06 producer Flashcards
(18 cards)
Topics of 06 Producer’s Slide:
How can we model technology?
What is the producer’s problem?
What do short run and long run mean?
How can we solve the producer’s problem analytically and
graphically?
- Concepts you should know and understand:
Production function
Producer’s problem (short run and long run)
Marginal product
Technical rate of substitution - Methods you should be able to apply:
Representation of production functions with isoquants
Graphical analysis of the producer’s problems (long run and short run)
Analysis of producer’s problems using first-order conditions (long run
and short run) - What you should be able to explain yourself:
Analogy between consumer theory and producer theory: what is analogous, what is different?
The production function determines..
the maximum possible output y as a function of the input bundle x used
f(x)=y
What are Isoquants?
Isoquants are curves showing all combinations of inputs (like labor and capital) that produce the same level of output.
-“f(x) = y” olan tüm x kombinasyonları bir isoquant oluşturur.
-aynı çıktıyı (y) sağlayan tüm girdi (x) kombinasyonları = bir isoquant.
What does the position of an isoquant on a graph indicate about the level of output?
Isoquants that are further up and to the right correspond to higher levels of production.
What is the marginal product (MP) of an input factor?
The additional output produced when one more unit of the input is used, other inputs constant.
∂f (x1, x2) / ∂x1 = MP1 (x1, x2)
MP1 > 0
What is decreasing marginal product?
Each additional unit of input contributes less to output than the previous one.
∂MP1(x1, x2) / ∂x1 < 0
Still increases output but less = Artış devam eder ama giderek küçülür.
What is the Technical Rate of Substitution (TRS)?
The TRS describes how much more of factor 2 is needed when a little of factor 1 is given up.
TRS = Bir girdiden vazgeçerken, diğer girdiyi ne kadar artırman gerekir ki üretim sabit kalsın?
TRS(x1,x2) = MP1 (x1, x2) / MP2 (x1, x2)
What is diminishing (decreasing) Technical Rate of Substitution (TRS)?
the more of one input is used, the less of another input is required to maintain the same output level.
d(TRS) / dx1 < 0
x1 arttıkça, TRS azalıyor → her yeni x1 birimi için daha az x2 tasarrufu oluyor.
The firm’s profit
the producer chooses the optimal input quantities x1 , . . . , xn to maximize their profit.
The firm’s profit is the difference between revenue and costs:
π = p. f(x1 , . . . , xn) - mΣj=1 wj xj
What is the profit function for a single-input producer?
π = p . f(x1) - w1x1
p: output price
f(x1): production function
w1: input cost
What is the first-order condition for profit maximization for a single-input producer??
p . MP(x1) = w1
→
MP(x1)= w1 / p
Value of marginal product (p⋅MP) must equal input cost w1.
How is the optimal input x1* determined?
- Solve MP(x1*)= w1 / p identify all candidates
- Compare profits for these candidates at x1* and x1=0 (corner solution)
What is the short-run profit maximization problem with a fixed input ˉx2?
max x1 π=p⋅f(x1,ˉx2)-w1x1-w2ˉx2
if x1* > 0 → p. MP1 (x1*,ˉx2) = w1 the value of the marginal product of x1 equals its cost.
x1* = x1*(p,w1,w2,x2 üzeri cizgi)
p: Price of the output good
w1: Price of input x1
w2: Price of other inputs
x2 cizgi: Fixed quantity of another input (short-run constraint).
Key Idea: The function captures how a profit-maximizing firm adjusts x1 when market conditions (p,w1,w2) or fixed inputs (x2 cizgi) change.
Intuition of the Optimality Condition (Short Run):
MP1(x1* ,¯ x2) is the marginal product of input x1, that is, the additional output produced by using one more unit of x1, given the fixed level of input¯ x2.
p· MP1(x*1 ,¯ x2) is the market value of this additional output. w1 is the cost of an additional unit of x1.
The optimal choice of x1* is achieved when the value of the additional output exactly equals the cost of the additional input unit.
If
p· MP1(x1* ,¯x2) > w1, it would be profitable to use more x1;
p· MP1(x*1 ,¯x2) < w1, less x1 should be used.
isoprofit lines to MP1(x1* ,¯ x2) = w1 / p graphically ?
iso-profit lines are sets of pairs (y , x1) for which the same profits can be achieved.
ayni kari saglayan farklı (y,x1) kombinasyonlarıni gösterir
y = π/p + w1/p x1+ w2/p x2 cizgi
Long run profit maximisation
max π = p. f(x1,x2) - w1x1 -w2x2
x1,x2
Solution: FOC
p. MP1(x1,x2) = w1
p. MP2(x1,x2) = w2
Thus, we have two equations with two unknowns. From this, we can often
determine the factor demand curves:
x1* = x1* (p, w1, w2)
x2* = x2* (p, w1, w2)
