Producer Theory

Question 1

Production Set

Answer 1

(Technology)

Denoted by Y, this is a subset of R^L and represents all the possible combinations of inputs and outputs that the firm can produce given its technology.

The production set is technically feasible for the firm.

Question 2

Transformation Function

Answer 2

This is a function F that represents the technology of the firm.

It maps a production plan to a real number.

The production set Y is defined by all the production plans y that satisfy F(y) ≤ 0:

Y = {y ∈ R | F(y) ≤ 0 }

This condition essentially means that the firm can only produce output levels that are technologically feasible, as determined by the transformation function.

Question 3

Marginal Rate of Transformation (MRT)

Answer 3

This is the rate at which the firm can substitute between two goods.

It’s defined as the negative ratio of the marginal product of one good to the marginal product of another good:

MRT= -MP/MP
MRT = (δF(y) / δy_l) / (δF(y)/ δy_k)

This ratio tells us how many units of good k the firm must give up to produce one more unit of good l, holding the level of technology constant.

This is an important concept in production theory:

• it reflects the trade-offs in production
• and the opportunity cost of producing one more unit of a good in terms of other goods.

Question 4

Properties of Production Set:
Y is non-empty

Answer 4

There is at least one feasible production plan in the set.

Question 5

Properties of Production Set:
Y is closed

Answer 5

The production set includes its boundary points; it is “closed” in a topological sense.

Question 6

Properties of Production Set:
No free lunch

Answer 6

You can’t get output without input; the only way to get zero output is to have zero input.

y ∈ Y and y ≥ 0 ⇒ y = 0

Question 7

Properties of Production Set:
Possibility of inaction

Answer 7

Not producing anything (zero input and output) is a feasible choice.

0 ∈ Y

Question 8

Properties of Production Set:
Possibility of inaction and fixed & sunk cost

Answer 8

Inaction is possible as long the origin still belongs to the production set.

If the firm experiences fixed costs, the firm is using an amount of input 1 without obtaining any output in return. Inaction, however, is still possible since the origin still belongs to the production set.

If the costs that the firm must incur (e.g., setup costs) are sunk, then the firm cannot move towards the origin 0.

Question 9

Properties of Production Set:
Free Disposal

Answer 9

You can dispose of excess production without cost; the production set extends towards lower outputs.

Y - R ∈ Y

Question 10

Properties of Production Set:
Irreversibility

Answer 10

Once production occurs, you cannot go back to the original inputs; there’s no “undo” in the production set.

Suppose that production plan y belongs to production set Y (and that it does not coincide with the origin). Then, production plan –y cannot belong.

There is no way back.

y ∈ Y and y ≠ 0 ⇒ -y ∉ Y

Question 11

Properties of Production Set:
Non-increasing returns to scale

Answer 11

Increasing input proportionally increases output by the same or lesser amount.

If production plan y belongs to Y, then a scaling down of production plan y, αy for α∈[0,1], is also part of the production set Y.

y ∈ Y ⇒ αy ∈ Y for all α ∈ [0;1]

Question 12

Properties of Production Set:
Non-increasing returns to scale and fixed & sunk cost

Answer 12

Nonincreasing returns to scale maintain an interesting relationship with the presence of fixed and sunk costs. In particular, the presence of any of these costs implies that the firm’s production set violates nonincreasing returns to scale.

Question 13

Properties of Production Set:
Non-decreasing returns to scale

Answer 13

Increasing input proportionally increases output by the same or greater amount.

If production plan y belongs to Y, then a scaling up of production plan y, αy for α ≥ 1, is also part of the production set Y.

In contrast the figure on the right shows a production set that violates nondecreasing returns to scale: scaling up production plan y yields a new production plan that does not belong to production set Y.

Question 14

Properties of Production Set:
Non-decreasing returns to scale and fixed & sunk cost

Answer 14

Unlike our previous discussion about the relationship between nonincreasing returns to scale and fixed and sunk costs, nondecreasing returns to scale can be satisfied even when firms incur fixed and sunk costs. The next two figures illustrate this point: scaling up production plan y yields a new production plan that belongs to production set Y, both when firms incur fixed costs (left figure) and when they incur sunk costs (right figure).

Question 15

Properties of Production Set:
Constant returns to scale

Answer 15

Inputs increased by any proportion result in outputs increasing by the same proportion.

If production plan y belongs to Y, then production plan αy also belongs to Y, for any α ≥ 0.

Question 16

Constant, increasing and decreasing returns to scale using isoquants

Answer 16

Question 17

Properties of Production Set:
Additivity (Free Entry)

Answer 17

Combining two feasible production plans yields another feasible plan.

Question 18

Properties of Production Set:
Convexity

Answer 18

A production set is convex if, when you take any two production points within this set, any linear combination or ‘average’ of these two points is also within the set.

The convexity of the production set implies that you can create a new production plan that involves producing, for instance, half of what plan A produces and half of what plan B produces, and this new plan will also be feasible

Question 19

The elasticity of substitution

Answer 19

it measures the percentage change in the two inputs used in response to a percentage change in their prices

• or measures the proportionate change in the ratio k/l relative to the proportionate change in the MRTS along an isoquant.
• also measures the curvature of an isoquant = the substitutability between inputs (or goods) for the other
• Note that the value of the elasticity of substitution is positive because when k/l decreases (increases) the MRTS decreases as well (increases as well, respectively).
  Indeed, if we move along an isoquant towards higher amounts of labor, then the ratio k/l decreases and the isoquant becomes flatter (reducing MRTS).

Question 20

PMP

Answer 20

The goal is to maximize profit, which is the revenue (price vector p times the production plan y), subject to the production technology constraints (the production set Y
and transformation function F).

max p y s.t. F(y) ≥ 0

Question 21

PMP Graphically

Answer 21

Question 22

PMP Single Output Case

Answer 22

Question 23

Profit Function

Answer 23

The value function resulting from the PMP, π (p), denoted as the “profit function” of the firm, associates every p with the highest amount of profits, chosen by the profit-maximizing production plan y(.), known as the supply correspondence.

π = p * q - w*z

or

π(p)= max {p y}

Question 24

Properties of profit function:
π (.) is homogenous of degree one

Answer 24

Means that an increase in prices of in- or output produces a proportional increase in profit.

Question 25

Properties of profit function:
π (.) is convex

Answer 25

Meaning that as you move between two different sets of prices, the profit doesn’t decrease at an increasing rate.

This is a common assumption, indicating that there are diminishing marginal gains in profit as prices rise.

Question 26

Properties of profit function:
If Y is convex, then Y = {y ∈ R | p * y ≤ π (p) for all p»0}

Answer 26

If Y is convex, then Y = {y ∈ R | p * y ≤ π (p) for all p»0}

• If the production set Y is convex, all production vectors for all prices (p y) generate less profits than the optimal profit function π (p).
• This describes the feasible production plans that are not more profitable than what is achievable at price p.

Question 27

Properties of profit function:
y(.) is homogenous of degree 0 in p

Answer 27

• The supply function y(⋅) is invariant to proportional changes in all prices.
• If all prices double, the supply function doesn’t change; it suggests that the firm’s choice of what to supply is based on relative prices, not absolute prices.
• Example: Inflation, deflation

Question 28

Properties of profit function:
Convexity of set y(p)

Answer 28

If Y is convex, then y(p) is a convex set for all p.

• If two different production plans are profitable at some price vector, then any combination of those production plans will also be profitable
• follows from the fact that the profit function is linear in quantities, and linear functions preserve convexity.

Moreover, if Y is strictly convex, then y(p) is single valued.

• If Y is strictly convex, there are no flat regions on the boundary of Y, which doesnt allow for multiple production plans yielding the same profit level
• so there is a unique profit-maximizing production plan, making y(p) a function (single-valued) rather than a correspondence (multi-valued)

Question 29

Properties of profit function:
Hotelling’s Lemma

Answer 29

Connects a firm’s profit maximization behavior with its supply decisions.

• It specifically relates the gradient of the profit function with respect to prices to the firm’s supply function
• It is a direct consequence of the duality theorem
• it can be established by the related arguments of
  the envelope theorem and of first-order conditions.

1) Single Valued Supply Function:

This states that if the supply function y(p) is single-valued (it consists of a single point), then the gradient of the profit function with respect to prices ∇ π (p) gives you the supply at those prices.

2) Gradient of the profit function

The gradient of the profit function ∇ π (p) with respect to prices is a vector where each component = the partial derivative of profit with respect to the price of one good.
This gradient tells us how much the profit function changes as the prices of the goods change.

Firm’s Response to Price Changes:
Hotelling’s Lemma gives us a way to understand how a firm’s profit responds to changes in prices. If we know the firm’s supply at some prices, we can predict how profits will change if those prices change.
Dual Relationship:
It shows a dual relationship between the profit function and the supply function. The firm’s supply choices at certain prices directly inform us about the slope of the profit function at those prices.

Question 30

Cost Minimisation Problem

Answer 30

Assuming free disposal:

min z≥0 wz s.t. f(z) ≥ q

The firm selects a vector of inputs (or factors of production), z, that minimizes total costs, wz, subject to productive feasibility.

The optimal vector of inputs is denoted as z(w,q), and it is usually referred to as the conditional factor demand correspondence.

Question 31

Conditional factor demand correspondence

Answer 31

z(w,q)

This term refers to the set of input combinations that will produce a certain quantity q at the lowest cost, given the input prices w.

The term “correspondence” is used because there might be multiple ways (sets of inputs) to produce q units at the lowest cost, especially when there are multiple inputs.

Question 32

Cost Function

Answer 32

c(w,q)

This function represents the minimum cost of producing a given quantity of output q given a vector of input prices w.

The cost function encapsulates the firm’s technology and the prices of inputs.

It tells us the lowest possible cost required to produce q units of output, given the prices for inputs like labor, materials, etc.

There are many properties of the cost function.

Question 33

Properties of the cost function

Answer 33

Assume that Y is closed and satisfies free disposal, then:

1) Homogeneity of Degree One
2) nondecreasing in q and w
3) c(.) is a concave function of w

4) z(.) is homogenous of degree zero in w

5) Convexity of Input Combinations and Uniqueness of Conditional Factor Demand

6) Shephard’s Lemma

7) If f(.) is homogenous of degree one, then c(.) and z(.) are homogenous of degree one q.

8) If f(.) is concave, then c(.) is convex function of q.

Question 34

Properties of the cost function:
Homogeneity of Degree One in w and nondecreasing in q

Answer 34

If you double all the input prices w, the minimum cost to produce a certain quantity q will also double.

As the quantity q you wish to produce increases, the minimum cost does not decrease. It either stays the same or increases, which is intuitive—producing more generally costs more.

Question 35

Properties of the cost function:
c(.) is a concave function of w

Answer 35

Concavity in this context means that the cost function exhibits diminishing returns to input prices. If you picture the graph of the cost function, a concave function will have a shape that curves downwards.

If you keep increasing the input prices, the cost of production does not increase as rapidly beyond a certain point.

The concavity of the cost function in input prices w indicates that a firm can respond to changes in input prices in a way that the resulting cost is less sensitive to price changes than if costs were a linear function of w

Question 36

Properties of the cost function:

Answer 36

If the set of all input combinations that can produce at least a certain quantity (set 1) is convex, then (set 2)

the firm’s production set is just the collection of all production plans where the revenue is at most equal to the firm’s maximum profit for any positive price level.

Essentially, this says that the firm’s feasible production plans are shaped by the technology that allows mixing inputs efficiently to maximize profits.

The conditional factor demand z(w,q) corresponds to the input bundle that minimizes the cost function c(w,q).

When a firm is solving its cost minimization problem, it selects the quantity of inputs (the input bundle) that will cost it the least to produce the desired output q.

The result of this minimization problem is reflected in the cost function. For a given q and input prices w, the cost function tells you the cost of producing q, and the conditional factor demand tells you which inputs and in what quantities will achieve that cost.

Question 37

Properties of the cost function:
z(.) is homogenous of degree zero in w

Question 38

Properties of the cost function:

Answer 38

If for any output level q, the input combinations that can produce at least q are convex (mixtures of inputs are feasible)= set 1,

then the conditional factor demand z(w,q) at given prices and output level is a convex set.

If the set is strictly convex, then there is a unique input combination for each w and q.

Question 39

Properties of the cost function: Shepherds Lemma

Answer 39

Shepard’s Lemma states that if the cost function is differentiable, then the gradient of the cost function with respect to input prices w at given prices and output q will give the conditional factor demand z(w,q).

Question 40

Properties of the cost function:

Answer 40

If the conditional factor demand is differentiable, then the Hessian matrix of the cost function with respect to w is symmetric and negative semidefinite.

This implies that costs increase at a decreasing rate with respect to factor prices.

Question 41

Properties of the cost function:
If f(.) is homogenous of degree one, then c(.) and z(.) are homogenous of degree one in q.

Answer 41

If the production function exhibits constant returns to scale, then both the cost function and conditional factor demand function are homogeneous of degree one in output q, meaning if output scales up, costs and input demands scale up proportionally.

Question 42

Properties of the cost function:
If f(.) is concave, then c(.) is convex function of q

Answer 42

If the production function is concave, then the cost function is a convex function in q, implying that the marginal costs are nondecreasing with output.

Question 43

Properties of profit function

Answer 43

Assume Y is closed and satisfied the free disposal property:

1) π (.) is homogenous of degree one in p

2) π (.) is convex in p

3) If Y is convex, then Y = {y ∈ R | p * y ≤ π (p) for all p»0}

4) y(.) is homogenous of degree zero

5) If Y is convex, then y(p) is a convex set for all p. Moreover, if Y is strictly convex, then y(p) is single valued.

6) Hotelling’s Lemma