I am reading a paper that derives a theoretical retirement model. There is a utility function and a budget constraint forming an optimal control problem. The solution to this problem states that \begin{align} U(C(t),0) &= \lambda \quad\text{for }0 \le t \le R,\\ U(C(t),1) &= \lambda \quad\text{for }R \le t \le T, \end{align} where $\lambda$ is the Lagrange multiplier. $U$ is the utility function where the arguments of the function are consumption ($C$) and leisure ($L$). $R$ is the retirement age, and $T$ is the years the agent lives.

The first condition says that the utility is constant during the working years where leisure available during working years is $0$. The second condition says that the utility is constant during the retirement years where leisure available during retirement years is $1$.

The paper then mentions that "This implies that the individual chooses a constant level of consumption for his pre-retirement years, $C_0$, and another (perhaps different) level of consumption, $C_1$, for his post-retirement years. $C_1$ will exceed (be smaller than) $C_0$ if \begin{equation} \frac{\partial^2U}{\partial C\;\partial L}>0\quad(<0) \end{equation} The last expression denotes "cross partial derivative of $U$ with respect to $L$ and then $C$".

I am struggling to interpret the inequality. That is, the cross partial derivative says that if the change in the marginal utility of leisure with respect to a change in consumption is positive, then the consumption level during retirement should be larger than that during the working years. Why?

  • $\begingroup$ It may help if you provide a link to the paper in question. Also, please consider using MathJax to format mathematical expressions. $\endgroup$
    – Herr K.
    Jul 8 '19 at 16:08
  • $\begingroup$ Burbidge, J. B., Robb, A. L., 1980. Pensions and retirement behaviour. The Canadian Journal of Economics 13 (3), 421–437. I will consider tex for formulas. $\endgroup$
    – Snoopy
    Jul 8 '19 at 17:10

The partial derivative $\frac{\partial^2 U(\cdot)}{\partial C\partial L}$ tells you how a change in leisure affects the marginal utility of consumption. Therefore if it is positive it means that the marginal utility of consumption is higher when $L=1$. In that case, the agent plans her life to be able to consume more when she is retired and leisure is bigger. In contrast, if leisure reduces the marginal utility for consumption, $\frac{\partial^2 U(\cdot)}{\partial C\partial L}<0$, then it is better to consume more while working, than when leisure is big.

  • $\begingroup$ Why do we read the cross partial derivative as "how leisure affects the marginal utility of consumption" and not as "how consumption affects the marginal utility of leisure"? Because, does not the stated cross partial derivative as first take the partial derivative with respect to leisure (which gives the marginal utility of leisure) and then take the partial derivative with respect to consumption? $\endgroup$
    – Snoopy
    Jul 8 '19 at 17:00
  • $\begingroup$ Oh, if the function is continuously differentiable, those two things are the same. That is, $\frac{\partial ^2 U(\cdot)}{\partial C\partial L}=\frac{\partial ^2 U(\cdot)}{\partial L\partial C}$. So both interpretations are valid. I simply chose the most convenient, at the end of the day, when the cross partial derivative is positive, it means that it is better to consume these two goods together, and this is why the consumer ends up consuming more when retired. $\endgroup$
    – Regio
    Jul 8 '19 at 20:50
  • $\begingroup$ Thanks for the helpful answers. $\endgroup$
    – Snoopy
    Jul 9 '19 at 15:33

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