# Help solving utility optimization problem from Connelly 1992 (Economics)

I am working through a paper by Connelly 1992 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:

$$$$\max_{\{?\}} U(x_m, Q, t_l)$$$$ s.t. $$$$Q = Q(t_q,t_{cc}q)$$$$ $$$$t_m + t_Q + t_l = 1$$$$ $$$$t_m*W + V = x_m + p_{cc}t_{cc}$$$$ $$$$t_Q + t_{cc} < 1$$$$

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.

• Is this paper "The Effect of Child Care Costs on Married Women's Labor Force Participation"? Commented Feb 9, 2022 at 21:25

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.

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.