## Calibration of a Land Cover Supply Function Using Transition Probabilities

By

S. Amer Ahmed Thomas W. Hertel and Ruben Lubowski



## GTAP Research Memorandum No. 14

October 20082 Calibration of Land Use Supply Function Using Transition Probabilities
By
S. Amer Ahmed, Thomas W. Hertel, and Ruben Lubowski

1. Introduction
An important question that arises frequently in the economic analysis of environmental and energy policies is, how does the supply of la nd across various uses change in response to policies? The GTAP modeling framework of Hertel (1997) addre sses this question by determining the supply of land across different uses through a Constant Elasticity of Transformation (CET) supply function. In the st andard GTAP model, the only type of land explicitly modeled is agricultural land, and this is distributed across us es with a one-level, CET function. However, in the GTAP-AEZ framewor k, data are available on a wider range of activities, including forestry (Sohnge n et al., forthcoming). Given the recent availability of data on land cover (Ramankutty et al., 2007) and harves ted cropland (Monfreda et al., forthcoming) the land supply decision is more naturally divi ded into the allocation of land cover across forestry, grazing and crops, followed by the allocation of harveste d area across cropping activities.
In this Research Memorandum, we focus on the former problem, namely the allocation of land cover between these three competing commercial uses. Naturally, the quality of the land cover responses produced by GTAP-based simulations is contingent on th e value of the CET parameter. And the value of this parameter is likely to depend on the length of run for the analysis in question. This research memorandu m describes the empirically based calibration strategy used to determine the value for the C ET parameter, based on recent research in the United States.
The following section reviews th e theory of the CET paramete r and how it is relevant for modeling land supply. Sections 3 and 4 describe the calibration methodology and data, while the final section discusses the calibration results.
2. CET Supply Function
Before getting into how the CET parameter is de termined, it is worth understanding what needs to go into a land supply function and the role of the CET functional form here. In a production context, we need a supply function that allows us determine multiple outputs from a set of inputs, or a single input. This latter cas e is the case of land supply, where there is a single input that supplies land specific to different uses. In the GTAP-BIO model, the multiple outputs that areproduced would be the three types of land cover – forest, pasture, and cropland. Represented algebraically, this function would be as follows:

3 

## ) i

⎥ 

## ( X Y Q =f Q (1)

Q X is the quantity of a single input x (total land), while Q Yi refers to the vector of land used in a specific use i (i:{1,2...N}). We conceptualize the land s upply by use problem as one of revenue maximization, in which the owner allocates land to different uses in order maximize the total value of rents. For this to be gua ranteed, the input requirement function f must be strictly convex in Yi . The CET form adopted in equation 2, gives us the necessary convexity (Powell and Gruen, 1968).

1/ ρ ρ X iY i i Q = α β Q ⎡ ⎤ ⎢ ⎣ ⎦



## ∑ (2)

The exponent ρ is used to determine the elasticity of transformation, and they are algebraically related through equation 3:

1

= 1- ρ σ < 0 (3)

A property of the CET form is that ρ must be strictly greater th an one, which implies that the transformation elasticity itself must be negative 1

. 3. Methodology
The key concept in the methodology used is the ow n return elasticities of land quantity, for each use ( ε j,t+5 ). This is the estimated own return elasticity of land use j , determined as the percentage difference of the estimated perturbed land use quan tity and the benchmark land use quantity for a given five year period.
However, two factors must be considered wh en studying land use changes in response to a policy. The first is that land uses respond to the market. If the net return per unit of land in a given use – i.e. the profitability of that use – in creases, then the quantity of land supplied for that use will increase. The second factor to consider is that there ar e secular transitions in land use over time, which are independent of changes in the profitability of a particular land use.

The method used for the estimation of own return elasticities of land qua ntity takes into account the two factors described above. The first is the matrix of la nd use transition probabilities 2 for a

1 For more details on the CET, please see Shumway and Powell (1984).

2 This is the probability that a unit of land will remain in its current use or move to an alternative use.five year period and the second is the set of ow n return elasticities of transition probability. Following Lubowski (2002) these two sets of da ta can then be used to simulate land use transitions over a baseline time path where there are no changes to use returns, and then a time path of land use transitions where the transition pr obabilities in each period changed due to a one percent increase in returns to each use. With a time series of land use acreages in a base case, and a time series of land use acreages with the own returns to a use increasing by a percent in each period, the percent difference between land quantities by use in each path will yield the requisite own return elasticities of land quantity.
Now the relationship between the own return elasticities of land use the CET must be determined. Let us return to the changes in la nd quantities and prices. The revenue generated by aggregate land endowment ( R X ) is the product of an average land price pe r unit and total land quantity. R X must be equal to the sum of the revenue s generated by the indi vidual uses of land ( R Yi ). These revenues are in turn products of the prices commanded by land uses j ( P j ) and the land quantities ( Q Yj ). The revenue share for a given use, θ j , is now P j Q Yj /P X Q X , where Q X is total land supply.
Now, the compensated (input constant) price elasticities ( ε ij ) can be written in terms of revenue shares and the CET elasticity, such that:

ij j ε = θ * σ (4)

Homogeneity of supply implies that:

ii ij j i ε =- ε

≠ 

# ∑ (5)

From equations 4 and 5, it now possible to de termine a relationship between the own price elasticity and the σ :

ii ij j i ε =- ε

≠ 

# ∑ 

# ( ) j j i =- θ * σ

≠ 

## ∑

j

j i =- σ θ

≠ 

# ∑

( ) i =- σ 1- θ

4 

## ) ( ii

i

ε σ =- 1- θ ⇒ (6)Equation 3 thus provides an estimate of th e CET parameter value associated with use j in the five year period considered. The revenue share weight ed average of the CET values for each use gives us the average CET value (Equation 7) over all the uses.

ii,t+5 t+5 i i i ε σ =- θ * 1- θ ⎛ ⎞ ⎜ ⎝ ⎠ 

## ∑ ⎟ (7)

4. Data

There are two sets of data for the calibration. The first set is the revenue shares are from the GTAP Land Use Database by Lee, Hertel, Rose, and Avetisyan ( forthcoming ), and are 0.7489, 0.0975, and 0.1023 for the revenue shares of cropland, pasture, and forest in the United States, respectively. The second set is the baseline and perturbed time pa ths of land quantities by use over a 100 year period. As menti oned earlier, these projections ar e based on matrices of land use transition probabilities and own return elasticities of transition probabilities. The transition probabilities matrix is from Lubowski (2002), while the own return elasticities of probabilities are estimated following a methodology similar to Lubowski et al (2006), but with a critical innovation.
The methodological innovation of Lubowski (2002) is th at instead of using national level data for the USA, the estimates are based on plot le vel data from USDA (2000). This approach tracks the amount of land of each quality in each use at each point in time, and ultimately an elasticity estimate that is more appropriate for a longer tim e frame. Thus, at each five-year step, the overall amount of land transitioning between all the uses changes based on the new amounts of land of different qualities allocated to each of the different uses, in both the baseline time path, as well as the time path where own returns to use are perturbed by one percent in every period. Figure 1 illustrates the di fferences in land use acreages over time, between the baseline path and the path where own-returns for each use are perturbed by one percent.

5Figure 1: Estimated Land Use Quantities at t : Difference Between Perturbed and Baseline Paths Source: Authors’ Simulations
These land use quantity time paths are determined for crops, forest, and pasture, which are the same three categories of land cover in th e GTAP Land Use Database. The average CET parameter in a given period is thus based on elas ticities calculated from projected changes in acreages by use, followed by taking their revenue share weighted averaged across the three land cover uses in the GTAP land data.

5. Calibration Results
As was seen in Figure 1, the land use response to increases in own-retu rns rises throughout the timeframe. Accordingly, the own-re turn elasticities also rise ove r this period (Figure 2), while the calibrated CET parameter falls (Figure 3) . The revenue share-weighted CET parameter decreases by at a decreasing rate every five-year period, indicative of an asymptotic relationship in the long run. However, in the short to me dium run, this can be approximated by a linear relationship. Figure 4 illustrates how the estimated CET values compare to the CET values that would have been produced by a linear interpolati on using the five-year, te n-year, and fifteen-year CET values generated by the calibration method. We can use this linear interpolation to estimate the average CET parameter associated with any desi red length of run. This parameter is uses in the GTAP model to govern the mobility of land between cropland cover, pastureland and accessible forest. In the US, the latter category is assumed to correspond to the privately held land sampled in the National Resources Inve ntory utilized by Lubowski (2002) in his econometric estimation.

6Figure 2: Own-Return Elas ticities of Land USe at t for Use i Source: Authors’ Simulations

Figure 3: CET Calibration Estimates by Land Use at time t , for t=5 to t=100 Source: Authors’ Simulations

7Figure 4: Comparison of Estimated CET Values with Linearly Interpolated Approximations of CET at time t Source: Authors’ Simulations

89 REFERENCES
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