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# System prepended metadata

title: Producer's Behavior
tags: [micro]

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tags: micro
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Producer's Behavior
===
## Production
### Production Vector

>In an economy with **L** commodities, a **production vector/production plan** is a vector $y=(y_1,...,y_L)\in R^L$. It is convenient to arrange the net input commodities before net output commodities. For instance, a production vector with only net output will be like $y=(-z_1,...,-z_{L-1},q)$, where $z_i\ge 0$ and $q\ge0$.

### Production Set

>A **production set** $Y$ is the collection of all feasible production plans for a specific firm. In other words, $y \in Y$ and $Y \subset R^L$.

### Properties of Production Sets

1. $Y$ is nonempty.

2. $Y$ is closed.

    To ensure the existence of production function.

3. No free lunch.

    If $y \in Y$ and $y \ge 0$, then $y=0$, i.e., the firm could not use zero inputs to produce and positive output.
4. Possibility of inaction.

    $0 \in Y$, i.e., the firm can complete shutdown. It may not hold with sunk cost, but it is plausible in the long run.

5. Free disposal.

    If $y \in Y$ and $y' \le y$, then $y' \in Y$, i.e., the firm can always waste inputs and throw out outputs.
    
6. Irreversibility.

    If $y \in Y$ and $y \ne 0$, then $-y \notin Y$.
7. Nonincreasing return to scale.

    If $y \in Y$, then $\alpha y \in Y$ for $\alpha \in [0,1]$. It is the small part of CRTS.
8. Nondecreasing return to scale.

    If $y \in Y$, then $\alpha y \in Y$ for $\alpha \ge 1$. It is the big part of CRTS.
9. Constant return to scale.

    If $y \in Y$, then $\alpha y \in Y$ for $\alpha \ge 0$. It is CRTS itself.
10. Additivity/free entry.

    If $y, y' \in Y$, then $y+y' \in Y$.
11. Convexity.


    If $y, y' \in Y$, then $\alpha y+(1-\alpha)y' \in Y$ for $\alpha \in [0,1]$. It assumes that "unbalanced" input combinations are not more productive than "balanced" ones.
    
12. $Y$ is a convex cone.

    If $y, y' \in Y$, then $\alpha y+ \beta y' \in Y$ for $\alpha, \beta \in [0,1]$.
    

## Profit Maximization
### The Profit Maximization Problem
Given a price vector $p\gg 0$ and a production vector $y \in \mathbb{R}^L$, the profit of operation $y$ is $p \cdot y = \sum_{i=1}^L p_i y_i$. Hence the firm's profict maximization problem (PMP) given a production set $Y$ is
$$\max_{y}  p \cdot y \\ \text{such that $y$ is in $Y$}$$

If the prodcution is a single-optput technology with production function $f(z)$, the profit mazimization problem with output price scalar $p$ and input price vector $w$ is

$$\max_{z}  pf(z)-w \cdot z$$

### F.O.C.

If $z^*$ is the optimal input vector, it must satisfies the following first-order conditions,

$$p \frac{\partial f(z^*)}{\partial z_i} \le w_i, i = 1,...,L-1, \text{with equality if } z_i^* > 0$$ 

In matrix notation,

$$p \nabla f(z^*) \le w \text{  and  } [\nabla f(z^*) - w]z^* = 0$$

### Profit Function

If we can find the optimal $y(p)$ of different price vector in PMP, we can define the profit function 
$$\pi(p) = p \cdot y(p)$$, which is the value function of the profit maximazation problem.

### Properties
Suppose that $\pi(\cdot)$ is the profit function of the production set $Y$ and $y(\cdot)$ is the associated supply correspondence. Assume that $Y$ is closed and satisfies the free disposal property, then

1. Supply correspondence is homogeneous of degree zero.

2. Profit functiont is convex.

3. If $Y$ is convex, then $Y = \{y\in \mathbb{R}^L: p\cdot y \le \pi (p) \forall p>>0\}$

4. Profit function is homogeneous of degree one.

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 a single-valued function.

6. (Hotelling's lemma) If $y(p)$ is function differentiable at $p^*$, then $\nabla\pi(p^*)=y(p^*)$.

7. If $y(p)$ is a function differentiable at $p^*$, then $Dy(p^*)=D^2\pi(p^*)$ is a symmetric and positive semidefinite matrix with $Dy(p^*)p^*=0$

### Explainations

1. It means that porpotional changes of input and output prices will not change a firm's optimal porduciton plan.
2. Given two price vectors $p_1, p_2$, and $p_3 = \alpha p_1 + (1-\alpha) p_2$, we have 
$$ \pi (p_1) = p_1 \cdot y(p_1) \ge p_1 \cdot y(p_3) \\
 \pi (p_2) = p_2 \cdot y(p_2) \ge p_2 \cdot y(p_3) \\
\alpha \pi (p_1) + (1-\alpha) \pi (p_2) \ge p_3 \cdot y(p_3) = \pi(p_3)$$

There are two economic intuitions. The first is that the profit is linear in $p$ without changing production plan, hence the real profit will be better than linear, that is a convex function. The second is the firms can hugely change its input with skew price vectores, but they can do little in a averge price vector.

3. It is the duality between profit function and production set.

4. Following the H.D.0 of optimal production plan $y(p)$, the profit function $\pi(\cdot)$ is H.D.1.

5. Suppose we have two production plans $y_1, y_2$ that generate the same profit under price $p$. The linearity of profit means every convex combination of $y_1, y_2$ will generate the same profit under $p$. Hence, the convexity of $Y$ just ensure the convex combinations of $y_1, y_2$ are feasible. The reslut of strictly convex production set can be proved by contracdiction. Actually, the optimal production plan is in the intersection of production set and the hyperplan generated by the price vector $p$.

6. The Hotelling's lemma is an application of the envelope theorem. There is a more elegant proof: Given a fixed price vector $p^*$, define $g(p) = \pi (p ) - p \cdot y(p^*)$. Clearly, $g(p) \ge 0$ and achieve its minimal at $p=p^*$. After taking partial derivative of $g(p)$ with respect to $p_i$, we have 
$$\frac{\partial g(p)}{\partial p_i} |_{p=p^*}= \frac{\partial \pi(p)}{\partial p_i} |_{p=p^*}-  y_i(p^*) = 0, i=1,...,L$$

7. The Hessian matrix of a convex function is potive semi-definitive. The diagonal temrs of P.S.D matrix is non-negative, which implies that the optput will increase when the optput price increase and the input will decrease when the input price increase. The Euler's formula states that $Df(x)x = 0$ if $f(x)$ is H.D.0. The implication is that substitutes always exists.

## Cost Minimization
### The Cost Minimization Problem


If the prodcution is a single-optput technology with production function $f(z)$, the cost minimization problem (CMP) with input price vector $w \gg 0$ and the output $q$ is

$$\min_{z}  w \cdot z\\ \text{such that $f(z) \ge q$}$$

### F.O.C.

If $z^*$ is the optimal input vector, it must satisfies the following first-order conditions,

$$  w_i \ge \frac{\partial f(z^*)}{\partial z_i} , i = 1,...,L-1, \text{with equality if } z_i^* > 0$$ 

In matrix notation,

$$w \ge \nabla f(z^*)  \text{  and  } [w - \nabla f(z^*) ]z^* = 0$$

### Cost Function

If we can find the optimal $z(w,q)$ of different input price vector and ouput in CMP, we can define the cost function 
$$c(w,q) = w \cdot z(w,q)$$, which is the value function of the cost minimization problem.

### Properties
Suppose that $c(w,q)$ is the cost function of a single-ouput technology $Y$ with production function $f(z)$ and that $z(w,q)$ is the associated conditional factor demand correspondence. Assume that $Y$ is closed and satisfies the free disposal property, then,

1. $z(w,q)$ is homogeneous of degree zero in $w$.

1. Cost function is homogeneous of degree one in $w$ and nondecreasing in $q$.

2. Cost function is a concave function of $w$.

3. If the set $\{z \ge 0 | f(z \ge q)\}$ are convex for every $q$, then $Y=\{(-z,q)|w \cdot z \ge c(w,q) \forall w \gg 0\}$.


5. If the sex $\{z  \ge 0 | f(z) \ge q\}$ is convex, then $z(w,q)$ is a convex set. Moreover, if $\{z  \ge 0 | f(z) \ge q\}$ is a strictly convex set, then $z(w,q)$ is single-valued.

5. (Shephard's lemma) If $c(w,q)$ is a differentiable function at $w^*$, then $\nabla_{w}c(w^*,q) = z(w^*,q)$.

6. If $z(w,q)$ is differentiable at $w^*$, then $D_{w}z(w^*,q) = D^2_{w}z(w^*,q)$ is a symmetric and negative semi-definite matrix with $D_w z(w^*,q)w^*=0$.

7. If the production function is homogeneous of degree one/constant return to scale, then the cost function and conditional factor demand correspondence are homogeneous of degree one in $q$.

8. If the production function is concave, then the cost function is a convex function of $q$.

### Explainations

1. Clearly, the porpotional change of input prices will not change the optimal input.

2. Since the factor demand correspondence is H.D.0 in $w$, the cost function is H.D.1 in $w$. The nondecreasing in $q$ property is the direct result of free disposal of production set.

3. Given two input prices $w_1, w_2$ and fixed $q$, let $w_3 = \alpha w_1 + (1-\alpha) w_2$, we have 

$$c(w_1,q) = w_1 \cdot z(w_1, q) \le w_1 \cdot z(w_3,q) \\ c(w_2,q) = w_2 \cdot z(w_1, q) \le w_2 \cdot z(w_3,q) \\ \alpha c(w_1,q) + (1-\alpha) c(w_2,q) \le w_3 \cdot z(w_3,q) = c(w_3,q)$$

The two economic intuitions are similar to profit function. The first is that the cost is linear in $w$ without changing input, hence the real cost will be less than linear, that is a concave function. The second is the firms can hugely change its input with skew price vectores, but they can do little in a averge price vector.

4. This is the duality between cost function and production function.

5. Just prove it by contradiction.

6. Define $h(w) = c(w,q) - w^*z(w^*,q)$. Clearly, $h(w) \le 0$ and it achives its maximum when $w = w^*$. After taking partial deritives, we have

$$\frac{\partial h(w)}{\partial w_i} |_{w=w^*} = \frac{\partial c(w,q)}{\partial w_i}  |_{w=w^*}- z_i^*(w^*,q) = 0, i=1,...,L-1$$

7. The Hessian matrix of a concave function is negative semi-definite and the Euler formula states that property of H.D.0 function. The result implies law of demand and the existence of substitutes.

8. The proof is straightforward.

9. Given two output $q_1, q_2$ and $q_3 = \alpha q_1 + (1-\alpha) q_2$. Let $z_1, z_2$ are the optimal input given $q_1, q_2$. We have 

$$f(z_3) \ge \alpha f(z_1) + (1-\alpha) f(z_2) = q_3$$

and $$w \cdot z_3 = \alpha w \cdot z_1 + (1-\alpha) w \cdot z_2 = \alpha c(w,q_1)  + (1-\alpha) c(w,q_2)$$ 
Hence, 
$$c(w,q_3) \le \alpha c(w,q_1)  + (1-\alpha) c(w,q_2)$$

## Duality
### Observed Data
For each time period $t=1,...,T$, we observe the price vector $p^t$ and a firms behavior $y^t$. How to construct the technology that generates the observed behavior $(p^t,y^t)$ as profit-mzximizing bahavior?


### Inner and Outer Bound
For each time period $t=1,...,T$, we observe the price vector $p^t$ and a firms behavior $y^t$. Assume the production set is convex and monotoic(free disposal), the following two sets are the inner and outer bound of the true production set.

$$YI =\text{ convex, monotonic hull of} \{y^t | t=1,...,T\}$$

Clearly, the true production set should cantain all observed technology $y^t$, and $YI$ is the smallest potential production set by the definition.

$$YO = \{y | p^t \cdot y \le p^t \cdot y^t \text{ for all } t=1,...,T\}$$

According to the profit-maximizing assumption, any technology can generate higher profit is not feasible, otherwise the firm would choose the alternative. Hence, $NOTY = \{y | p^t \cdot y > p^t \cdot y^t \text{ for any } t=1,...,T\}$ is not feasible, and its complement $YO$ is the outer bound of true production set.

### Duality
We can calculate a firm's profit funciton\cost function from its production set\production function, but can we reconstruct a firm's production set\production function fromt its profit funciton\cost function? Unlike observed data, where we only have finite observations, we assume we can have infinite information.

### Input Requirement Set

Assume the firm has single-output, and define the input requirement set as
$$V(y)=\{x|(y,-x) \in Y\}$$

We try to construct the potential input requirement set based on the cost function,

$$V^*(y) = \{x|w  \cdot  x \ge w  \cdot  x(w,y) = c(w,y) \text{ for all } w \ge 0\}$$

### Duality of Input Requirement Set
Suppsoe $V(y)$ is a closed, convex, monotonic(free disposal) techlonoly and the firm is profit-maximizing, then $V(y) = V^*(y)$.

**Proof**  Since the firm is profit-maximizing,  $V^*(y)$ contains $V(y)$, i.e. $V(y) \subset V^*(y)$ . Suppose there is a $x \in V^*(y)$ which is not in $V(y)$. Because $V(y)$ is closed and monotonic, we can apply the seperating hyperplane theorem, and there exists a vector $w^* \ge 0$ such that $$w^* \cdot x < w^* \cdot z = c(w^*, y) \text{ for all } z \in V(y)$$  

Let $z^*$ be a point in $V(y)$ that minimizes cost at the prices $w^*$, then $w^* \cdot x < w^* \cdot z^* = c(w^*, y)$. However, $x$ could not be an element of $V^*(y)$ by the definition.

### The Criteria of a Cost Function
Let $\phi (w, y)$ be a differentiable function statifying

1. $\phi (tw, y) = t\phi(w, y) \text{ for all } t \ge 0$

2. $\phi (w, y) \ge 0 \text{ for } w\ge 0 \text{ and } y \ge 0$

3. $\phi (w', y) \ge \phi (w, y) \text{ for  } w' \ge w$

4. $\phi (w, y)$ is concave in $w$

Then $\phi (w, y)$ is the cost funtion for the technology defined by $V^*(y) = \{x \ge 0|w  \cdot  x \ge \phi(w,y), \text{ for all } w \ge 0\}$

**Proof** Given a $w \ge 0$, we define

$$x(w,y) = (\frac{\partial \phi(w, y)}{\partial w_1}, ..., \frac{\partial \phi(w, y)}{\partial w_n})$$

and note that since $\phi(w, y)$ is H.D.1 in $w$, Euler's law implies that 

$$\phi(w,y) = \sum_{i=1}^n w_i \frac{\partial \phi(w, y)}{\partial w_i} = w \cdot x(w,y) $$

We need to show that for any give $w' \ge 0$, $x(w', y)$ actually minimizes $w' \cdot x$ over all $x \in V^*(y)$, i.e.,

$$\phi(w',y) = w' \cdot x(w', y) \le w' \cdot x \text{ for all } x \in V^*(y).$$

First, we show that $x(w',y) \in V^*(y)$. By the concavity of $\phi(w, y)$ in $w$, we have 

$$\phi(w,y) \le \phi(w',y) + D\phi(w', y)(w - w') \text{ for all } w \ge 0$$

Applying Euler's law, we have

$$\phi(w, y) \le w' \cdot x(w',y) + x(w',y) (w -w') = w \cdot x(w',y) \text{ for all } w \ge 0$$

By the definition of $V^*(y)$, $x(w',y) \in V^*(y)$.

Next, we show that $x(w,y)$ minimizes $w \cdot x$ over all  $x \in V^*(y)$.

If $x$ is in $V^*(y)$, then by the definition it must satisfy 

$$w \cdot x \ge \phi(w,y)$$

But by Euler's law,

$$\phi(w,y) = w \cdot x(w,y)$$

Hence,

$$w \cdot x \ge w \cdot x(w,y) \text{ for all } x \in V^*(y)$$

*Note:* The second criterion rules out free lunch. The third criterion is probably redundent. This theorem is equivalent to the relationship between a convex set and its supporting hyperplane.