Model

Set Up

Consider an agency run by a bureaucrat. Each year $t$ she allocates a budget $b_t$ to monthly expenditure $e_{t1},...,e_{t12}$ and a surplus $s_t$ that she can partially roll-over to the next year. If she decides to roll-over $s_t$ dollars, her budget in year $t+1$ will be

$$b_{t+1}(s_t)=\max_{x\in[0,s_t]}\alpha (s_t-x)+\frac{(x+1)^{1-\gamma}}{1-\gamma}-\frac{1}{1-\gamma}+\overline{B},$$

where $x$ represent the dollars she roll-over by accumulating inventories, $\alpha$ represents the fraction of the non-expended budget the legislature allows her to transfer to the following year, and $\overline{B}$ represent a fixed amount the agency receives each year independently of $s_t$. Note that when transferring resources through inventories, she faces a declining marginal rate in the dollars that are transferred to the next year's budget. The rate at which it declines is governed by the parameter $\gamma>0$. This is a convenient way of introducing a fixed asset into the model without increasing the number of state variables, and captures the fact that transferring resources through the accumulation of inventories is only effective for relatively modest levels of roll-over. Finally, note that the bureaucrat will combine both methods of transferring the surplus to the next year (the legislature allowed roll-over vs accumulating inventories) to maximize next year's budget.

The bureaucrat will allocate her budget $b_t$ between the monthly expenditures $e_1,...,e_{12}$ and the surplus transferred to the following year $s_t$ to maximize the present value of her payoffs. She dislikes the labor associated with the monthly expenditures at a constant marginal cost $c$, but benefits from having a good result on her year-end performance review. The payoff from her review depends on monthly expenditures as follows

$$P(e_{t1},...,e_{t12})=\frac{A}{1-\rho}\left(\Pi_{j=1}^{12}e_{tj}^\frac{1}{12}\right)^{1-\rho},$$

where $A$ is a parameter that scales her payoff and $\rho>0$ a parameter that governs the economies of scale of the benefit function (the larger $\rho$, the stronger the diseconomies of scale of the payoff function). Having diseconomies of scale in the payoff function incentivizes expenditure smoothing from year to year, while the complementary structure of monthly expenditures incentivizes expenditure smoothing within a year.

Finally, the bureaucrat discounts the future with a quasi-hyperbolic function with parameters $(\beta,\delta)$, and is naive with regard to her present bias. Hence, the problem faced by the bureaucrat when deciding how to spend her budget in year $t$ and month $m\leq11$ is

\[ \begin{eqnarray*} V(b_{t},e_{t1},...,e_{tm-1})&=&\max_{e_{tm}\in[0,b_{t}-e_{t1}-...-e_{tm-1}]} -ce_{tm}+\beta\delta \hat{V}(b_{t},e_{t1},...,e_{tm}),\\ \hat{V}(b_{t},e_{t1},...,e_{tm-1})&=&\max_{e_{tm}\in[0,b_{t}-e_{t1}-...-e_{tm-1}]} -ce_{tm}+\delta \hat{V}(b_{t},e_{t1},...,e_{tm}),\\ \end{eqnarray*} \]

and, for $m=12$, \[ \begin{eqnarray*} V(b_{t},e_{t1},...,e_{t11})&=&\max_{e_{t12}\in[0,b_{t}-e_{t1}-...-e_{t11}]} -ce_{t12}+P(e_{t1},...,e_{t12})+\beta\delta \hat{V}(b_{t+1}(b_t-e_{t1}-...-e_{t12})),\\ \hat{V}(b_{t},e_{t1},...,e_{t11})&=&\max_{e_{t12}\in[0,b_{t}-e_{t1}-...-e_{t11}]} -ce_{t12}+P(e_{t1},...,e_{t12})+\delta \hat{V}(b_{t+1}(b_t-e_{t1}-...-e_{t12})),\\ \end{eqnarray*} \]

Solution

To find the optimal strategy of the bureaucrat, we first focus on solving the continuation value at the end of the year, $\hat{V}(b_{t+1}(b_t-e_{t1}-...-e_{t12}))$. As this value function has a geometric discount, in can be solved using standard dynamic programming methods. The problem is further simplified if we divide it in two steps: first the bureaucrat decides at the beginning of the year how much of its budget she will spend on that year, and then decides how to to assign this amount throughout the year.

Lets start with the second part, that is, given an amount $u_t\leq b_t$ that has been assigned to be spent in the current year, how much does the bureaucrat decides to spend in each month. As the continuation value is fixed by the decision of how much is assigned to this year, we can focus on the current's year problem:

\[ \begin{eqnarray*} \max_{e_{t1},...,e_{t12}}& &-c\sum_{m=1}^{12} \delta^{m-1}e_{tm}+\delta^{11} P(e_{t1},...,e_{t12})\\ s.t:&& \sum_{m=1}^{12}e_{tm}=u_t. \end{eqnarray*} \]

The first order conditions of this problem are

\[ \begin{eqnarray*} -c &=& P_1(e_{t1},...,e_{t12}),\\ -c\delta &=& P_2(e_{t1},...,e_{t12}),\\ &\vdots&\\ -c\delta^{11} &=& P_{12}(e_{t1},...,e_{t12}),\\ \end{eqnarray*} \]

from where it follows that

\[ e_{t1}=e_{t2}\delta=...=e_{t12}\delta^{11}, \]

which together with the restriction yields

\[ \begin{eqnarray*} e_{t1} &=& \frac{\delta^{11}u_t}{\sum_{i=1}^{12}\delta^{i-1}},\\ e_{t2} &=& \frac{\delta^{10}u_t}{\sum_{i=1}^{12}\delta^{i-1}},\\ &\vdots&\\ e_{t12} &=& \frac{u_t}{\sum_{i=1}^{12}\delta^{i-1}}.\\ \end{eqnarray*} \]

Denote the optimal level of expenditure in month $m$ as a function of $u_t$ as $e_m(u_t)$. Then, the value function at the beginning of the year is given by the following recursive problem:

\[ \hat{V}(b_t)=\max_{u_t\leq b_t} -c\sum_{m=1}^{12} \delta^{m-1}e_{m}(u_t)+\delta^{11} P(e_1(u_t),...,e_{12}(u_t))+\delta^{11}\hat{V}(b_{t+1}(b_t-u_t)) \]

We find the value function numerically by discretizing the state space and iterating the value function.

With a numerical approximation of $\hat{V}(b_t)$, we can solve the problem of the present bias bureaucrat. First, note that given the expenditure assigned for this year $u_t$, the first order condition of the present bias bureaucrat yields:

\[ \begin{eqnarray*} e_{t1} &=& \frac{\beta\delta^{11}u_t}{\sum_{i=1}^{11}\delta^{i-1}+\beta\delta^{11}},\\ e_{t2} &=& \frac{\delta^{10}u_t}{\sum_{i=1}^{11}\delta^{i-1}+\beta\delta^{11}},\\ &\vdots&\\ e_{t12} &=& \frac{u_t}{\sum_{i=1}^{11}\delta^{i-1}+\beta\delta^{11}}.\\ \end{eqnarray*} \] With this result and $\hat{V}(b_t)$, we start solving forward for a year with budget $\overline{B}$. In January we find how much she spends by numerically finding the optimal level of expenditure for the year, $u_t$, taking into account the optimal plan given by the equations above and the continuation value function $\hat{V}(b_t)$. Then, in February we do the same exercise but starting from the budget that is left, that is, $\overline{B}-e_{t1}$. We continue in the same fashion until December.