Economics of technology and management - Appunti

Economics of Technology and Management (3CFU)

Prof. Tiziana D'Alfonso - Ric. 9-11, Via Avisto

Part I: Microeconomics; Markets

Part II: Accounting Methods

Part III: Investments Decisions

Evaluation: (4-5 es.) + (1-2 open questions)

No lesson: 13/9, 22/924/10, 26/10

L 15

Input and output

Input: x in R^m, x = (x₁, ..., x_m) [for us: m=2] [x = (x₁, x₂) = (L, K)] labour capital

Output: y in R^k, y = (y₁, ..., y_k) [for us: k=1] [y = y₁ = Q] Quantity produced

Let's see some other useful definitions:

Production set

Production set: set of technologically feasible combinations of inputs and outputs

(y, x) in R^m+k

Ex: 1 BICYCLE {2 TIRES, 1 FRAME} => (1,2,1) is feasible

(1,3,1) is feasible

(1,1,0) is not feasible

Input requirement set

Input requirement set: V(y) = {x in R^m: (y, x) in Production set}

Ex: BICYCLE

y = 1 -> (2,1) in x in V(1)

(0,2) in x not in V(1)

given all possible combinations for a feasible input

Production function

Production function: the maximum feasible output that can be produced with a given input vector.

Q = P(L, K)

Production set

Production function

Economics of Technology and Management (3CFU)

Prof. Tiziano D'Alfonso - Ric. 9-11, Via Ariosto

Part I: Microeconomics; Markets

Part II: Accounting Methods

Part III: Investments Decisions

Evaluation: (4-5 es.) + (1-2 open questions)

No lesson: 13/9, 22/924/9, 26/9

L15

Input and output

Input: x in R^m, x = (x₁, ..., x_m)

for us: m=2 [ x = (x₁, x₂) = (L, K) ] Labour Capital

Output: y in R^k, y = (y₁, ..., y_k)

for us: k=1 [ y = y₁ = Q ] Quantity produced

Let's see some other useful definitions:

Production set

Production set: set of technologically feasible combinations of inputs and outputs

(y, x) in R^m+k

Ex: 1 bicycle { 2 tires, 1 frame } => (1,2,1) is feasible

(1,3,1) is feasible

(1,1,0) is not feasible

Input requirement set

Input requirement set: V(y) = { x in R^m : (y, x) is Production set }

Ex: bicycle

y = 1 -> (2,1) in x in V(1)

(0,2) in x not in V(1)

given all possible combinations for a feasible input

Production function

Production function: the maximum feasible output that can be produced with a given input vector.

Q = P(L, K)

Properties of the prod. funct.

• Continuous - small input changes -> small output changes
• Str. Increasing - If input -> output
• Str. Quasi-concave - the function become concave after a certain point.

The Q increasing due to an L increments follows a decreasing rate ← economically relevant!

Total factor productivity

Total Factor Productivity: relation between output and a given input, keeping the others constant

Ex. K = Kbar → Q_L = R(L, Kbar)

Q = K^1/2L^1/2

K = 1 → Q_L = L^1/2

Marginal productivity

Marginal Productivity: rate at which output changes per additional unit of inputs employed

Q = R(L, K) MP_L = ∂f(L, K)/∂L MP_K = ∂f(L, K)/∂K

Average productivity

Average Productivity: how much you can produce on average using one input

Q = R(L, K) AP_L = f(L, K)/L AP_K = f(L, K)/K

Relation between productivity function, MP and AP

Mettiamo in relazione la productivity function con MP e AP.

La MP indica il tasso di crescita di Q in funzione di L. Quindi MP_L è positiva fintanto che Q è crescente e presenterà un massimo nel punto di flesso in L.

Per un certo L^*, l'AP corrisponde alla pendenza di una retta che collega l'origine con il punto sulla production function relativo a L^* (per definizione).

Quindi AP raggiunge il massimo in un punto intermedio tra L^* e L^**; in questo punto (A) si ha MP_L = AP_L.

AP_L

MP_L

2

Dimostrazione

Dimostriamo questo ultimo aspetto:

L⁺ = argmax_L AP(L)

AP(L) = Q(L,K̃)/L

∂AP(L)/∂L |_L=L* = 0

∂AP(L)/∂L = (∂Q(L,K̃)/∂L · L - Q(L,K̃))/L² = (MP_L · L - Q(L,K̃))/L² => ∂AP(L)/∂L |_L=L* = (MP_L|L=L* · L* - Q(L*,K̃))/L*² ! = 0

=> MP(L)|_L=L* = Q(L*,K̃)/L*

=> MP(L)|_L=L* = AP(L)|_L=L*

MP_L e AP_L si intersecano in 0 e L*.

Ulteriori considerazioni

Facciamo ulteriori considerazioni:

Per L₁: L < L₂ => MP > AP_L

L > L₂ => MP < AP_L

la produttività degli input tende a diminuire con il loro aumento.

Two inputs prod. function - isoquants

Two inputs prod. function - ISOQUANTS

curva 3D

Per rappresentare la funzione possiamo fare riferimento alle curve ISOQ