<p>I guess the text considers the problem</p>
<p><span class="math-container">$$\max_{x,Q,t_L,t_Q,t_{cc},t_m,t_L,q} u(x,Q,t_L),$$</span></p>
<p>subject to</p>
<p><span class="math-container">$$Q(t_Q,t_{cc}q)- Q=0$$</span></p>
<p><span class="math-container">$$1-t_Q-t_{cc}\geq0$$</span></p>
<p><span class="math-container">$$1-t_m-t_L-t_Q\geq 0$$</span>
<span class="math-container">$$t_mW+V-x-p_{cc}t_{cc}\geq 0$$</span></p>
<p>the Lagrangian for which is</p>
<p><span class="math-container">$$\mathcal L(.) = u(x,Q,t_L) + \lambda_1(Q(t_Q,t_{cc}q)-Q) + \lambda_2(1-t_Q-t_{cc}) + \lambda_3(1-t_m-t_L-t_Q) + \lambda_4 (t_mW+V-x-p_{cc}t_{cc})$$</span></p>
<p>The derivatives of the function <span class="math-container">$Q(.,.)$</span> with respect to the first argument are denoted <span class="math-container">$Q_1$</span> and for the second argument <span class="math-container">$Q_2$</span>. Differentiating the Lagrangian function with respect to the choice variables results in the following first order conditions</p>
<p><span class="math-container">$$\frac{\partial \mathcal L}{\partial x} = \frac{\partial \mathcal u}{\partial x} - \lambda_4 = 0$$</span>
<span class="math-container">$$\frac{\partial \mathcal L}{\partial Q} = \frac{\partial \mathcal u}{\partial Q} - \lambda_1 = 0$$</span>
<span class="math-container">$$\frac{\partial \mathcal L}{\partial t_L} = \frac{\partial \mathcal u}{\partial t_L} - \lambda_3 = 0,$$</span></p>
<p>assuming all three marginal utilities to be strictly positive in optimum it follows from the KKT complementarity conditions that constraint 4,1 and 3 are binding (as the constraints are also stated to be in the text).</p>
<p>Further first-order conditions are</p>
<p><span class="math-container">$$\frac{\partial \mathcal L}{\partial t_Q} = \lambda_1 Q_1 - \lambda_2 - \lambda_3 = 0$$</span>
<span class="math-container">$$\frac{\partial \mathcal L}{\partial t_{cc}} = \lambda_1 Q_2q - \lambda_2 - \lambda_4p_{cc} = 0$$</span>
<span class="math-container">$$\frac{\partial \mathcal L}{\partial t_m} = -\lambda_3 + \lambda_4 W = 0,$$</span></p>
<p>the only Lagrange coefficient not determined as a marginal utility is <span class="math-container">$\lambda_2$</span> but we can get</p>
<p><span class="math-container">$$\lambda_2 = \lambda_1 Q_2q-\lambda_4 p_{cc} = \lambda_1 Q_1 - \lambda_3,$$</span>
and therefore</p>
<p><span class="math-container">$$\lambda_3 = \lambda_1 (Q_1 - Q_2q) + \lambda_4 p_{cc},$$</span></p>
<p>divide with <span class="math-container">$\lambda_4$</span> - which we know is strictly positive - and use that <span class="math-container">$\lambda_3/\lambda_4 = W$</span> to get</p>
<p><span class="math-container">$$W = \frac{\lambda_1}{\lambda_4}(Q_1 - Q_2q) + p_{cc}$$</span></p>
<p>which gives the results upon substitution of expressions for <span class="math-container">$\lambda_1$</span> and <span class="math-container">$\lambda_4$</span> in terms of marginal utilities.</p>
<p>Finally, treat <span class="math-container">$p_{cc}$</span> as a function of <span class="math-container">$q$</span> and differentiate Lagrangian with respect to <span class="math-container">$q$</span> to get</p>
<p><span class="math-container">$$\frac{\partial L(.)}{\partial q} = \lambda_1 Q_2 t_{cc} -\lambda_4t_{cc} \frac{\partial p_{cc}}{ \partial q} =0,$$</span></p>
<p><span class="math-container">$$t_{cc}\lambda_4 \left(\frac{\lambda_1}{\lambda_4} Q_2  - \frac{\partial p_{cc}}{ \partial q}\right)=0,$$</span></p>
<p>hence assuming internal solution for <span class="math-container">$t_{cc}&gt;0$</span> it follows that</p>
<p><span class="math-container">$$\frac{\lambda_1}{\lambda_4} Q_2  = \frac{\partial p_{cc}}{ \partial q}$$</span> where in the text it simply says <span class="math-container">$Q_2  = \frac{\partial p_{cc}}{ \partial q}$</span> I'm guessing this is simply because <span class="math-container">$\frac{\lambda_1}{\lambda_4}$</span> is ignored as some in optimum irrelevant utility scaling constant.</p>

CC BY-SA 4.0
2022-02-10T13:52:42.823Z
2022-02-10T13:52:42.823Z
Jesper Hybel
<p>Mather choose her time allocations to maximize her utility.</p>
<p><span class="math-container">$\frac{U_{l}}{U_{x}}=W$</span>: this part is just the marginal utility of using time on consumption equals to the marginal utility of using the same time on leisure.</p>
<p><span class="math-container">$W=\frac{U_{Q}}{U_{x}}\left(Q_{1}-Q_{2} q^{*}\right)+p_{c c}^{*}$</span>: this part is the marginal utility of using time on consumption equals to the marginal utility of using the same time on child. This is more complicated than the leisure case because increase in time on child also reduce the time spent on childcare service, which affect mother's utility through change in child quality and saved money that can be used for consumption.</p>

CC BY-SA 4.0
2022-02-09T21:29:50.080Z
2022-02-09T21:29:50.080Z
Alalalalaki
<p>I am working through a <a href="https://www.jstor.org/stable/2109545" rel="nofollow noreferrer">paper by Connelly 1992</a> in RESTAT and I'm hoping to get some assistance in working through the optimization problem that she sets up. I apologize that it might be simple for many of you. I'm not an econ student and it's not my strength. The problem is as follows:</p>
<p><span class="math-container">\begin{equation}
\max_{\{?\}} U(x_m, Q, t_l)
\end{equation}</span>  s.t.
<span class="math-container">\begin{equation}
Q = Q(t_q,t_{cc}q)
\end{equation}</span>
<span class="math-container">\begin{equation}
    t_m + t_Q + t_l = 1
\end{equation}</span>
<span class="math-container">\begin{equation}
    t_m*W + V = x_m + p_{cc}t_{cc}
\end{equation}</span>
<span class="math-container">\begin{equation}
t_Q + t_{cc} &lt; 1
\end{equation}</span></p>
<p>She jumps to the next part, saying that some of the FOCs lead to: <span class="math-container">$\frac{U_l}{U_x} = W = \frac{U_Q}{U_x}(Q_1-Q_2q^{*})+p_{cc}^{*}$</span>, where <span class="math-container">$q^{*}$</span> is the optimally-chosen level of <span class="math-container">$q$</span> and <span class="math-container">$p_cc^{*}$</span> is the price at the optimally chosen <span class="math-container">$q$</span>.</p>
<p>I've left out the context of the parameters, which I hope does not really matter. I'm mostly hoping someone can help me understand how to break down the utility max problem. In particular, I get some hints at which parameters the consumer is choosing, but it is not entirely clear (hence the ? I left in the max). Also, the third expression of the three-way equality is not obvious to me.</p>
<p>Thank you very much in advance for your patience and help.</p>

CC BY-SA 4.0
2022-02-09T21:25:59.543Z
Is this paper "The Effect of Child Care Costs on Married Women's Labor Force Participation"?
Giskard
<utility><consumer-theory>
Help solving utility optimization problem from Connelly 1992 (Economics)
salve_regina
# Help solving utility optimization problem from Connelly 1992 (Economics)

(Asked by: username_0 on 2022-02-09)

I am working through a [paper by Connelly 1992](https://www.jstor.org/stable/2109545) in RESTAT and I'm hoping to get some assistance in working through the optimization problem that she sets up. I apologize that it might be simple for many of you. I'm not an econ student and it's not my strength. The problem is as follows:

\begin{equation}
\max\_{\{?\}} U(x\_m, Q, t\_l)
\end{equation} s.t.
\begin{equation}
Q = Q(t\_q,t\_{cc}q)
\end{equation}
\begin{equation}
t\_m + t\_Q + t\_l = 1
\end{equation}
\begin{equation}
t\_m\*W + V = x\_m + p\_{cc}t\_{cc}
\end{equation}
\begin{equation}
t\_Q + t\_{cc} < 1
\end{equation}

She jumps to the next part, saying that some of the FOCs lead to: $\frac{U\_l}{U\_x} = W = \frac{U\_Q}{U\_x}(Q\_1-Q\_2q^{\*})+p\_{cc}^{\*}$, where $q^{\*}$ is the optimally-chosen level of $q$ and $p\_cc^{\*}$ is the price at the optimally chosen $q$.

I've left out the context of the parameters, which I hope does not really matter. I'm mostly hoping someone can help me understand how to break down the utility max problem. In particular, I get some hints at which parameters the consumer is choosing, but it is not entirely clear (hence the ? I left in the max). Also, the third expression of the three-way equality is not obvious to me.

Thank you very much in advance for your patience and help.

**Tags:** utility, consumer-theory
> **Comments:**
> **username_1**: Is this paper "The Effect of Child Care Costs on Married Women's Labor Force Participation"?  

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## 2 answers

### Answer by username_2 on 2022-02-10. Score: 5

I guess the text considers the problem

$$\max\_{x,Q,t\_L,t\_Q,t\_{cc},t\_m,t\_L,q} u(x,Q,t\_L),$$

subject to

$$Q(t\_Q,t\_{cc}q)- Q=0$$

$$1-t\_Q-t\_{cc}\geq0$$

$$1-t\_m-t\_L-t\_Q\geq 0$$
$$t\_mW+V-x-p\_{cc}t\_{cc}\geq 0$$

the Lagrangian for which is

$$\mathcal L(.) = u(x,Q,t\_L) + \lambda\_1(Q(t\_Q,t\_{cc}q)-Q) + \lambda\_2(1-t\_Q-t\_{cc}) + \lambda\_3(1-t\_m-t\_L-t\_Q) + \lambda\_4 (t\_mW+V-x-p\_{cc}t\_{cc})$$

The derivatives of the function $Q(.,.)$ with respect to the first argument are denoted $Q\_1$ and for the second argument $Q\_2$. Differentiating the Lagrangian function with respect to the choice variables results in the following first order conditions

$$\frac{\partial \mathcal L}{\partial x} = \frac{\partial \mathcal u}{\partial x} - \lambda\_4 = 0$$
$$\frac{\partial \mathcal L}{\partial Q} = \frac{\partial \mathcal u}{\partial Q} - \lambda\_1 = 0$$
$$\frac{\partial \mathcal L}{\partial t\_L} = \frac{\partial \mathcal u}{\partial t\_L} - \lambda\_3 = 0,$$

assuming all three marginal utilities to be strictly positive in optimum it follows from the KKT complementarity conditions that constraint 4,1 and 3 are binding (as the constraints are also stated to be in the text).

Further first-order conditions are

$$\frac{\partial \mathcal L}{\partial t\_Q} = \lambda\_1 Q\_1 - \lambda\_2 - \lambda\_3 = 0$$
$$\frac{\partial \mathcal L}{\partial t\_{cc}} = \lambda\_1 Q\_2q - \lambda\_2 - \lambda\_4p\_{cc} = 0$$
$$\frac{\partial \mathcal L}{\partial t\_m} = -\lambda\_3 + \lambda\_4 W = 0,$$

the only Lagrange coefficient not determined as a marginal utility is $\lambda\_2$ but we can get

$$\lambda\_2 = \lambda\_1 Q\_2q-\lambda\_4 p\_{cc} = \lambda\_1 Q\_1 - \lambda\_3,$$
and therefore

$$\lambda\_3 = \lambda\_1 (Q\_1 - Q\_2q) + \lambda\_4 p\_{cc},$$

divide with $\lambda\_4$ - which we know is strictly positive - and use that $\lambda\_3/\lambda\_4 = W$ to get

$$W = \frac{\lambda\_1}{\lambda\_4}(Q\_1 - Q\_2q) + p\_{cc}$$

which gives the results upon substitution of expressions for $\lambda\_1$ and $\lambda\_4$ in terms of marginal utilities.

Finally, treat $p\_{cc}$ as a function of $q$ and differentiate Lagrangian with respect to $q$ to get

$$\frac{\partial L(.)}{\partial q} = \lambda\_1 Q\_2 t\_{cc} -\lambda\_4t\_{cc} \frac{\partial p\_{cc}}{ \partial q} =0,$$

$$t\_{cc}\lambda\_4 \left(\frac{\lambda\_1}{\lambda\_4} Q\_2 - \frac{\partial p\_{cc}}{ \partial q}\right)=0,$$

hence assuming internal solution for $t\_{cc}>0$ it follows that

$$\frac{\lambda\_1}{\lambda\_4} Q\_2 = \frac{\partial p\_{cc}}{ \partial q}$$ where in the text it simply says $Q\_2 = \frac{\partial p\_{cc}}{ \partial q}$ I'm guessing this is simply because $\frac{\lambda\_1}{\lambda\_4}$ is ignored as some in optimum irrelevant utility scaling constant.


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### Answer by username_3 on 2022-02-09. Score: 3

Mather choose her time allocations to maximize her utility.

$\frac{U\_{l}}{U\_{x}}=W$: this part is just the marginal utility of using time on consumption equals to the marginal utility of using the same time on leisure.

$W=\frac{U\_{Q}}{U\_{x}}\left(Q\_{1}-Q\_{2} q^{\*}\right)+p\_{c c}^{\*}$: this part is the marginal utility of using time on consumption equals to the marginal utility of using the same time on child. This is more complicated than the leisure case because increase in time on child also reduce the time spent on childcare service, which affect mother's utility through change in child quality and saved money that can be used for consumption.


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