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 2008 
2 

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 land across various uses change in response to 

policies? The GTAP modeling framework of Hertel (1997) addresses this question by 

determining the supply of land across different uses through a Constant Elasticity of 

Transformation (CET) supply function. In the standard GTAP model, the only type of land 

explicitly modeled is agricultural land, and this is distributed across uses with a one-level, CET 

function. However, in the GTAP-AEZ framework, data are available on a wider range of 

activities, including forestry (Sohngen et al., forthcoming). Given the recent availability of data 

on land cover (Ramankutty et al., 2007) and harvested cropland (Monfreda et al., forthcoming) 

the land supply decision is more naturally divided into the allocation of land cover across 

forestry, grazing and crops, followed by the allocation of harvested 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 the 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 memorandum describes the empirically based calibration 

strategy used to determine the value for the CET parameter, based on recent research in the 

United States. 

The following section reviews the theory of the CET parameter 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 determined, 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 case 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 are 
produced would be the three types of land cover – forest, pasture, and cropland. Represented 

algebraically, this function would be as follows: 

3 

i)

⎥

Q =f Q X Y ( (1) 

QX is the quantity of a single input x (total land), while QYi refers to the vector of land used in a 

specific use i (i:{1,2...N}). We conceptualize the land supply 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 guaranteed, 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 iYi

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 than one, which implies that the 

transformation elasticity itself must be negative1

. 

3. Methodology 

The key concept in the methodology used is the own return elasticities of land quantity, for each 

use (εj,t+5). This is the estimated own return elasticity of land use j, determined as the percentage 

difference of the estimated perturbed land use quantity and the benchmark land use quantity for a 

given five year period. 

However, two factors must be considered when 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 – increases, then the quantity of land supplied for that 

use will increase. The second factor to consider is that there are 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 quantity takes into account 

the two factors described above. The first is the matrix of land use transition probabilities2

 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 own return elasticities of transition probability. 

Following Lubowski (2002) these two sets of data 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 probabilities 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 land quantities and prices. The revenue generated by 

aggregate land endowment (RX) is the product of an average land price per unit and total land 

quantity. RX must be equal to the sum of the revenues generated by the individual uses of land 

(RYi). These revenues are in turn products of the prices commanded by land uses j (Pj) and the 

land quantities (QYj). The revenue share for a given use, θ j, is now Pj QYj/PXQX, where QX is total 

land supply. 

Now, the compensated (input constant) price elasticities (εij) 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 determine 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 ≠

∑

=-σ( )i 1-θ

4 

( )

ii

i

ε

σ =-

1-θ

⇒ (6) 
Equation 3 thus provides an estimate of the CET parameter value associated with use j in the five 

year period considered. The revenue share weighted 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 paths of land quantities by use 

over a 100 year period. As mentioned earlier, these projections are 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 that instead of using national level data 

for the USA, the estimates are based on plot level 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 time 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 differences in land use acreages over time, between the baseline path and 

the path where own-returns for each use are perturbed by one percent. 

5 
Figure 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 the GTAP Land Use Database. The average CET 

parameter in a given period is thus based on elasticities 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-returns rises throughout the 

timeframe. Accordingly, the own-return elasticities also rise over 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 medium 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 interpolation using the five-year, ten-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 desired 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 Inventory utilized by Lubowski (2002) in his 

econometric estimation. 

6 
Figure 2: Own-Return Elasticities 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 

7 
Figure 4: Comparison of Estimated CET Values with Linearly Interpolated 

Approximations of CET at time t

Source: Authors’ Simulations 

8 
9 

REFERENCES 

Hertel, Thomas W. (ed) (1997), Global Trade Analysis: Modelling and Applications. Cambridge: 

Cambridge University Press. 

Lee, Huey Lin, Thomas W. Hertel, Steven Rose, and Misak Avetisyan (forthcoming) “An 

Integrated Global Land Use Data Base for CGE Analysis of Climate Policy Options”, in 

Hertel T.W., Rose, S., and Tol, R. (eds.) Economic Analysis of Land Use in Global 

Climate Change Policy, Routledge. 

Lubowski, Ruben (2002) “Determinants of Land Use Transitions in the United States: 

Econometrics Analysis of Changes among the Major Land-Use Categories”. PhD 

Dissertation, Harvard University: Cambridge, MA. 

Lubowski, Ruben, Andrew J. Plantinga, and Robert N. Stavins (2006) “Land-Use Changes and 

Carbon Sinks: Econometric Estimation of the Carbon Sequestration Supply Function”, 

Journal of Environmental Economics and Management, 51(2006) 

Monfreda, Chad, Navin Ramankutty, and Thomas W. Hertel (forthcoming) “Global Agricultural 

Land Use Data for Climate Change Analysis”, in Hertel T.W., Rose, S., and Tol, R. (eds.) 

Economic Analysis of Land Use in Global Climate Change Policy, Routledge. 

Powell, Alan A. and F. H. G. Gruen (1968) “The Constant Elasticity of Transformation 

Production Frontier and Linear Supply System”. International Economic Review, Vol. 9, 

No. 3 (Oct., 1968), pp. 315-328 

Ramankutty, Navin, Amato T. Evan, Chad Monfreda, and Jonathan A. Foley (2007) “Farming 

the Planet Part 1: The 1 Geographic Distribution of Global Agricultural Lands in the Year 

2000”. Global Biogeochemical Cycles, Feb 2007. 

Shumway, C. Richard and Alan A. Powell (1984) “A Critique of the Constant Elasticity of 

Transformation (CET) Linear Supply System”. Western Journal of Agricultural 

Economics. Vol. 9, No. 2, pp 314-321. 

Sohngen, Brent, Colleen Tennity, Marc Hyntka, and Karl Meeusen (forthcoming) “Global 

Forestry Data for the Economic Modeling of Land Use”, in Hertel T.W., Rose, S., and 

Tol, R. (eds.) Economic Analysis of Land Use in Global Climate Change Policy, 

Routledge. 

U.S. Department of Agriculture (2000) Summary Report: 1997 National Resources Inventory 

(revised December 2000), Natural Resources Conservation