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Multi-Year Dynamics. McCarl and Spreen Chapter 8. Dynamic Concerns Arise:. binding financial constraints which change with time (e.g. cash flow) production situations in which decisions made in one period affect choices or outcomes in the next (e.g. crop rotations)
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Multi-Year Dynamics McCarl and Spreen Chapter 8
Dynamic Concerns Arise: • binding financial constraints which change with time (e.g. cash flow) • production situations in which decisions made in one period affect choices or outcomes in the next (e.g. crop rotations) • a need to adjust over time to changing conditions • exhaustible resources
Dynamics Background • Length of total time period for the model and starting date must be specified • The length of explicitly modeled sub-periods must be determined • Initial and final inventory conditions must be specified (important!) • Activity life must be specified • Discount rate must be specified
Should dynamics be explicit? Dynamic situations may not require multiple time periods. In a steady state dynamic equilibrium model the same decision is assumed to be made over and over again and the model uses a “representative” single period. These models do not look at adjustment paths, and it is assumed that model parameters are constant across time.
Dynamic Equilibrium Models Dynamic equilibrium models are used when we are willing to assume: a) resource, technology, and price data are constant; b) a single long-run steady state solution is acceptable. Disequilibrium models are used when these assumptions do not hold.
The Two Types of Errors • Unnecessarily modeling explicit dynamics when a steady-state model would suffice • Improperly omitting them
Examples • Crop rotations can often be modeled as steady-states with various rotations entered as single-decision variables. • Under old-style farm bills that used 3-year and 5-year averages to determine program “base,” the adjustment paths to changes in prices etc. were important and dynamics needed to be explicit.
How Long? • Trade-offs between time coverage and size of model. • Time horizon must be long enough so that alterations in its duration do not impact the initial period solution, if the initial period solution is of interest to the modeler. • If first period is not of interest, model period must be long enough that the variables of interest are not affected by extension of the model.
How Long, continued? • A problem: size limits on model may make the previous goals impractical. • An alternative is to define a feasible size and then introduce terminal conditions for in-process inventories or left-over resources. • McCarl and Spreen provide references for further discussion of model period length.
Time Intervals • Periods may be of uniform or variable duration. • Years, quarters, months, weeks, have all been used as time intervals. • Model size increases rapidly as time intervals become shorter. • It is possible to have weekly detail on some variables but coarser detail on others.
Initial and Terminal Conditions • Terminal conditions should be set to reflect the value of in-process inventories – e.g. trees in the field, cattle being fed, etc. • Initial conditions should reflect likely inventory on hand, cash on hand, etc. • Often initial conditions are specified as exogenous (so many acres of trees), but they can also be modeled as decision choices with cash on hand used to purchase inputs. • Terminal values are harder to specify correctly than initial conditions.
Enterprise Life Enterprise life refers to the length of time an activity lasts. If enterprise life is not a fixed value, but can change depending on decisions, then activities and constraints in the model must be used to keep track of the age of items on hand.
Time Value of Money A dollar today is preferred to a dollar in the future • The dollar could be invested to earn interest • If dollar is spent on consumption, we’d prefer to get the enjoyment now • Risk is also a factor as unforeseen circumstances could prevent us from getting the dollar • Inflation may diminish the value of the dollar over time
Present Value and Future Value • Present Value (PV): the number of dollars available or invested at the current time or the current value of some amount to be received in the future • Future Value (FV): the amount to be received at some future time or the amount a present value will be worth at some future date when invested at a given interest rate
Present Value FV 1 or FV PV = (1 + i )n (1 + i ) n i is chosen to represent the decision-maker’s discount rate. The higher the value of i , the more money now is preferred to money in the future. n is the number of periods into the future.
Risk Risk is usually a factor in dynamic models, but this chapter deals with the certainty case. Risk is covered in another chapter of the book.
Disequilibrium with Known Life • Problem: how to commit resources over a number of time periods. • Once an item is begun, it will require resources for a known number of periods. • Formulation must consider: resource availability, choice of variables, and continuing use over the life of the enterprise. • Time preferences, and initial and terminal conditions must also be specified.
Strawberry Example • five-year time horizon • two crops: wheat and strawberries • strawberries take 3 years to produce • wheat is annual • farm has 700 acres, 50 acres currently in first-year strawberries • 1200 acre-feet of water available
Strawberry Problem Berry Information Wheat is an annual with net returns of $340/acre and annual water needs of 1 acre foot. Strawberries sell for $140 ton.
Approaches We can model annually, or use the “committed” land approach we saw earlier with the activity sequences for crop production.
Terminal values for strawberry acres in progress are provided in the objective function as given in the book. Initial conditions are 60 acres planted already, 50 in 1 year old strawberries and 10 in 2 year old berries. Land available is reduced accordingly in years 1 and 2. When you solve this problem, 60 acres of profit from the pre-committed strawberry land are not included in the objective function.
How this Works • The objective value includes all returns for 3 years of production ($1230) if we have enough time to complete 3 years of strawberries. • If we have less than 3 years left, strawberry returns are reduced and a terminal value assigned explicitly. • We could, instead, add that terminal value directly to the objective function in years 4 and 5. But if we do, it is harder to change the terminal value and track the effect.
Another way to solve this problem. The objective value will be different because this model includes all 700 acres in the first and second year. Decision variables have the same values as in the other solution.
Disequilibrium with Unknown Life In some situations, the length of the activity must be determined by the model. For example, it may be possible to keep an acre of strawberries for either 3 years or 4 years, depending on which is more profitable for the situation at hand.
Revised Strawberry Information Initial conditions: 50 acres 1yo, 10 acres 2yo.
Here, 2 year old strawberries in the first year are limited by the 50 acres that were planted the year before. 3-year-old strawberries are limited to the 10 acres that were planted 2 years ago.
Equilibrium with Known Life Disequilibrium models are large and often results are sensitive to hard-to-estimate terminal (or even initial) values. Equilibrium models provide an alternative approach. Here, the firm is assumed to operate in a steady-state; hence initial inventories will equal terminal inventories and may be set to optimal levels determined by the model.
Conditions • activity lasts for a designated number of periods • resource use depends on elapsed age of the activity
General Form of Problem (4 period example) From period 4 on, we have resource use for every age category. Under an equilibrium assumption, the activity level is constant, so we can sum up the resource use across time.
Known-Life Strawberry ProblemAn Equilibrium Approach(3 periods) Equilibrium assumption: For each acre in strawberries now, there will be one that was started last year and one started two years ago. We sum up resource use, and returns, over the three years.
Solution 479 acres of wheat 74 3-acre units of strawberries
Same Problem, Using Averages Here, we use the average annual return for an acre of strawberries and average annual water use. Solution: 479 acres of wheat and 221 1-acre units of berries. 221/3=74
Equilibrium with Unknown Life The strawberry example with length of activity determined in model.
Equilibrium Model Extended An equilibrium model can include a sequence of activities. Sequencing problems are discussed in section 7.5 of the text. We did not explicitly cover them in class before.
Sequencing (7.4) • Some activities must be completed before others are begun • Sequencing may be controlled within a variable or between variables using constraints • The technique used depends on the situation
Example from Chapter 7 A firm produces one output using two tasks (events). There is a 3-week production horizon, and the two events can take place any time during that period, but the second event must take place after the first event – although the two events can take place in the same week
First Case: Returns Independent of Timing X1, X2, X3: amount of activity 1 done in weeks 1, 2, 3, respectively Y1, Y2, Y3: amount of activity 2 done in weeks 1, 2, 3, respectively. For week 1: Y1 LE X1 For week 2: Y2 LE X1 + X2 – Y1 For week 3: Y3 LE X1+X2+X3-(Y1+Y2) Sequencing handled by constraints
Second Case:Returns Depend on Timing If the returns depend on timing, we must define a variable for each combination of timing possible. Example: A1B2 is the variable meaning that activity A (first activity) is done in week 1and activity B (second activity) is done in week 2.
Chapter 8 :Crop Rotation Example • Two crops: corn and soybeans • Crop yield varies with dates of planting and harvesting • Time is disaggregated into six annual periods: spring before planting, two periods during planting, two during harvest, and one in fall after harvest • Cultural operations (in order): plowing, planting, harvesting
Operations We can plow any time after harvest, including in the same time period as the harvest (pd4 or pd5). We can plant in the same period that we plow, if the plowing is done in one of the allowed planting periods (pd2 and pd3).
Technical Data The farm has 400 acres
Equilibrium Assumptions The equilibrium assumption is that this year's plowing and crop mix equals last year's. spring plow plant harvest fall plow
Recursive Programming Recursive programming models involve problems in which some model coefficients depend on earlier model solutions. The solution in period t generates data to input for period t+1. The model is optimized for each time period with the data generator updating the model at each round.
