producer theory Flashcards
what is a production plan?
A production plan is a vector y ∈ R^L whose components indicate
the amounts of the various inputs and outputs used in the production process.
By convention, if yj >0 (<0) then commodity j is an output (input).
For example, if L = 3, y = (−2,−3,7) means that the firm uses 2 units of commodity one and 3 units of commodity two to produce 7 units of commodity three.
why we need to talk about technological constraints?
Nature imposes technological constraints on firms: only certain
combinations of inputs are feasible ways to produce a given output.
To understand a firm’s choices, we need a convenient way to summarize the production possibilities for the firm.
The production set Y ⊂ R^L is the collection of all possible production plans that a firm can use
- any y ∈ Y is feasible
any y ∉ Y is not feasible
what differs when we have a specialized firm?
The production set is the most general way to characterize the firm’s
technology because it allows for multiple inputs and multiple outputs.
However, an important and simpler case is when a firm is “specialized” and only produces one output using many inputs.
In this case:
- We call the single output q , and the vector of inputs is x ∈ R₊^L, (each input is non-negative, you can’t use negative amounts of inputs)
We describe the firm’s technology (how the firm turns inputs into outputs) using a production function.
What happens when we have a firm that works with one input and one output?
- saying that the point y’ is in Y means that it is technologically possible to produce q’ amount of output with x’ amount of input
- since inputs are costly, it’s important to focus on producing the maximum output for a given input. If y’ is inside the production set but not on the upper boundary, it means the production is inefficient.
- The upper boundary of the production set is called the production function, and it shows the maximum output that can be produced for a given amount of input.
How does the production function acts for an economy with L commodities?
- In an economy with L commodities, there are many different goods that can be used as inputs for production.
- The production function f takes a specific combination of inputs (represented by a vector ) and tells us how much output can be produced from those inputs.
- The phrase : f maps R₊^{L-1} into R₊^L means that the production function takes a set of positive inputs (like how much of each good you use) and gives a positive output (like how much of the final product you can make)
R₊^{L-1}= input space R₊^L= output space
-The notation q=f(x) means that if you use the inputs from the vector , you can produce a maximum of units of output. - IMPORTANT: the production function is a cardinal function, which means it gives specific numerical values for how much output can be produced based on the inputs used.
what are the charactheristics of the production function?
The production function f is:
1. continuous: small changes in the vector of inputs lead to small changes in output
2. strictly increasing: using more of every input leads to more output
3. strictly quasi-concave: a production function is strictly quasi-concave when the output Y is strictly convex, because any convex combination of two input vectors produces at least as much as of the original two.
4. f(0)=0 If no inputs are used, , there will be no output. Positive output requires positive inputs
What is the Marginal Product?
The marginal product of input i, is the derivative of the production function.
MPi represents the extra output you get from adding one extra unit of input i
- For most inputs, this value is positive, meaning that using more of input i generally leads to an increase in output.
- Even though the MPi is positive, it can become smaller as you keep increasing the amount of input i while keeping all other inputs the same.
What does the law of diminishing marginal product states?
in all productive processes, the MP of an input will diminish if we add more and more of that input, keeping all other inputs constant.
what is an isoquant?
A further way of describing technological constraints is via an isoquant map.
An isoquant is the set of all possible input vectors producing the same
quantity of output. Denoting this set by I (q), then I (q) ≡{x ≥0 : f (x) = q}.
soquants are similar to indifference curves used in consumer theory. While indifference curves show combinations of goods that give the same level of satisfaction (utility), isoquants show combinations of inputs that produce the same level of output.
Isoquants have cardinal labeling, meaning they are labeled with specific output quantities. This is different from the ordinal labeling of indifference curves, which rank preferences without assigning specific numerical values.
what is the leontief production function?
It’s a particular production function that works with fixed proportions, meaning you cannot substitute one input for another.
q= f (x1,x2) = min{ax1,bx2}
where a >0 and b >0
Suppose that the inputs “tyres” (x1) and “steering wheels” (x2) are
used in the production of cars (and nothing else, for simplicity).
Then the Leontief production function is q = min{1/4x1,x2}.
The isoquants are L shaped
perfect substitutes
The inputs can be substituted freely at a constant rate so that the amount of output produced depends only on the total amount of inputs used.
The production function is, therefore, of the form: q= f (x1,x2) = ax1 + bx2. where a,b>0
Suppose that to produce one hamburger we need 100g of meat, which can be either Canadian beef (x1) or US beef (x2).
Since the kind of meat does not matter, the hamburgers made depend on the
total amount of beef available. Thus, we write the production function as q= x1 + x2.
The isoquants are just like the case of perfect substitutes in consumer
theory: straight lines
coubb-douglas function
It allows inputs to be substituted for each other, but not at a constant rate. The production function has the form: q= f (x1,x2) = Ax₁ªx₂ᴮ
where A, a, and b are >0
Although it is like the Cobb Douglas utility function, the parameters
have now a meaning:
- A measures the scale of production: how much output we would get
if we used one unit of each input.
− The parameters a and b measure how the amount of output
responds to changes in the inputs.
- The Cobb-Douglas isoquants have the same nice, well-behaved shape as
the C-D indifference curves
MRTS
the Marginal Rate of Rechnical Subsitution in the producer theory is analog to the MRS in consumer theory.
It measures the rate at which one input can be substituted for another without changing the amount of output produced.
MRTS=-MPi/MPj
(MP=∂f/∂x)
when preferences are strictly convex, how does MRTS acts?
Strict convexity of the technology implies that |MRTS|decreases as x₁ increases along the same isoquant.
Strict convexity means that as you use more of one input the additional benefit of that input gets smaller.
The slope of the isoquant decreases, making the curve flatter as you move along it.
what is the main difference between MP and Returns to scale?
- Marginal products show how output changes when you increase one input while keeping others fixed
- Returns to scale refer to how output varies as all inputs change in the same proportion
