CHAPTER 8: A MULTI-PERIOD MAYALAND CROP MODEL
The easiest way of developing models that incorporate time is to extend the linear models developed in earlier exercises. Using this format, multi-period models can be thought of simply as a series of single-period models linked by dynamic constraints or "equations of motion" that link the periods. These linking flows can incorporate any multi-period phenomena: capital accumulation, growth in livestock herds, harvesting forests, and degradation of the environment.
To capture multi-period activities in an Excel linear programming model, the same activities conducted in different time periods are treated as different activities. For example, the only difference in the model between corn grown in Year 1 (CORN1) and corn grown in Year 2 (CORN2) is in the activity name. The technical coefficients that describe the activities and the basic resource availability in each period are the same, although they could be altered over time to reflect technical change if that was deemed to be desirable.
The magnitude of elements in the objective function in Year 2 (CORN2) are different from Year 1 because they must be discounted. Net returns in the future are worth less than returns in the future by the discount factor 1/(1+ i)t.
By remaining within a linear framework, the power of the Simplex algorithm continues to be available. There is no significant problem regarding the number of state and control variables although, as will become obvious, Excel with its COPY command makes it relatively simple to generate very large models by simply copying groups of cells that represent different time periods along a diagonal. But because the transfer rows, i.e., the equations of motion, are explicitly linear, solution times for models with 50 period constraints and 20 periods remain reasonable on modern microcomputers.
The development of the multi-period model proceeds in several stages. With the exception of the first, each is a possible stopping point in the construction of an operational model. By breaking down the whole into a series of parts, it will be easier to assimilate the rather substantial accumulation of variables and equations shown in the final version of the model. There are a number of possible extensions, not all of which have been included in the discussion below.
(3) Adding capital and labor markets to simulate opening the household to trade.
(3) Extending the model to include a marginal propensity to consume income and a capital market
Creating Multi-periods from a One-Period Model
Table 8.1 shows a single period model (Model 8.1) derived from the original Mayaland tutorial. It differs somewhat in the magnitudes of the net revenues incorporated in the objective function and capital has been substituted for mules in the constraint set. The Sensitivity Report 8.1 provides information on the optimal solution.
Table 8.2 shows a three-period version of the single period model (Model 8.2). At this stage there are no linkages between the years and the optimal resource allocation remains the same, i.e., the levels of the activities are the same for each period. However, the value of the objective function declines from one period to the next because it has been discounted. As the Sensitivity Report 8.2 shows, the shadow prices of the binding constraints (capital and the peanut marketing constraint) decline accordingly.
The fact that the levels of the activities do not change is to be expected since the discount factor affects all activities in the primal solution by the same amount, i.e., they are affected proportionately. The shadow prices of the constraints, however, decline because the value of the objective function is reduced due to discounting.
Adding Capital Constraints
Perhaps the most obvious linking constraint between time periods is capital. This follows from the assumption that investment in period t + 1 is constrained by the amount of savings in period t. For example, in the version presented below, it is assumed that capital needed to fund the variable costs in period t+1 is obtained from profits generated in year t.
Adding a multi-year capital constraint requires a number of additions to the model. Chief among these are the following:
(1) Initial and terminal conditions must be specified. All dynamic models share the need to specify how the growth element in the model is to be set in motion and how the accumulated "inventory" is to be valued.
(2) A series of equations are required to produce the capital use and accumulation mechanisms.
The capital use and accumulation equation above says, in words, that the demand for capital is made up of (1) the cost of hiring labor, (2) variable production costs, (3) consumption needs. The supply of capital comes from (1) crops sold in the previous year, and (2) off-farm earnings
(3) Adding consumption requires the addition of a constraint that sets aside a minimum amount of capital for household maintenance. Minimum consumption levels must also be included in the model's specification; consumption is an important substitute for investment.
A terminal period needs to be computed and added to the objective function. Without it, the model gets no credit for saving in the next to last period. In this case, the value for a unit of the terminal activity is simply the value of the capital inventory, i.e., the net revenue of crops that have been grown in the next to last period.
With the addition of the capital constraints, the model no longer simply replicates a single period. The model is initially limited by the amount of capital available from resources saved in periods before the models initial period. Since there is no capital market, the total land planted to crops is small in the first period. In subsequent years, savings out of earnings from the previous year become available and additional acreage is devoted to crop.
Any dynamic model of the household requires information about consumption. The fact that consumption influences capital accumulation can be seen from the conventional formulation of the growth models in which the savings of one period provide the investment funds for the subsequent. As consumption goes up, savings (and subsequently investment) go down. Consumption can be incorporated in the multi-period model by inserting an activity that simulates consumption and requiring that a certain amount of capital be available to the household in each period to meet specified consumption requirements.
A variety of additional extensions of the model help to create a more realistic picture of a firm's growth opportunities. Obvious examples include:
1. A credit market in which borrowing and lending make it possible to alleviate the shortage of capital. (Imperfect credit markets are readily simulated by adjusting the magnitude of the transaction costs between borrowing and lending.
2. A labor market in which labor hired-out contributes to capital availability.
3. A formulation of the consumption activity in which the marginal propensity to consume declines as income increases.
4. A formulation of the labor and incomes activities in which the propensity to consume leisure, i.e., reduce labor availability declines as income increases.