Excess of locally D-optimal designs for Cobb–Douglas model

Abstract

In this paper we study the problem of homothety’s influence on the number of optimal design support points under fixed values of a regression model’s parameters. The Cobb–Douglas two-dimensional nonlinear in parameters model used in microeconomics is considered. There exist two types of optimal designs: saturated (i.e. design with the number support points equal to the number of parameters) and excess design (i.e. design with greater number of support points). The optimal designs with the minimal number of support points are constructed explicitly. Numerical methods for constructing designs with greater number of points are used.

Appendix

Throughout this section without losing generality, we can assume that $b_1=b_2=\gamma =1$. Also, due to the fact that D-optimal design for model (3.2) doesn’t depend on the parameter $\theta _0$, we put $\theta _0=1.$

Lemma 1

Under the assumptions of this section a saturated D-optimal design for model (3.2) is concentrated at points

$$\begin{aligned} \{\breve{A}=(0,0), \ \breve{B}=(\alpha _1, 0), \ \breve{C}=(0, \alpha _2) \}, \ \alpha _i \in (0,1]. \end{aligned}$$

(4.1)

Proof of Lemma 1

Note that by Theorem 1 D-optimality of the design $\xi ^*$ is equivalent to

$$\begin{aligned} \max _{(x_1,x_2)\in \mathcal {X}}d((x_1,x_2),\xi ^*)=p,\ \hbox {where}\ d((x_1,x_2),\xi )=f^T((x_1,x_2))M^{-1}(\xi )f((x_1,x_2)). \end{aligned}$$

For model (3.2) we have $p=3$ and function $d((x_1,x_2),\xi )$ has a form:

$$\begin{aligned} d((x_1,x_2),\xi )=e^{-2(x_1\theta _1+x_2\theta _2)}(x_1^2d_{22} +(2x_2d_{23}-2d_{12})x_1+x_2^2d_{33}-2x_2d_{13}+d_{11}), \end{aligned}$$

where $d_{ij}$ are elements of the matrix $M^{-1}(\xi )$. An analysis of this function shows that it has no more than 4 maxima which are concentrated at points

$$\begin{aligned} \{\breve{A}=(0,0), \ \breve{B}=(\alpha _1, 0), \ \breve{C}=(0, \alpha _2),\ \breve{D}=(\beta _1, \beta _2) \}, \ \alpha _i,\beta _i \in [0,1]. \end{aligned}$$

(4.2)

Indeed, for any given non-singular design $\xi$ and fixed $x_1=x_1^{*}\in [0,1]$ function $d((x_1^{*},x_2),\xi )$ has two local extrema at the point $x_2=0$ and at some point $x_2=x_2^{*}\in (0,1]$ on the interval [0, 1]. This immediately implies that function $d((x_1,x_2),\xi )$ has maxima at points of the form (4.2). It remains to show that the number of global maxima is less than or equal to 4. To do this, we prove that the function $d((x_1,x_2),\xi )$ can not have two global maxima at points of the form $\breve{D}$ when $\beta _i>0$. We prove it by reductio ad absurdum. To be more specific let

$$\begin{aligned} \max _{x_1,x_2}d((x_1,x_2),\ \xi )=d(X^{*}_1,\ \xi )=d(X^{*}_2,\ \xi ),\ X^{*}_i=(x^{*}_{i1},x^{*}_{i2})\in (0,1]\times (0,1],\ i=1,2. \end{aligned}$$

Consider the line $x_2=ax_1+b$ that passes through the points $X^{*}_i.$ We have

$$\begin{aligned} \max _{x_1}d((x_1,ax_1+b),\ \xi )=d((x^{*}_{11},ax^{*}_{11}+b),\ \xi )=d((x^{*}_{21},ax^{*}_{21}+b),\ \xi ). \end{aligned}$$

After the appropriate replacements, we obtain

$$\begin{aligned} d((x_1,ax_1+b),\xi )=e^{-x_1\widetilde{\theta }_1-\widetilde{\theta }_2}(\widetilde{a}x_1^2+\widetilde{b}x_1+\widetilde{c}). \end{aligned}$$

This function has no more than 2 global maxima at the points $x_1=0$ and $x_1=x^{*}_1\in (0,1]$ on the interval [0, 1]. We have obtained a contradiction. Thus, we have proved that the saturated D-optimal design is concentrated at points

$$\begin{aligned} \text{ supp }(\overline{\xi })= & {} \{\breve{B}=(\alpha _1, 0), \ \breve{C}=(0, \alpha _2),\ \breve{D}=(\beta _1, \beta _2)\}, \ \alpha _i \in (0,1],\ \beta _i \in [0,1]. \end{aligned}$$

Let us prove, by contradiction, that if $\overline{\xi }$ is optimal, then $\beta _1=\beta _2=0$. Consider all possible combinations of sets of parameter’s values $\{\alpha _1,\alpha _2,\beta _1,\beta _2\}$. Let’s start with the case $\{0<\alpha _1<1,0<\alpha _2<1,0<\beta _1<1, 0<\beta _2<1\}$. Due to the necessary optimality conditions of the design $\xi$ the unknown variables $\alpha _i$ and $\beta _i$ must satisfy the system of equations

$$\begin{aligned}&\left\{ \begin{array}{l} \displaystyle \frac{\partial d((x_1,x_2),\overline{\xi })}{\partial x_1}\Bigr |_{x_1=\alpha _1,x_2=0}=0\\ ] \displaystyle \frac{\partial d((x_1,x_2),\overline{\xi })}{\partial x_2}\Bigr |_{x_1=0, x_2=\alpha _1}=0\\ ] \displaystyle \frac{\partial d((x_1,x_2),\overline{\xi })}{\partial x_1}\Bigr |_{x_1=\beta _1, x_2=\beta _2}=0\\ ] \displaystyle \frac{\partial d((x_1,x_2),\overline{\xi })}{\partial x_2}\Bigr |_{x_1=\beta _1, x_2=\beta _2}=0\\ ] \end{array} \right. \Rightarrow \\&\quad \Rightarrow \left\{ \begin{array}{l} \displaystyle \alpha _1\alpha _2\theta _1-\alpha _1\beta _2\theta _1-\alpha _2\beta _1\theta _1-\alpha _2+\beta _2=0\\ ] \displaystyle \alpha _1\alpha _2\theta _2-\alpha _1\beta _2\theta _2-\alpha _2\beta _1\theta _2-\alpha _1+\beta _1=0\\ ] \displaystyle \alpha _1\alpha _2\theta _1-\alpha _1\beta _2\theta _1-\alpha _2\beta _1\theta _1+\alpha _2=0\\ ] \displaystyle \alpha _1\alpha _2\theta _2-\alpha _1\beta _2\theta _2-\alpha _2\beta _1\theta _2+\alpha _1=0\\ ] \end{array} \right. \Rightarrow \left\{ \begin{array}{l} \displaystyle 3\beta _1\theta _1=2\\ ] \displaystyle 3\beta _2\theta _2=2\\ ] \displaystyle \alpha _1=\frac{\beta _1}{2}\\ ] \displaystyle \alpha _2=\frac{\beta _2}{2}\\ ] \end{array} \right. \end{aligned}$$

This system has a unique solution: $\alpha _1 = \frac{1}{3\theta _1}, \alpha _2 = \frac{1}{3\theta _2}, \beta _1 = \frac{2}{3\theta _1}, \beta _2 = \frac{2}{3\theta _2}.$ A direct calculation shows that for such $\alpha _i$ and $\beta _i$ we have

$$\begin{aligned} d((0,0),\overline{\xi })=\frac{1}{3}(e^2+8)e^{2/3} > 3, \end{aligned}$$

i. e. the design $\overline{\xi }$ is not optimal by Theorem 1. The remaining cases are also checked in a similar way. For example, for $\{\alpha _1=1,\alpha _2=1,\beta _1=1, \beta _2=1\}$ we have

$$\begin{aligned} d((0,0),\overline{\xi })=3(e^{2\theta _1}+e^{2(\theta _1+\theta _2)}+e^{2\theta _2})> 3,\ \hbox {when} \ \theta _1,\theta _2>0. \end{aligned}$$

Thus, we have proved that the saturated D-optimal design is concentrated at points 4.1

Lemma 1 is proved. $\square$

Proof of Theorem 2

Optimality of designs (3.6)–(3.4) is verified directly by Theorem 1. For example, for the design $\overline{\xi }$ in form (3.6) we have

$$\begin{aligned} d((t_1,t_2),\overline{\xi })=3 e^{2(1-t_1\theta _1-t_2\theta _2)}[e^{-2}(t_1\theta _1+t_2\theta _2-1)^2+t_1^2\theta _1^2+t_2^2\theta _2^2]. \end{aligned}$$

On the design space $\mathcal {X}$, this function has a local minimum point $\left( \frac{2}{(2+e^{2})\theta _1}, \frac{2}{(2+e^{2})\theta _2}\right)$, saddle point $\left( \frac{1}{2\theta _1}, \frac{1}{2\theta _2}\right)$ and the global minimum point $\left( 1, 1\right)$ and takes the following values at these points, correspondingly:

$$\begin{aligned} d\left( \left( \frac{2}{(2+e^{2})\theta _1}, \frac{2}{(2+e^{2})\theta _2}\right) ,\overline{\xi }\right)= & {} 3e^{-8/(2+e^2)}<3,\\ d\left( \left( \frac{1}{2\theta _1}, \frac{1}{2\theta _2}\right) ,\overline{\xi }\right)= & {} \frac{3}{2}<3,\\ d\left( \left( 1, 1\right) ,\overline{\xi }\right)= & {} 3e^{(2-2\theta _1-2\theta _1)}(e^{-2}(\theta _1+\theta _2-1)^2+\theta _1^2+\theta _2^2). \end{aligned}$$

The analysis of the last expression allows us to conclude that $d\left( \left( 1, 1\right) ,\overline{\xi }\right)$ reaches maximum when $\theta _1=\theta _2=1:$

$$\begin{aligned} d\left( \left( 1, 1\right) ,\overline{\xi }\right) \Bigr |_{\theta _1=1,\theta _2=1}=3e^{-2}(e^{-2}+2)<3. \end{aligned}$$

The function $d((t_1,t_2),\overline{\xi })$ reaches maximum at points $A,\ B^\prime ,\ C^\prime$:

$$\begin{aligned} d\left( \left( 0, 0\right) ,\overline{\xi }\right) =d\left( \left( \theta _1^{-1}, 0\right) ,\overline{\xi }\right) =d\left( \left( 0, \theta _2^{-1}\right) ,\overline{\xi }\right) =3. \end{aligned}$$

The remaining cases are verified in a similar way.

The behavior of the function $d((t_1,t_2),\overline{\xi })$ for $\theta _1=\theta _2=1$, $\mathcal {X}=[0,\frac{3}{2}]\times [0,\frac{3}{2}]$ is depicted in Fig. 4.

The only thing we need to check now is that sets $\varLambda _3$ and $\varLambda _4$ are defined in a proper way. Let’s start with part (a). It follows from the continuity of the function $d((t_1,t_2),\overline{\xi })$ that these sets have a common boundary and there exist two types of the optimal designs for the parameters belonging to the boundary:

$$\begin{aligned} \xi ^{(I)}_{opt}=\left( \begin{array}{cccc} A &{} \quad B &{} \quad C &{} \quad D^\prime \\ \frac{1}{3} &{} \quad \frac{1}{3} &{} \quad \frac{1}{3} &{} \quad 0 \end{array}\right) \quad \hbox {and} \quad \xi ^{(II)}_{opt}=\left( \begin{array}{cccc} A &{} \quad B &{} \quad C &{} \quad D^{\prime \prime } \\ \frac{1}{3} &{} \quad \frac{1}{3} &{} \quad \frac{1}{3} &{} \quad 0 \end{array}\right) , \end{aligned}$$

such that

$$\begin{aligned} d(D^\prime ,\xi ^{(I)}_{opt})=d(D^{\prime \prime },\xi ^{(II)}_{opt})=3 \end{aligned}$$

It follows from the previous discussion that the function $d(D^{\prime \prime },\xi ^{(II)}_{opt})=d((t_2,1),\xi ^{(II)}_{opt})$ has no more than one maxima on the interval (0, 1].

Fig. 4

The behavior of the function $d((t_1,t_2),\overline{\xi })$ when $\theta _1=\theta _2=1$, $\mathcal {X}=[0,\frac{3}{2}]\times [0,\frac{3}{2}]$ for the case (3.6) from Theorem 2

Note that for fixed $t_2$ function $d(D^{\prime \prime },\xi ^{(II)}_{opt})=d((t_2,1),\xi ^{(II)}_{opt})$ is a monotonic function by $\theta _1$ since

$$\begin{aligned} d((t_2,1),\xi ^{(II)}_{opt})= 3e^{-2(\theta _1t_2+\theta _2)}\left[ t_2^2(e^{2\theta _1}+1)+e^{2\theta _2}\right] . \end{aligned}$$

Thus by the implicit function theorem the following system has a unique real solution, say, $\theta _2=\varPsi _1( \theta _1)$

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\partial d((x_1,1),\xi ^{(II)}_{opt})}{\partial x_1}\Bigr |_{x_1=t_2}&{}=&{}0,\\ \displaystyle d((t_2,1),\xi ^{(II)}_{opt})&{}=&{}3; \end{array} \right. \end{aligned}$$

as well as the equation $d((1,1),\xi ^{(II)}_{opt})=3$ has a unique real solution, say, $\theta _2=\varPsi _2( \theta _1)$. Functions $\varPsi _1$ and $\varPsi _2$ can be obtained by solving of corresponding equations. Now the results of part (a) immediately follows from the conditions $\theta _2 \le \theta _1$, $t_2\le 1$ and the fact that the boundary function is $\varPsi (\theta _1)=\min (\theta _1,\max (\varPsi _1(\theta _1),\varPsi _2(\theta _1))).$

Parts (b) and (c) are verified in a similar way.

Theorem 2 is proved. $\square$