replace a unit of L by a unit of K to maintain constant output. 3
4 2. f(λl, λk) 3 (λl) 1 /3 (λk) 1 /3 λ 1 /3 λ 1 /3 3L 1 /3 K 1 /3 }{{} λ 2 /3 f(k, L) < λf(k, L) f(k,l) Production function has decreasing returns to scale. 3. Functions of marginal productivity are: MgP L F L L 2 /3 K 1 /3 MgP K F K L1 /3 K 2 /3 Functions of medium productivity are: MeP L f(k, L) L 3L1 /3 K 1 /3 L 3L 2 /3 K 1 /3 MeP K f(k, L) K K 3L1/3 1/3 3L 1 /3 K 2 /3 K. The problem of maximization is: Max {L,K} Π : p f(k, L) wl rk First-order conditions are: I take second ecuation: Π L 0 p L 2 /3 K 1 /3 w Π K 0 p L1 /3 K 2 /3 r p L 1 /3 K 2 /3 r
5 I replace it in the rst ecuation: Then: L(w, r, p) L 1 /3 rk 2 /3 p L 2 /3 r2 K /3 p 2 pk 1 /3 ( r 2 K /3 p 2 ) w p 3 r 2 K w K(w, r, p) L 2 /3 p3 r 2 w ( ) /3 r 2 p 3 r 2 w p 2 ( ) r 3 p 3 2 r 2 w p 3 r3 p 6 r p 3 w 2 p3 rw 2 5. Quantity supplied of product: K(2, 1, 2) L(2, 1, 2) f(2, ) 3 (2) 1 /3 () 1 /3 6 5
6 Calculating TRS for f(l, K) L 1 / K 1 / 1 T RS (L,K) L 3 / K 1 / 3/ 1 L1 / K L K 1 / 3 / L 1 / K 3 / K L The TRS is the slope of the isoquant and measures the rate which you need to replace one factor production by another to keep production steady. In this case, we must replace a unit of L by a unit of K to maintain constant output. 2. f(λl, λk) (λl) 1 / (λk) 1 / λ 1 / λ 1 / L 1 / K 1 / }{{} λ 1 /2 f(k, L) < λf(k, L) f(k,l) Production function has decreasing returns to scale: 3. Functions of marginal productivity are: MgP L F L L 3/ K 1 / MgP K F K L1 / K 3/ Functions of medium productivity are: MeP L f(k, L) L L1 / K 1 / L L 3 / K 1 / MeP K f(k, L) K L1/ K 1/ L 1 / K 3 / K 6
7 . The maximization problem is: Max {L,K} Π : p f(k, L) wl rk First-order conditions are: I take second ecuation: Π L 0 p L 3/ K 1 / w Π K 0 p L1 / K 3/ r p L 1 / K 3 / r I replace it in rst ecuation: Then: L(w, r, p) L 1 / rk 3 / p L 3 / r3 K 9 / p 3 pk 1 / ( r 3 K 9 / p 3 ) w p r 3 K 2 w p2 K(w, r, p) r 3 /2 w 1 /2 L 3 / ( ) 9/ r 3 p 2 r 3 /2 w 1 /2 p 3 ( ) r p 2 3 r 3 /2 w 1 /2 p r p 6 r 9 /2 p w p2 3 /2 r 1 /2 w 3 /2 7
8 5. Quantity supplied of product: K(,, ) 2 3 /2 1 /2 1 L(,, ) 2 1 /2 3 /2 1 f(1, 1) (1) 1 / (1) 1 / 8
9 8. (a) First-order conditions are: Max {L,K} Π : p L 1 / wl rk Π L 0 p 1 L 3/ w Π K 0 p 1 2 L1 / K 1/2 r 9
10 Second ecuation: p L 1 / K 1 /2 2r K 1 /2 2r pl 1 / pl1 / I replace it in rst ecuation: p 1 ( ) L 3 / pl 1/ w 2r 2r p 2 L 1 /2 w 8r L 1 /2 8rw p 2 L 1 /2 L(w, r, p) p2 8rw p (8wr) 2 Then: ( ) 1/ p p (8wr) 2 2r ( p p (8wr) 1 /2 2r ) p 2 2r(8wr) 1 /2 K(w, r, p) p r 2 8rw p 32r 3 w Product supply: ( p )1/ ( p Y (w, r, p) (8wr) 2 32r 3 w )1/2 p (8rw) p 2 1 / /2 r 3 /2 w p 3 1 /2 8 1 / /2 }{{} r 2 w 16 10
11 3. Min {L,K} CT : wl + rk s.a. L 1 / Y We use the Lagrangian method: L(L, K, λ) wl + rk λ(l 1 / Y ) First-order conditions are: L L 0 w 1 λl 3 / 0 L K 0 r 1 2 λl1 / K 1 /2 0 L λ 0 L1 / Y 0 Multiplying rst ecuation by L and second by K: Then: wl 1 λl 3 / }{{ L } L 1 / } {{ } Y rk 1 2 λl1 / K 1 /2 K }{{} }{{} Y 1 λy 1 2 λy L λ Y w K λ Y 2r Substituting in third ecuation for clear λ: ( λ Y ) 1/ ( λ Y ) 1/2 Y w 2r λ (1 /+ 1 /2) Y 1 / (w) 1 / Y 1 /2 (2r) 1 /2 Y λ 3 / Y Y 1 / (w) 1 / Y 1 /2 (2r) 1 /2 Y Y 1 / (w) 1 / Y 1 /2 (2r) 1 /2 Y 1 / (w) 1 / (2r) 1 /2 11
12 ( ) /3 λ Y 1 / (w) 1 / (2r) 1 /2 Y 1/3 (w) 1 /3 (2r) 2 /3 Substituting λ in L and K : L(w, r, Y ) Y 1 /3 (w) 1 /3 (2r) 2 /3 Y w Y /3 (2r) 2 /3 (w) 2 /3 ( ) 2/3 1 Y /3 r 2 /3 2 w 2 /3 K(w, r, Y ) Y 1 /3 (w) 1 /3 (2r) 2 /3 Y 2r Function cost in the long-term: Y /3 (w) 1 /3 (2r) 1 /3 2 1 /3 Y /3 w 1 /3 r 1 /3 T C lt w L(w, r, Y ) + r K(w, r, Y ) w ( ) 2/3 1 Y /3 r 2 /3 + r 2 Y 1 /3 w 1 /3 /3 2 w 2 /3 r 1 /3 Out common factor: ( ) 2/3 1 Y /3 r 2 /3 w 1 / /3 Y /3 r 2 /3 w 1 /3 2 ( (1 ) 2/3 ( ) ( ) T C st (w, r, Y ) + 2 /3) 1 Y /3 r 2 /3 w 1 /3 1 2/ /3 2 2 /3 Y /3 r 2 /3 w 1 /3 3 Y /3 r 2 /3 w 1 / /3 2 2 /3. Supply function. ( 3 Max {L,K} Π : p Y p /3 c.p.o Π Y 0 ) Y /3 r 2 /3 w 1 /3 } {{ } T C /3 Y 1 /3 r 2 /3 w 1 /3 0 Y 1 /3 p 2 2 /3 r2 /3 w 1 /3 12
13 Y (w, r, p) p3 16 r 2 w Is the same supply function as that found in rst paragraph. 5. We are talking about elasticity: ε T C,w T C /T C w/w ) ε T Clt,w ( /3 ( /3 T C w w T C Y /3 r 2 /3 w 2 /3 w ) 1 Y /3 r 2 /3 w 1 /3 3 If wage w increases 1 %, long-term cost of producing Y units increases in 1/3%. 6. In the short-term capital is constant: We isolate L in the constraint: Min T C wl + rk s.a. L 1 / Y L K 2 Y We substituting it in objective function to nd function cost in the shortterm: T C st (w, r, Y ) w Y K 2 + rk 7. T C st (1, 1, Y ) T C lt (1, 1, Y ) ( 3 1 Y /3 ) Y /3 1 2 /3 1 1 /3 13
14 (b) 1) 2. First-order conditions are: Max {L,K} Π : p L 1 /3 K 2 /3 wl rk Π L 0 p 1 3 L 2/3 K 2 /3 w Π K 0 p 2 3 L1 /3 K 1/3 r Multiplying the rst ecuation by L and second ecuation by K: 1
15 p 1 3 L 2 /3 }{{ L } K 2 /3 wl L 1 /3 p 2 3 L1 /3 K 1 /3 K }{{} K 2 /3 rk On the assumption that Y L 1 /3 K 2 /3 we reformulate these expressions: p 1 3 Y wl p 2 3 Y rk Solve for L and K to nd the optimal demands of the factors: L py 3w K 2pY 3r We introduce the optimal demands of the factors in the production function: L 1 /3 K 2 /3 Y Out common factor: ( ) 1/3 ( ) 2/3 py 2pY Y 3w 3r ( Y 1 /3+ 2 /3 p ) 1/3 ( ) 2/3 2p Y 3w 3r ( p ) 1/3 ( ) 2/3 2p 0 3w 3r What's the problem? When the company has constant returns to scale, supply function is not well dened. This company is indierent to his production level. 15
16 (c) First-order conditions are: Max {L,K} Π : p L 3 / K 3 / wl rk Π L 0 p 3 L 1/ K 3 / w Π K 0 p 3 L3 / K 1/ r Multiplying rst ecuation by L and the second by K: 16
17 p 3 L 1 / }{{ L } K 3 / wl L 3 / p 3 L3 / K 1 / K }{{} K 3 / rk On the assumption that Y L 3 / K 3 / we reformulate these expressions: p 3 Y wl p 3 Y rk Solve L and K to nd optimal demands of the factors: L 3 K 3 We introduce the optimal demands of the factors in the production function: py w py r L 3 / K 3 / Y ( ) 3/ ( ) 3/ 3 py 3 py Y w r Y 3 /+ 3 / ( 3 Y 1 /2 Y 1 /2 Y (w, r, p) ( 3 ( 3 ) 3/ ( p 3 w ( 3 ) 3/ ( p 3 w ) 3/ ( p 3 w ) 3/2 ( p 3 w ) 3/ p Y r ) 3/ p r ) 3/ p r ) 3/2 p (w /2 ( r)3 r 3 p) 3 17
18 3. Min {L,K} T C : wl + rk s.a. L 3 / K 3 / Y Lagrange method: L(L, K, λ) wl + rk λ(l 3 / K 3 / Y ) First-order conditions are: L L 0 w 3 λl 1 / K 3 / 0 L K 0 r 3 λl3 / K 1 / 0 L λ 0 L3 / K 3 / Y 0 Multiplying rst ecuation by L and second ecuation by K: Then: wl 3 λl 1 / }{{ L } L 3 / K 3 / } {{ } Y rk 3 λl3 / K 1 / K }{{} K 3 / }{{} Y L λ 3 Y w K λ 3 Y r Substituting in the third ecuation to clear λ: λ 3 /2 3 λy 3 λy ( λ 3Y ) 3/ ( λ 3Y ) 3/ Y w r ( ) 3/2 3 Y 3 /2 w 3 / r 3 / Y λ 3 /2 ( ) 3/2 3 Y 1 /2 w 3 / r 3 / 18
19 λ Substituting λ in L and K : L(w, r, Y ) K(w, r, Y ) Function cost in the long-term: ( ) 1 3 Y 1 /3 w 1 /2 r 1 /2 ( ) 1 3 Y 1 /3 w 1 /2 r 3 Y 1 /2 w Y 2 /3 w 1 /2 r 1 /2 ( ) 1 3 Y 1 /3 w 1 /2 r 3 Y 1 /2 r Y 2 /3 w 1 /2 r 1 /2 CT lt w L(w, r, Y ) + r K(w, r, Y ) w Y 2 /3 w 1 /2 r 1 /2 + r Y 2 /3 w 1 /2 r 1 /2 2Y 2 /3 w 1 /2 r 1 /2. Supply function: Max {L,K} Π : p Y 2Y 2 /3 w 1 /2 r 1 /2 }{{} CT c.p.o Π Y 0 p 3 Y 1 /3 w 1 /2 r 1 /2 Y 1 /3 3 w1 /2 r 1 /2 p 1 Y (w, r, p) (w r)3 /2 ( 3 p) 3 Is the same supply function as that found in rst paragraph. 19
20 5. We are talking about elasticity: ε T Clt,w 1 2 2Y 2 /3 w 1 /2 r 1 /2 w 1 2Y 2 /3 w 1 /2 r 1 /2 2 If wage w increases 1 %, long-term cost of producing Y units increases in 1/2%. 6. Short-term capital is constant: Min T C wl + rk s.a. L 3 / K 3 / Y We isolate L in the restriction: L K 1 Y /3 Substituting in objective function to nd the cost function in the short-term: T C st (w, r, Y ) w K 1 Y /3 + rk 7. T C st (1, 1, Y ) T C lt (1, 1, Y ) Y / Y 2 /3 Y /3 2Y 2 / We have a quadratic ecuation: Y 2 ± (1)(1)
21 (d) The company has constant returns to scale. 21
22 (e) The company has constant returns to scale. 8.5 Max {L} Π : pl 1 /2 wl rk c.p.o : Π L 0 1/2 pl 1 /2 w 0 22
23 L 1 /2 2w p L 1 /2 p 2w L(w, p) p2 K (2w) 2 Supply function: Y (w, p) ( p 2 )1/2 K ( )1/2 K (2w) 2 p pk 2w 2w How does a change of w in the oer? Y w pk 2 p and K are positives therefore an increase of w decreases the production. How does a change of p in the oer? Y p K 2w w and K are positives therefore an increase of p increases the production. 8.6 Then: Max {Y } Π : py Y 3 + 7Y 2 17Y 66 c.p.o : Π Y 0 p 3Y 2 + 1Y 17 0 Rewrite: 23
24 3Y 2 + 1Y (17 p) 0 We have a quadratic equation: Y 1 ± 1 2 ( 3)( 17 + p) 2( 3) Graphic: p 8, para p 6 2 / f(λl) (λl) α λ α L α Is a homogeneous function of degree α. 2. Y L α 2
25 L Y 1 /α T C w Y 1 /α 3. MgC 1 α w Y (1 α) /α. Yes it's true. MeC w Y 1 /α Y w Y (1 α) /α Min {L,K} 2L + K s.a. 27L 2 K Y Lagrangian method: L(L, K, λ) 2L + K λ(27l 2 K Y ) First-order conditions: 2 2λ27LK 0 27LK 1 λ 1 λ27l L 2 1 λ 27L 2 K Y 0 27L 2 K Y Equate the rst and the second equation: 27LK 27L 2 25
26 K L Substituting in third equation: 27L 2 L Y Total Costs: L K ( ) 1/3 Y 27 ( ) 1/3 Y 27 ( ) 1/3 ( ) 1/3 ( ) 1/3 Y Y Y T C
27 2. ( ) 1/3 1 MeC 3 Y 2 /3 27 MgC ( ) 1/3 1 Y 2 / Min {L,K} 2L + K s.a. 27L 2 K Y We isolate K in constraint: K Y 27L 2 27
28 Substituting in objective function: 2L + Y 27L 2 Deriving with respect to L and equate to zero: 2 2Y 27L 0 L Y 27 Total costs: T C 2 27 Y + 1. MeC Y 28
29 MgC
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