Topic 6_Producer Theory

1. Production function

Production function is some kinds of function that transforms inputs into outputs.

• Properties of production function

  The production function, $f:{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$$f: \mathbb{R}_{+}^n \rightarrow \mathbb{R}_{+}$, is continuous, strictly increasing, and strictly quasiconcave on ${\mathbb{R}}_{+}^n$$\mathbb{R}_{+}^n$, and $f(\mathbf{0})=0$$f(\mathbf{0})=0$.

  • $f(0)=0$$f(0) = 0$

  • if continuous and differentiable, and strictly increasing. then marginal product is always positive. $\frac{\partial f(x)}{\partial x}>0$$\frac{\partial f(x)}{\partial x}>0$

• Isoquant curve

  $$\begin{array}{}\text{(1)}& Q(y)\equiv \{\mathbf{x}\ge 0\mid f(\mathbf{x})=y\}.\end{array}$$

  For an input vector $\mathbf{x}$$\mathbf{x}$, the isoquant through $\mathbf{x}$$\mathbf{x}$ is the set of input vectors producing the same output as $\mathbf{x}$$\mathbf{x}$, namely, $Q(f(\mathbf{x}))$$Q(f(\mathbf{x}))$.

  

• MRTS

  Measures the rate at which one input can be substituted for another without changing the amount of output produced.

  $$\begin{array}{}\text{(2)}& {\mathrm{MRTS}}_{ij}(\mathbf{x})=\frac{\partial f(\mathbf{x})/\partial x_i}{\partial f(\mathbf{x})/\partial x_j}=\frac{M{P}_i}{M{P}_j}\end{array}$$

  In the two-input case, as depicted in Fig. 3.1, ${\mathrm{MRTS}}_{12}\left({\mathbf{x}}^1\right)$$\operatorname{MRTS}_{12}\left(\mathbf{x}^1\right)$ is the absolute value of the slope of the isoquant through ${\mathbf{x}}^1$$\mathbf{x}^1$ at the point ${\mathbf{x}}^1$$\mathbf{x}^1$.

   

• Elasticity of substitution

  $$\begin{array}{}\text{(3)}& \sigma =\frac{\mathrm{\%}\text{ change in }\frac{x_2}{x_1}}{\mathrm{\%}{\text{ change in MRTS }}_{21}}=\frac{\mathrm{d}\left(\frac{x_2}{x_1}\right)}{\mathrm{d}\left(\frac{f_1}{f_2}\right)}\cdot \frac{\left(\frac{f_1}{f_2}\right)}{\left(\frac{x_2}{x_1}\right)}=\frac{\mathrm{d}\ln \left(\frac{x_2}{x_1}\right)}{\mathrm{d}\ln \left(\frac{f_1}{f_2}\right)}\end{array}$$

  Also, rigorously,

  $$\begin{array}{}\text{(4)}& {\sigma }_{ij}(\mathrm{x})=\frac{d\ln \left(x_j/x_i\right)}{d\ln MRT{S}_{ij}}=\frac{d\ln \left(x_j/x_i\right)}{d\ln \left(M{P}_i/M{P}_j\right)}=\frac{\frac{d\left(x_j/x_i\right)}{x_j/x_i}}{\frac{d\left(M{P}_i/M{P}_j\right)}{\left(M{P}_i/M{P}_j\right)}}\end{array}$$

  • % change in MRTS vs. % change in factor ratio at $\mathrm{x}$$\mathrm{x}$

  • $\sigma$$\sigma$ measures how easily input factors can be substituted for one another (holding other inputs and output constant). (The curvature of the isoquant curve at $x$$x$)

  • If $\sigma$$\sigma$ is large, then it is easier to substitute between the input factors.

  • Strong (weak) substitution: when increasing $x_1$$x_1$ and reducing $x_2$$x_2$ along one isoquant, MRTS ${}_{ij}$$_{i j}$ - the ability of $x_1$$x_1$ to substitute for $x_2$$x_2$ - drops a little(a lot)

  • $\sigma \in [0,+\mathrm{\infty })$$\sigma \in[0,+\infty)$

  

  Example

  Elasticity of substitution of a CES production function is $\sigma =\frac{1}{1-\rho }$$\sigma = \frac{1}{1-\rho}$.

  $$\begin{array}{}\text{(5)}& y={(x_1^{\rho }+x_2^{\rho })}^{1/\rho }\end{array}$$

  Then, by definition,

  $$\begin{array}{}\text{(6)}& \begin{array}{rl}\frac{d\ln MRT{S}_{12}(\mathbf{x}(r))}{d\ln r}& =\frac{d\ln r^{1-\rho }}{d\ln r}\\ & =(1-\rho )\frac{d\ln r}{d\ln r}\\ & =1-\rho .\end{array}\end{array}$$

  Hence,

  $$\begin{array}{}\text{(7)}& \sigma =\frac{1}{1-\rho }\end{array}$$

• JR Theorem 3.1 Homogeneous production functions (h.d. <1) is concave

  Proof: Suppose first that $\alpha =1$$\alpha=1$, i.e., that $f$$f$ is homogeneous of degree one. Take any ${\mathbf{x}}^1\gg 0$$\mathbf{x}^1 \gg 0$ and ${\mathbf{x}}^2\gg \mathbf{0}$$\mathbf{x}^2 \gg \mathbf{0}$ and let $y^1=f\left({\mathbf{x}}^1\right)$$y^1=f\left(\mathbf{x}^1\right)$ and $y^2=f\left({\mathbf{x}}^2\right)$$y^2=f\left(\mathbf{x}^2\right)$. Then $y^1,y^2>0$$y^1, y^2>0$ because $f(0)=0$$f(0)=0$ and $f$$f$ is strictly increasing. Therefore, because $f$$f$ is homogeneous of degree one,

  $$\begin{array}{}\text{(8)}& f\left(\frac{{\mathbf{x}}^1}{y^1}\right)=f\left(\frac{{\mathbf{x}}^2}{y^2}\right)=1\end{array}$$

  Because $f$$f$ is (strictly) quasiconcave,

  $$\begin{array}{}\text{(9)}& f(\frac{t{\mathbf{x}}^1}{y^1}+\frac{(1-t){\mathbf{x}}^2}{y^2})\ge 1\text{ for all }t\in [0,1].\end{array}$$

  Now choose $t^{\ast }=y^1/(y^1+y^2)$$t^*=y^1 /\left(y^1+y^2\right)$ and $(1-t^{\ast })=y^2/(y^1+y^2)$$\left(1-t^*\right)=y^2 /\left(y^1+y^2\right)$. Then from (P.1),

  $$\begin{array}{}\text{(10)}& f(\frac{{\mathbf{x}}^1}{y^1+y^2}+\frac{{\mathbf{x}}^2}{y^1+y^2})\ge 1.\end{array}$$

  Again invoking the linear homogeneity of $f$$f$ and using (P.2) gives

  $$\begin{array}{}\text{(11)}& f({\mathbf{x}}^1+{\mathbf{x}}^2)\ge y^1+y^2=f\left({\mathbf{x}}^1\right)+f\left({\mathbf{x}}^2\right).\end{array}$$

  Thus, (P.3) holds for all ${\mathbf{x}}^1,{\mathbf{x}}^2\gg \mathbf{0}$$\mathbf{x}^1, \mathbf{x}^2 \gg \mathbf{0}$. But the continuity of $f$$f$ then implies that (P.3) holds for all ${\mathbf{x}}^1,{\mathbf{x}}^2\ge \mathbf{0}$$\mathbf{x}^1, \mathbf{x}^2 \geq \mathbf{0}$.

  To complete the proof for the $\alpha =1$$\alpha=1$ case, consider any two vectors ${\mathbf{x}}^1\ge \mathbf{0}$$\mathbf{x}^1 \geq \mathbf{0}$ and ${\mathbf{x}}^2\ge$$\mathbf{x}^2 \geq$ 0 and any $t\in [0,1]$$t \in[0,1]$. Recall that linear homogeneity ensures that

  $$\begin{array}{}\text{(12)}& \begin{array}{rl}f\left(t{\mathbf{x}}^1\right)& =tf\left({\mathbf{x}}^1\right),\\ f((1-t){\mathbf{x}}^2)& =(1-t)f\left({\mathbf{x}}^2\right).\end{array}\end{array}$$

  If we apply (P.3) and use (P.4) and (P.5), we conclude that

  $$\begin{array}{}\text{(13)}& f(t{\mathbf{x}}^1+(1-t){\mathbf{x}}^2)\ge tf\left({\mathbf{x}}^1\right)+(1-t)f\left({\mathbf{x}}^2\right),\end{array}$$

  as desired. Suppose now that $f$$f$ is homogeneous of degree $\alpha \in (0,1]$$\alpha \in(0,1]$. Then $f^{\frac{1}{\alpha }}$$f^{\frac{1}{\alpha}}$ is homogeneous of degree one and satisfies Assumption 3.1. Hence, by what we have just proven, $f^{\frac{1}{\alpha }}$$f^{\frac{1}{\alpha}}$ is concave. But then $f={\left[f^{\frac{1}{\alpha }}\right]}^{\alpha }$$f=\left[f^{\frac{1}{\alpha}}\right]^\alpha$ is concave since $\alpha \le 1$$\alpha \leq 1$.

• Return to scale

  A production function $f(\mathbf{x})$$f(\mathbf{x})$ has the property of (globally):

  1. Constant returns to scale if $f(t\mathbf{x})=tf(\mathbf{x})$$f(t \mathbf{x})=t f(\mathbf{x})$ for all $t>0$$t>0$ and all $\mathbf{x}$$\mathbf{x}$;

  2. Increasing returns to scale if $f(t\mathbf{x})>tf(\mathbf{x})$$f(t \mathbf{x})>t f(\mathbf{x})$ for all $t>1$$t>1$ and all $\mathbf{x}$$\mathbf{x}$;

  3. Decreasing returns to scale if $f(t\mathbf{x})<tf(\mathbf{x})$$f(t \mathbf{x})<t f(\mathbf{x})$ for all $t>1$$t>1$ and all $\mathbf{x}$$\mathbf{x}$.

  (Local) Returns to Scale The elasticity of scale at the point $\mathbf{x}$$\mathbf{x}$ is defined as

  $$\begin{array}{}\text{(14)}& \mu (\mathbf{x})\equiv \underset{t\to 1}{lim}\frac{d\ln [f(t\mathbf{x})]}{d\ln (t)}=\frac{\sum _{l=1}^n{f}_l(\mathbf{x})x_l}{f(\mathbf{x})}.\end{array}$$

  Returns to scale are locally constant, increasing, or decreasing as $\mu (\mathbf{x})$$\mu(\mathbf{x})$ is equal to, greater than, or less than one. The elasticity of scale and the output elasticities of the inputs are related as follows:

  $$\begin{array}{}\text{(15)}& \mu (\mathbf{x})=\sum _{i=1}^n{\mu }_l(\mathbf{x})\end{array}$$

2. Cost minimization

If the object of the firm is to maximize profits, it will necessarily choose the least costly, or cost-minimizing, production plan for every level of output.

The cost function, defined for all input prices $\mathbf{w}\gg \mathbf{0}$$\mathbf{w} \gg \mathbf{0}$ and all output levels $y\in f\left({\mathbb{R}}_{+}^n\right)$$y \in f\left(\mathbb{R}_{+}^n\right)$ is the minimum-value function,

If $\mathbf{x}(\mathbf{w},y)$$\mathbf{x}(\mathbf{w}, y)$ solves the cost-minimization problem, then

Marginal rate of technical substitution between two inputs is equal o the ratio of their prices

Solutions to this cost minimization problem are called conditional input demand ${\mathbf{x}}^{\ast }(w,y)$$\mathbf{x}^*(w,y)$

Example

CES production function

Mathematically, cost function is identical to the expenditure function in the consumer theory.

• Properties of the Cost function (identical to expenditure function)

  If $f$$f$ is continuous and strictly increasing, then $c(\mathbf{w},y)$$c(\mathbf{w}, y)$ is

  1. Zero when $y=0$$y=0$,

  2. Continuous on its domain,

  3. For all $\mathbf{w}\gg 0$$\mathbf{w} \gg 0$, strictly increasing and unbounded above in $y$$y$,

  4. Increasing in $\mathbf{w}$$\mathbf{w}$,

  5. Homogeneous of degree 1 in $\mathbf{w}$$\mathbf{w}$,

  6. Concave in $\mathbf{w}$$\mathbf{w}$.

  Moreover, if $f$$f$ is strictly quasiconcave we have

  7. Shephard's lemma: $c(\mathbf{w},y)$$c(\mathbf{w}, y)$ is differentiable in $\mathbf{w}$$\mathbf{w}$ at $({\mathbf{w}}^0,y^0)$$\left(\mathbf{w}^0, y^0\right)$ whenever ${\mathbf{w}}^0\gg 0$$\mathbf{w}^0 \gg 0$, and

  $$\begin{array}{}\text{(19)}& \frac{\partial c({\mathbf{w}}^0,y^0)}{\partial w_i}=x_i({\mathbf{w}}^0,y^0),i=1,\dots ,n\end{array}$$
  8. $$\begin{array}{}\text{(20)}& \frac{{\partial }^2{C}^{\ast }}{\partial w_1\partial w_2}=\frac{\partial x_1^{\ast }}{\partial w_2};\frac{{\partial }^2{C}^{\ast }}{\partial w_2\partial w_1}=\frac{\partial x_2^{\ast }}{\partial w_1}\end{array}$$

     And because:

     $$\begin{array}{}\text{(21)}& \frac{{\partial }^2{C}^{\ast }}{\partial w_1\partial w_2}=\frac{{\partial }^2{C}^{\ast }}{\partial w_2\partial w_1}\end{array}$$

     we have:

     $$\begin{array}{}\text{(22)}& \frac{\partial x_1^{\ast }}{\partial w_2}=\frac{\partial x_2^{\ast }}{\partial w_1}\end{array}$$

• Properties of Conditional Input Demands

  1. $\mathbf{x}(\mathbf{w},y)$$\mathbf{x}(\mathbf{w}, y)$ is homogeneous of degree zero in $\mathbf{w}$$\mathbf{w}$,

  2. The substitution matrix, defined and denoted

  $$\begin{array}{}\text{(23)}& \begin{array}{c}{\sigma }^{\ast }(\mathbf{w},y)\equiv \left(\begin{array}{ccc}\frac{\partial x_1(\mathbf{w},y)}{\partial W_1}& \cdots & \frac{\partial x_1(\mathbf{w},y)}{\partial W_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial x_n(\mathbf{w},y)}{\partial w_1}& \cdots & \frac{\partial X_n(\mathbf{w},y)}{\partial w_n}\end{array}\right),\end{array}\end{array}$$

  is symmetric and negative semidefinite. In particular, the negative semi-definiteness property implies that $\partial x_i(\mathbf{w},y)/\partial w_i\le 0$$\partial x_i(\mathbf{w}, y) / \partial w_i \leq 0$ for all $i$$i$.

   

  • Hicksian Third Law

     

• When the production function is homothetic, what happens on cost and input demand, then

  • The cost function is multiplicatively separable in input prices and output and can be written $c(\mathbf{w},y)=h(y)c(\mathbf{w},1)$$c(\mathbf{w}, y)=h(y) c(\mathbf{w}, 1)$, where $h(y)$$h(y)$ is strictly increasing and $c(\mathbf{w},1)$$c(\mathbf{w}, 1)$ is the unit cost function, or the cost of 1 unit of output;

  • The conditional input function are also multiplicatively separable in input prices and output can be written $\mathbf{x}(\mathbf{w},y)=h(y)\mathbf{x}(\mathbf{w},1)$$\mathbf{x}(\mathbf{w}, y)=h(y) \mathbf{x}(\mathbf{w}, 1)$, where $h^{\prime }(y)>0$$h^{\prime}(y)>0$ and $\mathbf{x}(\mathbf{w},1)$$\mathbf{x}(\mathbf{w}, 1)$ is the conditional input demand for 1 unit of output.

  Proof

  Let $F$$F$ denote the production function. Because it is homothetic, it can be written as $F(\mathbf{x})=f(g(\mathbf{x}))$$F(\mathbf{x})=f(g(\mathbf{x}))$, where $f$$f$ is strictly increasing, and $g$$g$ is homogeneous of degree one.

  For simplicity, we shall assume that the image of $F$$F$ is all of ${\mathbb{R}}_{+}$$\mathbb{R}_{+}$. Consequently, as you are asked to show in Exercise 3.5, $f^{-1}(y)>0$$f^{-1}(y)>0$ for all $y>0$$y>0$. So, for some $y>0$$y>0$, let $t=f^{-1}(1)/f^{-1}(y)>0$$t=f^{-1}(1) / f^{-1}(y)>0$. Note then that $f(g(\mathbf{x}))\ge y⟺g(\mathbf{x})\ge f^{-1}(y)⟺$$f(g(\mathbf{x})) \geq y \Longleftrightarrow g(\mathbf{x}) \geq f^{-1}(y) \Longleftrightarrow$ $g(t\mathbf{x})\ge t{f}^{-1}(y)=f^{-1}(1)⟺f(g(t\mathbf{x}))\ge 1$$g(t \mathbf{x}) \geq t f^{-1}(y)=f^{-1}(1) \Longleftrightarrow f(g(t \mathbf{x})) \geq 1$. Therefore, we may express the cost function associated with $F$$F$ as follows.

  $$\begin{array}{}\text{(24)}& \begin{array}{l}c(\mathbf{w},y)=\underset{\mathbf{x}\in {\mathbb{R}}_{+}^n}{min}\mathbf{w}\cdot \mathbf{x}\text{ s.t. }f(g(\mathbf{x}))\ge y\\ =\underset{\mathbf{x}\in {\mathbb{R}}_{+}^n}{min}\mathbf{w}\cdot \mathbf{x}\text{ s.t. }f(g(t\mathbf{x}))\ge 1\\ =\frac{1}{t}\underset{\mathbf{x}\in {\mathbb{R}}_{+}^n}{min}\mathbf{w}\cdot t\mathbf{x}\text{ s.t. }f(g(t\mathbf{x}))\ge 1\\ =\frac{1}{t}\underset{\mathbf{z}\in {\mathbb{R}}_{+}^n}{min}\mathbf{w}\cdot \mathbf{z}\text{ s.t. }f(g(\mathbf{z}))\ge 1\\ =\frac{f^{-1}(y)}{f^{-1}(1)}c(\mathbf{w},1)\text{, }\end{array}\end{array}$$

  where in the second to last line we let $\mathbf{z}\equiv t\mathbf{x}$$\mathbf{z} \equiv t \mathbf{x}$.

   

   

   

   

• Long run/ short run cost function

  $$\begin{array}{}\text{(25)}& sc(\mathbf{w},\bar{\mathbf{w}},y,\bar{\mathbf{x}})\equiv \underset{\mathbf{x}}{min}\mathbf{w}\cdot \mathbf{x}+\bar{\mathbf{w}}\cdot \bar{\mathbf{x}}\text{ s.t. }f(\mathbf{x},\bar{\mathbf{x}})\ge y.\end{array}$$

  If $\mathbf{x}(\mathbf{w},\bar{\mathbf{w}},y,\bar{\mathbf{x}})$$\mathbf{x}(\mathbf{w}, \overline{\mathbf{w}}, y, \overline{\mathbf{x}})$ solves this minimisation problem, then

  $$\begin{array}{}\text{(26)}& sc(\mathbf{w},\bar{\mathbf{w}},y,\bar{\mathbf{x}})=\mathbf{w}\cdot \mathbf{x}(\mathbf{w},\bar{\mathbf{w}},y,\bar{\mathbf{x}})+\bar{\mathbf{w}}\cdot \bar{\mathbf{x}}.\end{array}$$

  The optimized cost of the variable inputs, $\mathbf{w}\cdot \mathbf{x}(\mathbf{w},\bar{\mathbf{w}},y,\bar{\mathbf{x}})$$\mathbf{w} \cdot \mathbf{x}(\mathbf{w}, \overline{\mathbf{w}}, y, \overline{\mathbf{x}})$, is called total variable cost. The cost of the fixed inputs, $\bar{\mathbf{w}}\cdot \bar{\mathbf{x}}$$\overline{\mathbf{w}} \cdot \overline{\mathbf{x}}$, is called total fixed cost.

3. Profit maximization

Then the first-order conditions require that marginal cost equals marginal benefit of production.

the MRTS between any two inputs is equated to the ratio of their prices.

• Profit function

  $$\begin{array}{}\text{(30)}& \pi (p,\mathbf{w})\equiv \underset{(\mathbf{x},y)\ge 0}{max}py-\mathbf{w}\cdot \mathbf{x}\text{ s.t. }f(\mathbf{x})\ge y.\end{array}$$

• Profit function has to actually exist. (This actually requires that a production function cannot be IRS)

  Suppose a production function exhibit increasing return to scale, and we assume that ${\mathbf{x}}^{\prime }$$\mathbf{x}^{\prime}$ and $y^{\prime }=f\left({\mathbf{x}}^{\prime }\right)$$y^{\prime}=f\left(\mathbf{x}^{\prime}\right)$ maximize profits at $p$$p$ and $\mathbf{w}$$\mathbf{w}$. With increasing returns,

  $$\begin{array}{}\text{(31)}& f\left(t{\mathbf{x}}^{\prime }\right)>tf\left({\mathbf{x}}^{\prime }\right)\text{ for all }t>1\end{array}$$

  Multiplying by $p>0$$p>0$, subtracting $\mathbf{w}\cdot t{\mathbf{x}}^{\prime }$$\mathbf{w} \cdot t \mathbf{x}^{\prime}$ from both sides, rearranging, and using $t>1$$t>1$ and the non-negativity of profits gives

  $$\begin{array}{}\text{(32)}& pf\left(t{\mathbf{x}}^{\prime }\right)-\mathbf{w}\cdot t{\mathbf{x}}^{\prime }>pf\left({\mathbf{x}}^{\prime }\right)-\mathbf{w}\cdot {\mathbf{x}}^{\prime }\text{ for all }t>1.\end{array}$$

  This says higher profit can always be had by increasing inputs in proportion $t>1-$$t>1-$ contradicting our assumption that ${\mathbf{x}}^{\prime }$$\mathbf{x}^{\prime}$ and $f\left({\mathbf{x}}^{\prime }\right)$$f\left(\mathbf{x}^{\prime}\right)$ maximised profit. Notice that in the special case of constant returns, no such problem arises if the maximal level of profit happens to be zero. In that case, though, the scale of the firm's operation is indeterminate because $(y^{\prime },{\mathbf{x}}^{\prime })$$\left(y^{\prime}, \mathbf{x}^{\prime}\right)$ and $(t{y}^{\prime },t{\mathbf{x}}^{\prime })$$\left(t y^{\prime}, t \mathbf{x}^{\prime}\right)$ give the same level of zero profits for all $t>0$$t>0$.

  如果生产函数是IRS，那么意味着企业总能通过扩大生产获得更多的利润，这个和一开始假设的条件不一致。

• Properties of production function

  If $f$$f$ satisfies Assumption 3.1, then for $p\ge 0$$p \geq 0$ and $\mathbf{w}\ge \mathbf{0}$$\mathbf{w} \geq \mathbf{0}$, the profit function $\pi (p,\mathbf{w})$$\pi(p, \mathbf{w})$, where well-defined, is continuous and

  1. Increasing in $p$$p$,

  2. Decreasing in $\mathbf{w}$$\mathbf{w}$,

  3. Homogeneous of degree one in $(p,\mathbf{w})$$(p, \mathbf{w})$,

  4. Convex in $(p,\mathbf{w})$$(p, \mathbf{w})$,

  5. Differentiable in $(p,\mathbf{w})\gg \mathbf{0}$$(p, \mathbf{w}) \gg \mathbf{0}$. Moreover, under the additional assumption that $f$$f$ is strictly concave (Hotelling's lemma),

  $$\begin{array}{}\text{(33)}& \frac{\partial \pi (p,\mathbf{w})}{\partial p}=y(p,\mathbf{w}),\text{ and }\frac{\partial \pi (p,\mathbf{w})}{\partial w_i}=-x_i(p,\mathbf{w}),i=1,2,\dots ,n\text{. }\end{array}$$

•  

• Properties of output supply and input demand function

   

1. Homogeneity of degree zero:

2. Own-price effects:

3. The substitution matrix is positive and semidefinite

   $$\begin{array}{}\text{(36)}& \begin{array}{c}\left(\begin{array}{cccc}\frac{\partial y(p,\mathbf{w})}{\partial p}& \frac{\partial y(p,\mathbf{w})}{\partial W_1}& \cdots & \frac{\partial y(p,\mathbf{w})}{\partial W_n}\\ \frac{-\partial x_1(p,\mathbf{w})}{\partial p}& \frac{-\partial x_1(p,\mathbf{w})}{\partial W_1}& \cdots & \frac{-\partial x_1(p,\mathbf{w})}{\partial W_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{-\partial x_n(p,\mathbf{w})}{\partial p}& \frac{-\partial x_n(p,\mathbf{w})}{\partial W_1}& \cdots & \frac{-\partial x_n(p,\mathbf{w})}{\partial W_n}\end{array}\right)\end{array}\end{array}$$