<p>To solve the utility maximization problem you need to follow the money.</p>
<p>The total income is <span class="math-container">$M$</span> which is split between the mother and father according to the Pareto weights <span class="math-container">$\mu$</span> and <span class="math-container">$(1-\mu)$</span>. This implies that <span class="math-container">$\mu M$</span> is used on <span class="math-container">$u_m$</span> and <span class="math-container">$(1-\mu)M$</span> on <span class="math-container">$u_f$</span>.</p>
<p>The income <span class="math-container">$\mu M$</span> is split by the mother on <span class="math-container">$X_1$</span> for child <span class="math-container">$1$</span> and <span class="math-container">$X_2$</span> for child <span class="math-container">$2$</span>. Let <span class="math-container">$s_1^m$</span> be the share of mother income <span class="math-container">$\mu M$</span> used on <span class="math-container">$X_1$</span> and <span class="math-container">$s_2^m$</span> the share of <span class="math-container">$\mu M$</span> used on <span class="math-container">$X_2$</span>. Similarly, let <span class="math-container">$s_1^f$</span> and <span class="math-container">$s_2^f$</span> be the same shares for the case of the father.</p>
<p>The mother therefore uses <span class="math-container">$s_1^m\mu M$</span> on <span class="math-container">$X_1^m$</span> (subscript to indicate it is <span class="math-container">$X_1$</span> paid by the mother) and <span class="math-container">$s_2^m\mu M$</span> on <span class="math-container">$X_2^m$</span>. Similarly, for the father <span class="math-container">$s_1^f(1-\mu)M$</span> used on <span class="math-container">$X_1^f$</span> and <span class="math-container">$s_2^f(1-\mu) M$</span> used on <span class="math-container">$X_2^f$</span>.</p>
<p>Finally, whatever is spent on <span class="math-container">$X_1$</span> is spent partly on <span class="math-container">$h_1$</span> and partly on <span class="math-container">$e_1$</span> by the shares <span class="math-container">$c$</span> and <span class="math-container">$(1-c)$</span> by both the mother and the father.</p>
<p>Hence, the mothers demand for <span class="math-container">$h_1$</span> is given as</p>
<p><span class="math-container">$$h_1^m = \frac{cs_1^m \mu M}{P_{h_1}},$$</span></p>
<p>and the mother demand for <span class="math-container">$e_1$</span> is given as</p>
<p><span class="math-container">$$e_1^m = \frac{(1-c)s_1^m \mu M}{P_{e_1}},$$</span></p>
<p>The mothers demand for <span class="math-container">$h_2$</span> is given as</p>
<p><span class="math-container">$$h_2^m = \frac{cs_2^m \mu M}{P_{h_2}},$$</span></p>
<p>and the mother demand for <span class="math-container">$e_1$</span> is given as</p>
<p><span class="math-container">$$e_2^m = \frac{(1-c)s_2^m \mu M}{P_{e_2}},$$</span></p>
<p>and the fathers' demands are the same just with <span class="math-container">$(1-\mu)$</span> instead of <span class="math-container">$\mu$</span> and the fathers shares <span class="math-container">$s_1^f$</span> and <span class="math-container">$s_2^f$</span>.</p>
<p>The shares <span class="math-container">$s_1^m,s_1^f,s_2^m,s_2^f$</span> are found by solving the CES utility maximization problem and the expenditure minimization problem of the CD utility.</p>
<p>The Marshall demand for a CES utility function of the form</p>
<p><span class="math-container">$$(kZ_1^\alpha + (1-k)Z_2^{1-\alpha})^{1/\alpha}$$</span></p>
<p>Can be written as</p>
<p><span class="math-container">$$Z_1^\star =\left[ \frac{k^\alpha P_{Z_1}^{1-\alpha}}{k^\alpha P_{Z_1}^{1-\alpha} + (1-k)^\alpha P_{Z_2}^{1-\alpha}}\right]  \frac{I}{P_{Z_1}} $$</span></p>
<p><span class="math-container">$$Z_2^\star =\left[ \frac{(1-k)^\alpha P_{Z_2}^{1-\alpha}}{k^\alpha P_{Z_1}^{1-\alpha} + (1-k)^\alpha P_{Z_2}^{1-\alpha}}\right]  \frac{I}{P_{Z_2}},$$</span></p>
<p>where the terms in the brackets are the shares (see for example this tutorial <a href="https://www.youtube.com/watch?v=WhxprnW2Sc4" rel="nofollow noreferrer">https://www.youtube.com/watch?v=WhxprnW2Sc4</a>).</p>
<p>To solve for the share <span class="math-container">$s_1^m$</span> do the following:</p>
<p>(1) Let <span class="math-container">$P_{Z_1}$</span> and <span class="math-container">$P_{Z_2}$</span> be <span class="math-container">$P_{X_1}$</span> and <span class="math-container">$P_{X_2}$</span></p>
<p>(2) Let <span class="math-container">$k=a$</span> and</p>
<p>(3) Let <span class="math-container">$\alpha=\rho$</span></p>
<p>To get</p>
<p><span class="math-container">$$s_1^m = \left[ \frac{a^\rho P_{X_1}^{1-\rho}}{a^\rho P_{X_1}^{1-\rho} + (1-a)^\rho P_{X_2}^{1-\rho}}\right],$$</span></p>
<p>Finally, to get the price of <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> you need the expenditure function of the CD functions <span class="math-container">$A_1h_1^ce_1^{1-c}$</span> and <span class="math-container">$A_2h_2^ce_2^{1-c}$</span>. I will derive this by assuming the income is <span class="math-container">$E$</span> and then use the Marshall demands to get the value function which I then invert to get the expenditure function.</p>
<p>The value function for <span class="math-container">$A_1h_1^ce_1^{1-c}$</span> is</p>
<p><span class="math-container">$$V(P_{h_1},P_{e_1},E) = \frac{A_1c^c(1-c)^{1-c}}{P_{h_1}^cP_{e_1}^{1-c}} E,$$</span></p>
<p>so the expenditure function is</p>
<p><span class="math-container">$$E = \frac{P_{h_1}^cP_{e_1}^{1-c}}{A_1c^c(1-c)^{1-c}} V,$$</span></p>
<p>which tells you that the price for <span class="math-container">$X_1$</span> utility point <span class="math-container">$V$</span> is simply</p>
<p><span class="math-container">$$P_{X_1} = \frac{P_{h_1}^cP_{e_1}^{1-c}}{A_1c^c(1-c)^{1-c}},$$</span></p>
<p>by the same argument the price <span class="math-container">$P_{X_2}$</span> is found to be</p>
<p><span class="math-container">$$P_{X_2} = \frac{P_{h_2}^cP_{e_2}^{1-c}}{A_2c^c(1-c)^{1-c}},$$</span></p>
<p>you can then back insert these into the expression for <span class="math-container">$s_1^m$</span>.</p>
<p>Note that the prices of <span class="math-container">$P_{X_1}$</span> and <span class="math-container">$P_{X_2}$</span> are the same for the mother and father and only depend on exogenously given constants.</p>
<p>You have to do this for the rest of the shares as well ... but I leave that as an exercise for you.</p>
<p>PS. To derive health ratio</p>
<p><span class="math-container">$$\lambda := \frac{h_1}{h_2} = \frac{h_1^f + h_1^m}{h_2^f + h_2^m}$$</span></p>
<p><span class="math-container">$$h_1^m + h_1^f= \frac{cM(s_1^m\mu  +s_1^f(1-\mu))}{P_{h_1}},$$</span></p>
<p><span class="math-container">$$h_2^m + h_2^f= \frac{cM(s_2^m\mu  +s_2^f(1-\mu))}{P_{h_2}},$$</span></p>
<p><span class="math-container">$$\lambda = \frac{P_{h_2}}{P_{h_1}}\frac{(s_1^m\mu  +s_1^f(1-\mu))}{(s_2^m\mu  +s_2^f(1-\mu))}$$</span></p>

CC BY-SA 4.0
2024-06-16T21:48:41.593Z
Thank you so, so much for providing this explanation with the utmost clarity.
Shen Shen
CC BY-SA 4.0
2024-06-16T21:58:50.890Z
You are welcome. Maybe you will also find this interesting: https://economics.stackexchange.com/questions/50478/two-stage-utility-maximization-problem/50482#50482. And if you like the answer you may consider accepting it.
Jesper Hybel
CC BY-SA 4.0
2024-06-17T03:16:17.830Z
Done! Thanks a lot again. If it is ok, I would like to followup with another (perhaps very dumb) question. If I was to derive the ratio of the amount of health input of child 1 to child 2, would the correct approach be substituting $P_{X_1}$ and $P_{X_2}$ back into $s_1^m$, $s_2^m$, $s_1^f$ and $s_2^f$ as you mentioned, then substitute those into $\frac{h_1}{h_2}$, obtained from $\frac{h_1^m+h_1^f}{h_2^m+h_2^f}$?
Shen Shen
CC BY-SA 4.0
2024-06-17T06:52:12.223Z
Yes, that is correct. Use exogenous parameters to calculate $P_{X_1}$ and $P_{X_2}$ then calculate shares using further exogenous parameters and prices and finally calculate the health ratio $\lambda$ as added that to the question.
Jesper Hybel
CC BY-SA 4.0
2024-06-17T12:54:30.837Z
Sorry, I no longer follow what you are trying to achieve. I suggest  you open a follow up question if further clarification is needed.
Jesper Hybel
CC BY-SA 4.0
2024-06-17T14:38:04.760Z
Yeah I realised it made little sense so I deleted it right after(seems like it went through anyway). Sorry for the trouble, and thank you again for the assistance.
Shen Shen
CC BY-SA 4.0
2024-06-15T19:01:23.157Z
2024-06-17T06:56:48.747Z
2024-06-17T06:56:48.747Z
Jesper Hybel
<p>I apologise for any improper use of terminology/notation as I only have a very rudimentary understanding of the topic. I would like to know how to solve the following utility maximisation problem concerning intrahousehold resource allocation:</p>
<p>The mother’s and father’s utilities enter into the collective utility function, which has a simple Cobb Douglas form, where <span class="math-container">$\mu$</span> denotes mothers Pareto weight, and <span class="math-container">$1-\mu$</span> is that of the father:</p>
<p>max <span class="math-container">$U=u_m^{\mu}u_f^{(1-\mu)}$</span>.</p>
<p>The parental sub-utility functions, nested in the household utility function, both take on the CES form, in which parents decide the amount of human capital X of child 1 and child 2 that would maximize their individual utility. The problem thus only concerns the allocation of resources towards two public goods. a and <span class="math-container">$\rho$</span> are the mother’s preference for child 1 and degree of inequality aversion, respectively. b and <span class="math-container">$\sigma$</span> serve the same roles in the father’s sub utility:</p>
<p><span class="math-container">$u_m=(aX_1^\rho+(1-a)X_2^\rho)^\frac{1}{\rho}$</span></p>
<p><span class="math-container">$u_f=(bX_1^\sigma+(1-b)X_2^\sigma)^\frac{1}{\sigma}$</span>.</p>
<p>s.t. the two children’s human capital production function, also of a Cobb Douglas functional form:</p>
<p><span class="math-container">$X_1=A_1h_1^ce^{1-c}_1$</span></p>
<p><span class="math-container">$X_2=A_2h_2^ce^{1-c}_2$</span></p>
<p>where A is the child’s genetic endowments, and h and e each denotes the amount of health and education inputs going into the production function of the child.</p>
<p>also s.t.
<span class="math-container">$M=P_{h_1}h_1+P_{h_2}h_2+P_{e_1}e_1+P_{e_2}e_2$</span>,</p>
<p>Where P denotes price of the input its subscript corresponds to. The share of the budget the mother has control over in her subutility maximization is <span class="math-container">$\mu$</span>, and for the father, <span class="math-container">$1-\mu$</span>.</p>
<p>I would like to solve for the ratio of health/education input of child 1 to child 2.</p>

CC BY-SA 4.0
2024-06-15T09:52:15.170Z
Hi: It's best to give the dimensions of the variables ( like $n \times 1$ or whatever ) and also their meanings. For example, $P_{x}$ probably denotes the projection but I'm not certain. Also, you should explain what's known and what's not. For example, what are the $h's$ and the $e's$. All of this will give you a greater chance of getting a response.
mark leeds
CC BY-SA 4.0
2024-06-15T11:57:42.857Z
Thank you so much for the suggestions.
Shen Shen
CC BY-SA 4.0
2024-06-15T13:25:28.317Z
It is super tedious. Have you derived the expenditure functions for the CES and CD utility?
Jesper Hybel
<microeconomics><utility>
Maximisation of nested utility function with 2 constraints
Shen Shen
# Maximisation of nested utility function with 2 constraints

(Asked by: username_0 on 2024-06-15)

I apologise for any improper use of terminology/notation as I only have a very rudimentary understanding of the topic. I would like to know how to solve the following utility maximisation problem concerning intrahousehold resource allocation:

The mother’s and father’s utilities enter into the collective utility function, which has a simple Cobb Douglas form, where $\mu$ denotes mothers Pareto weight, and $1-\mu$ is that of the father:

max $U=u\_m^{\mu}u\_f^{(1-\mu)}$.

The parental sub-utility functions, nested in the household utility function, both take on the CES form, in which parents decide the amount of human capital X of child 1 and child 2 that would maximize their individual utility. The problem thus only concerns the allocation of resources towards two public goods. a and $\rho$ are the mother’s preference for child 1 and degree of inequality aversion, respectively. b and $\sigma$ serve the same roles in the father’s sub utility:

$u\_m=(aX\_1^\rho+(1-a)X\_2^\rho)^\frac{1}{\rho}$

$u\_f=(bX\_1^\sigma+(1-b)X\_2^\sigma)^\frac{1}{\sigma}$.

s.t. the two children’s human capital production function, also of a Cobb Douglas functional form:

$X\_1=A\_1h\_1^ce^{1-c}\_1$

$X\_2=A\_2h\_2^ce^{1-c}\_2$

where A is the child’s genetic endowments, and h and e each denotes the amount of health and education inputs going into the production function of the child.

also s.t.
$M=P\_{h\_1}h\_1+P\_{h\_2}h\_2+P\_{e\_1}e\_1+P\_{e\_2}e\_2$,

Where P denotes price of the input its subscript corresponds to. The share of the budget the mother has control over in her subutility maximization is $\mu$, and for the father, $1-\mu$.

I would like to solve for the ratio of health/education input of child 1 to child 2.

**Tags:** microeconomics, utility
> **Comments:**
> **username_1**: Hi: It's best to give the dimensions of the variables ( like $n \times 1$ or whatever ) and also their meanings. For example, $P_{x}$ probably denotes the projection but I'm not certain. Also, you should explain what's known and what's not. For example, what are the $h's$ and the $e's$. All of this will give you a greater chance of getting a response.  
> **username_0**: Thank you so much for the suggestions.  
> **username_2**: It is super tedious. Have you derived the expenditure functions for the CES and CD utility?  

---

## 1 answers

### Answer by username_2 on 2024-06-15. Score: 2 | Accepted answer

To solve the utility maximization problem you need to follow the money.

The total income is $M$ which is split between the mother and father according to the Pareto weights $\mu$ and $(1-\mu)$. This implies that $\mu M$ is used on $u\_m$ and $(1-\mu)M$ on $u\_f$.

The income $\mu M$ is split by the mother on $X\_1$ for child $1$ and $X\_2$ for child $2$. Let $s\_1^m$ be the share of mother income $\mu M$ used on $X\_1$ and $s\_2^m$ the share of $\mu M$ used on $X\_2$. Similarly, let $s\_1^f$ and $s\_2^f$ be the same shares for the case of the father.

The mother therefore uses $s\_1^m\mu M$ on $X\_1^m$ (subscript to indicate it is $X\_1$ paid by the mother) and $s\_2^m\mu M$ on $X\_2^m$. Similarly, for the father $s\_1^f(1-\mu)M$ used on $X\_1^f$ and $s\_2^f(1-\mu) M$ used on $X\_2^f$.

Finally, whatever is spent on $X\_1$ is spent partly on $h\_1$ and partly on $e\_1$ by the shares $c$ and $(1-c)$ by both the mother and the father.

Hence, the mothers demand for $h\_1$ is given as

$$h\_1^m = \frac{cs\_1^m \mu M}{P\_{h\_1}},$$

and the mother demand for $e\_1$ is given as

$$e\_1^m = \frac{(1-c)s\_1^m \mu M}{P\_{e\_1}},$$

The mothers demand for $h\_2$ is given as

$$h\_2^m = \frac{cs\_2^m \mu M}{P\_{h\_2}},$$

and the mother demand for $e\_1$ is given as

$$e\_2^m = \frac{(1-c)s\_2^m \mu M}{P\_{e\_2}},$$

and the fathers' demands are the same just with $(1-\mu)$ instead of $\mu$ and the fathers shares $s\_1^f$ and $s\_2^f$.

The shares $s\_1^m,s\_1^f,s\_2^m,s\_2^f$ are found by solving the CES utility maximization problem and the expenditure minimization problem of the CD utility.

The Marshall demand for a CES utility function of the form

$$(kZ\_1^\alpha + (1-k)Z\_2^{1-\alpha})^{1/\alpha}$$

Can be written as

$$Z\_1^\star =\left[ \frac{k^\alpha P\_{Z\_1}^{1-\alpha}}{k^\alpha P\_{Z\_1}^{1-\alpha} + (1-k)^\alpha P\_{Z\_2}^{1-\alpha}}\right] \frac{I}{P\_{Z\_1}} $$

$$Z\_2^\star =\left[ \frac{(1-k)^\alpha P\_{Z\_2}^{1-\alpha}}{k^\alpha P\_{Z\_1}^{1-\alpha} + (1-k)^\alpha P\_{Z\_2}^{1-\alpha}}\right] \frac{I}{P\_{Z\_2}},$$

where the terms in the brackets are the shares (see for example this tutorial <https://www.youtube.com/watch?v=WhxprnW2Sc4>).

To solve for the share $s\_1^m$ do the following:

(1) Let $P\_{Z\_1}$ and $P\_{Z\_2}$ be $P\_{X\_1}$ and $P\_{X\_2}$

(2) Let $k=a$ and

(3) Let $\alpha=\rho$

To get

$$s\_1^m = \left[ \frac{a^\rho P\_{X\_1}^{1-\rho}}{a^\rho P\_{X\_1}^{1-\rho} + (1-a)^\rho P\_{X\_2}^{1-\rho}}\right],$$

Finally, to get the price of $X\_1$ and $X\_2$ you need the expenditure function of the CD functions $A\_1h\_1^ce\_1^{1-c}$ and $A\_2h\_2^ce\_2^{1-c}$. I will derive this by assuming the income is $E$ and then use the Marshall demands to get the value function which I then invert to get the expenditure function.

The value function for $A\_1h\_1^ce\_1^{1-c}$ is

$$V(P\_{h\_1},P\_{e\_1},E) = \frac{A\_1c^c(1-c)^{1-c}}{P\_{h\_1}^cP\_{e\_1}^{1-c}} E,$$

so the expenditure function is

$$E = \frac{P\_{h\_1}^cP\_{e\_1}^{1-c}}{A\_1c^c(1-c)^{1-c}} V,$$

which tells you that the price for $X\_1$ utility point $V$ is simply

$$P\_{X\_1} = \frac{P\_{h\_1}^cP\_{e\_1}^{1-c}}{A\_1c^c(1-c)^{1-c}},$$

by the same argument the price $P\_{X\_2}$ is found to be

$$P\_{X\_2} = \frac{P\_{h\_2}^cP\_{e\_2}^{1-c}}{A\_2c^c(1-c)^{1-c}},$$

you can then back insert these into the expression for $s\_1^m$.

Note that the prices of $P\_{X\_1}$ and $P\_{X\_2}$ are the same for the mother and father and only depend on exogenously given constants.

You have to do this for the rest of the shares as well ... but I leave that as an exercise for you.

PS. To derive health ratio

$$\lambda := \frac{h\_1}{h\_2} = \frac{h\_1^f + h\_1^m}{h\_2^f + h\_2^m}$$

$$h\_1^m + h\_1^f= \frac{cM(s\_1^m\mu +s\_1^f(1-\mu))}{P\_{h\_1}},$$

$$h\_2^m + h\_2^f= \frac{cM(s\_2^m\mu +s\_2^f(1-\mu))}{P\_{h\_2}},$$

$$\lambda = \frac{P\_{h\_2}}{P\_{h\_1}}\frac{(s\_1^m\mu +s\_1^f(1-\mu))}{(s\_2^m\mu +s\_2^f(1-\mu))}$$

> **Comments:**
> **username_0**: Thank you so, so much for providing this explanation with the utmost clarity.  
> **username_2**: You are welcome. Maybe you will also find this interesting: https://economics.stackexchange.com/questions/50478/two-stage-utility-maximization-problem/50482#50482. And if you like the answer you may consider accepting it.  
> **username_0**: Done! Thanks a lot again. If it is ok, I would like to followup with another (perhaps very dumb) question. If I was to derive the ratio of the amount of health input of child 1 to child 2, would the correct approach be substituting $P_{X_1}$ and $P_{X_2}$ back into $s_1^m$, $s_2^m$, $s_1^f$ and $s_2^f$ as you mentioned, then substitute those into $\frac{h_1}{h_2}$, obtained from $\frac{h_1^m+h_1^f}{h_2^m+h_2^f}$?  
> **username_2**: Yes, that is correct. Use exogenous parameters to calculate $P_{X_1}$ and $P_{X_2}$ then calculate shares using further exogenous parameters and prices and finally calculate the health ratio $\lambda$ as added that to the question.  
> **username_2**: Sorry, I no longer follow what you are trying to achieve. I suggest  you open a follow up question if further clarification is needed.  
> **username_0**: Yeah I realised it made little sense so I deleted it right after(seems like it went through anyway). Sorry for the trouble, and thank you again for the assistance.  

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