COMSOL and ANSYS Models

Based on the foregoing discussion and building on other empirical studies, we develop a crop production function in the specific context of Nigeria and data availability. Traditionally, empirical studies have estimated the relationship between agricultural output and land, labour and capital inputs. However, several other factors also affect agricultural products, such as weather, agronomic constraints, agricultural practices and farm characteristics. Following Blanc (2011) and in line with Frisvod and Ingram (1995), we specify anempirical agricultural output function of the form:

AGQ_t = f (LAB_t, IRGA_t, FIMV_t ARDL_t, CO2_t AMTP_t, AMRF_t,) 3.2 Where AGQt is the aggregate agricultural output time t and the inputs LABt, IRGA, FIMV and

ARLDt, are labor, irrigation,value of food importation and arable land respectively. More importantly we includecarbon emission (C02), annual temperature (AMTP) and annual rainfall (AMRF) as climatic factors that may affect agricultural production. Few empirical analyses onagricultural production functions consider acreage as an explanatory variable (e.g. Chen et al.,

2004). Most empirical studies on crop yield analyses consider experimental data where land expansion is not applicable, this does not pose an omitted variable bias in this studies. However, as this study employs national data reflecting actual cropping decisions, decreasing marginal productivity of land needs to be considered.

The Model in its stochastic form is expressed in the form:

AGQt= LAB^β1IRGA^β2FIMV^β3ARLD^β4CO2^β5AMTP^β6AMRF^β7U 3.3 Taking the log of equation (3.3) transforms the equation into linear form.

LAGQt=β0+β1LLABt+β2LIRGAt + β3LFIMV +β4LARDLt

+ β5CO2t+β6LAMTPt+β7LAMRFt + Ut 3.4

The log transformation of the equation is taken in order to standardize the value of the variables, achieve linearity as well as allow for the easy interpretation of their coefficients as elasticity.

The sum of exponents β₁………β₇, represents the return to scale. Each of the parameters β₁…………β₇ can be interpreted as a partial elasticity of the output with respect to the input in the production function. The U is the error term.

The above model is specified on the assumption that the impact of climate change on agricultural output is the same across products. But in reality, different agricultural outputs respond to climate change differently. This suggests that the responses of crop, livestock, fish and forestry to climate change in Nigeria may not be symmetry. To account for this, we specify alternative models for crop production, livestock, fishery and forestry. Specifically, we look at these variables as solely dependent on climate change. Two things informed our approach here: (1) we want to explicitly assess the impact of climate change on different agricultural outputs in Nigeria (2) and also investigate the agricultural output that is more prone to climate change. Thus from the baseline model specified in equation (3.4) we derive our agricultural product – specific model as follows:

LAGQit=β0+ XΠ +β1LCO2it+ β2LAMTPit+β3LAMRFit+ Ɛt 3.5

Where LAGQit is (n x 1) vector of explained variable i at time t., specifically, LAGQit includes:

crop production (LAQCY), livestock production (LAQLV), fishery (LAQFH) and forestry

(LAQFS). X is the matrix of other important explanatory variables in a particular function, and Π is the matrix of coefficients and ε is the error term. Thus, the following equations:

LAQCYt=β10+β11ARLDt+β12LIRGt+ β13LCO2t+β4LAMTPt+β14LAMRFt+Ɛ1t 3.5a LAQLVt=β20+β21LCO2t+β22LAMTPt+β23LAMRFt+Ɛ2t 3.5b

LAQFHt=β30+β31LCO2t+β32LAMTPt+β33LAMRFt+Ɛ3t 3.5c LAQFRt=β40+β41LCO2t+β42LAMTPt+β43LAMRFt+Ɛ4t 3.5d

Where the β‟s are the parameters and Ɛ1t, Ɛ2t, Ɛ3t and Ɛ4tare uncorrelated error terms. We added arable land and irrigation as explanatory variables in crop production equation because they are important explanatory variables in crop production. Other functions are expressed only in terms of climate change.

A priori expectation

Variable Equ (3.4) Equ (3.5a) Equ (3.5b) Equ (3.5c) Equ (3.5d) LLAB > 0 (+)

LIRGA > 0 (+) > 0 (+) LFIMV < 0 (-)

LARLD > 0 (+) > 0 (+)

LCo2 < 0 (-) < 0 (-) < 0 (-) < 0 (-) > 0 (+) LAMTP > 0 (+) > 0 (+) > 0 (+) > 0 (+) > 0 (+)

LAMRF > 0 (+) > 0 (+) > 0 (+) > 0 (+) > 0 (+) Source: researcher’s compilation

Model Justification

Our models are estimated in both static and dynamic forms. For instance, in examining the long run impact of climate change on agricultural output, we estimated a static OLS model, while a dynamic OLS model estimated in fixed effect is used to assess the short run impact of climate change on agricultural output. The static OLS approach is necessary on the assumption that the effect of climate change on agricultural products at farm level is instantaneous. On the other hand, when we relax the above assumption, we estimated dynamic models in VAR framework.

The choice of dynamic Models is based on the fact that they portray the time path of the dependent variable in relation to the current and past values of the independent variables. This is

necessary if we characterise the responses of agricultural products to climate change over time.

We also derive and estimate impulse response models of equations (3.4); (3.5a); (3.5b); (3.5c) and (3.5d). The impulse response functions(IRFs) follow the vector autoregressive (VAR) framework and have been variously used to characterise the response of one variable to an unexpected shock in another variable.Nordhaus and David (1997) is of the opinion that climate change is abrupt or unexpected, thus occasions shock. IRFs have been efficient in capturing the response of one variable to a unit shock in another variable. For instance, Bayoumi and Eichengreen (1994) use a VAR modelto analyse nominal and real shocks under different exchange rate regimes, also Hoffmann (2003) opines thatvector autoregression (VAR) approach can be utilised to test whether economies respond differently to shocks.

Variable Identification/Measurement

Aggregate agricultural output (AGQ) is measured as the total value of all agricultural product produced in a given year measured in constant market price in billions of naira.

Crop production (AQCY) is measured as the total value of all food crop produced in a year measured in constant market price in billions of naira.

Livestock production (AQLV) is the market value (in constant price) of all livestock produced in a given year measured in billions of naira.

Fishery (AQFH) is defined in similar manner as livestock production.

Forestry (AQFR) is the market value of all timbers produced in a given year measured in billions of naira

Arable land (ARLD) is defined as land under temporary crops (double – cropped are counted once), temporary meadows for moving or for pasture, land under market or kitchen gardens, and land temporary fallowed. It is measured in hectares.

Irrigation (IRGA) is a measure as the proportion of total arable land artificially provided with water for agricultural purposes. It is percentage of total arable land purposely provided with water, including land irrigated by controlled flooding.

Labour (LAB) is made up of employees who offer their services to the employers in the agricultural sector for the purpose of earning wages or other sources of income as their reward.

Annual temperature (AMTP) is measured as the degree of hotness or coldness. It can equally be seen as the intensity of heat present in a substance or object especially as expressed according to a comparative scale. Based on the subject on examination, the researcher will adopt the annual mean temperature (⁰C) which implies change in the state of the climate that can be identified by changes in the mean temperature that persists for an extended period, typically decades or longer (IPCC, 2007).

Annual rainfall (AMRF) is a measured as the quantity of rain fall within a given area at a given time. It provides suitable condition for much type of agricultural products. Based on the subject on examination, the researcher will adopt the annual mean rainfall (mm) which implies change in the state of the climate that can be identified by changes in the mean rainfall that persists for an extended period, typically decades or longer (IPCC, 2007).

Value of food importation (FIMV) is the total value of food imported into the country from overseas measured in billions of naira.

Carbon emission (CO2) is measured in mole fraction scale.