Estimating A Demand Function For Fruit And Vegetables

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In this undertaking I will analyze the quarterly informations set for FTVG20 from Ruritania between 1981 and 2010. I will happen a functional signifier which best fits the informations and so trial for undistinguished variables, structural interruptions, seasonality and homogeneousness. I will utilize Slutsky ‘s equation to cipher the income and permutation effects and so construe the theoretical account. The societal, economic, geographic and economic features of Ruritania are non known.

Literature Review

Ruel, Minot and Smith use household outgo studies in 10 Sub-saharan African states and used a Working-Lessor functional signifier to happen that the chief determiners of demand are per capita outgo, family size, families headed by a female, instruction and location ( urban vs rural ) .

A survey by Seale considered the consequence of monetary value and income on the demand for different nutrient classs. They found that the nutrient budget portion of fruit and vegetable ingestion is 10-25 % , which is much higher than that of Ruritania. They calculated the outgo snap of fruit and veggies for low income states ( LICs ) to be 0.636, middle-income states ( MICs ) 0.514 and high-income states ( HICs ) 0.281. The Frisch own-price snap of demand was -0.514 in LICs, -0.416 in MICs and -0.227 in HICs.

There have been several surveies sing non-economic factors that contribute to the demand for fruit and veggies. A survey by Nayga found that demand depends on socio-demographic factors such as location, age, household construction, ethnicity, kids and instruction, whilst Pollard, Kirk and Cade find societal desirableness, wonts, centripetal entreaty, convenience and advertisement to be explanatory variables. Block ‘s research in Indonesia finds that female parents with nutritionary cognition spend a greater proportion of their nutrient budget on nutrients rich in foods and minerals, such as fruit and veggies.

Functional Form

Studenmund says that ‘choice of a functional signifier is a critical portion of the specification of that equation. ‘ He goes on to advert that the usage of Ordinary Least Squares means that the equation should be linear in the parametric quantities instead than variables. In finding a demand map for fruit and veggies I will see the undermentioned functional signifiers:

In finding which functional signifier is preferred and which variables are important, I will utilize the statistical trials detailed below:

Changes In Demand

Roberta Cook ‘s research has shown that per capita fruit and veggies ingestion ( lbs ) in the United States has increased by 12.4 % from 1976-2006. Interestingly, in the same period there was a 28 % decrease in the sum of citrous fruit fruits consumed but growing was boosted by non-citrus fruits and veggies. Cook suggests that the addition in demand is due to alterations in life style such as the big addition in the figure of two-income families. This has led to a focal point on cooking rapidly hence utilizing more fresh green goods.

The spread secret plan below shows the alteration in measure demanded for fruit and veggies in Ruritania over the clip period 1981 to 2010. Quantity demanded was changeless between 1980 and 1991 before increasing exponentially. The information does non follow the consequences of Cook ‘s research but I am able to foretell that the informations will suit either a log-log or log-linear theoretical account.

Choosing the Functional Form

Although the additive theoretical account and the linear-log theoretical account pass the F-test, merely 85 % and 83 % of the fluctuation in the information is explained by the several arrested development theoretical account. Both theoretical accounts besides fail the Breusch-Godfrey trial, Durbin-Watson trial, White ‘s Test, Jarque Bera trial and the Ramsey RESET trial. From these consequences I can reason that the demand map for fruit and veggies is non in additive or linear-log signifier.

The log-log functional signifier and the log-linear functional signifier both explain around 93.5 % of the informations, which is comparatively high. They both pass the T-test, Durbin-Watson trial, White ‘s Test, Jarque Bera trial, Breusch-Godfrey trial and the Ramsey RESET trial at 5 % . Although they both pass the same trials, the log-log signifier passes the Ramsey RESET trial at 0.0028123 whilst the log-linear signifier base on ballss at 1.7429. Since the log-log theoretical account passes this more satisfactorily, the theoretical account will hold a lower opportunity of misspecification. Additionally, a log-log theoretical account allows easier reading as snap is changeless and equal to i?? at every point. I will hence take the log-log functional signifier as the demand map for fruit and veggies. For analysis, if an independent variable alterations by 1 % whilst other independent variables are held changeless, so the dependent variable will alter by the i?? value of the independent variable.

Testing single parametric quantities

Having identified the preferable functional signifier, I will now prove the significance of single parametric quantities at a 5 % significance degree.

Calculated utilizing a 2-tailed T-test

H0: i??0 = 0

H1: i??0 ?0

Test statistic ( T ) = i??0 – i??0 T ( N-2 ) where N = 120 so T ( 118 )

Se ( i??0 )

If – technetium ? T ? technetium fail to reject the void hypothesis and i??0 is non important

If t ? technetium or T ? -tc reject the void hypothesis and i??0 is important

The critical values for the T – trial are +/- 1.98.

From the t-test I have found that merely four of the parametric quantities are important at a 5 % significance degree. They are: monetary value of fruit and veggies, monetary value of tea, monetary value of all other goods and the degree of income. Since the monetary value of meat and fish, intercept and monetary value of travel are close to the critical value, I will maintain these in the theoretical account. I will now run a 2nd arrested development excepting the variables: monetary value of java, monetary value of beer, monetary value of vino and monetary value of leisure, and will utilize more t-tests to find which of the parametric quantities are important. The consequences are shown in the tabular array below.

Whilst the intercept is still undistinguished, I will go on to include it in the theoretical account as taking it can make prejudice in the regression.The monetary value of meat and fish and the monetary value of travel are still undistinguished in this arrested development so I will take them from the theoretical account.

The restricted arrested development theoretical account has the functional signifier:

ln ( QFTVG20 ) = i??0 + i??2ln ( PFTVG ) + i??3ln ( PTEA ) + i??9ln ( PALLOTH ) + i??10ln ( INCOME ) + et

To guarantee the remotion of the six parametric quantities improves the theoretical account, I will run an F-test on the restricted theoretical account:

F = ( SSRR-SSUR ) /r

( SSUR ) /n-k

Where R = figure of limitations in the theoretical account, n = figure of observations, k = figure of parametric quantities in the unrestricted theoretical account ( including the intercept )

The void hypothesis is:

H0: i??1 = i??4 = i??5 = i??6 = i??7 = i??8 = 0

Hour angle: Null hypothesis is untrue

At 5 % significance degree, critical value F ( 6,109 ) = 2.18

F = ( 16.7224302 – 16.0433624 ) /6

( 16.0433624 ) /120-11

F = 0.7689409526 & A ; lt ; 2.18

Since the trial statistic is less that the critical value, I fail to reject the void hypothesis so the variables are jointly undistinguished and can now be removed.

Further Trials

I will see whether there are structural interruptions and seasonal alterations.

Structural Breaks

I have chosen to chart QFTVG20 over clip instead than lnQFTVG20 as there is a pronounced addition in fruit and veggies ingestion after 1998 which does non look on the graph for lnFTVG. This addition in ingestion may be due to a structural alteration. I will therefore split the arrested development theoretical account into two, and carry out a Chow Test, where:

H0 = no structural alteration

HA = structural alteration

Chow Test

n1 = figure of observations in the first arrested development

n2 = figure of observations in the 2nd arrested development

K = figure of parametric quantities including the invariable

SSRR = RSS from original theoretical account

SSUR = RSS from arrested development 1 + RSS from arrested development 2

F = ( 16.7224302 – 10.7905333 – 5.82816287 ) /5 = 0.1373241701

( 10.7905333 + 5.82816287 ) / ( 72 + 48 – 2×5 )

At a 5 % significance degree, the critical value is F ( 5,110 ) = 2.29

Since 0.137 & A ; lt ; 2.29 I fail to reject the void hypothesis and can reason that there is no structural alteration when tested at the 5 % significance degree.

Seasonal Dummy Variables

Since fruit and veggies grow on a seasonal footing, it is prudent to include seasonal dummy variables to see whether the information follows seasonality. To make this, I will make four silent person variables, nevertheless, I will merely include three dummy variables so as to avoid falling into the silent person variable trap. This avoids obtaining perfect multicollinearity. The three silent persons refer to the difference between themselves and the omitted ( mention ) silent person variable.

With the inclusion of three silent person variables, the theoretical account becomes:

ln ( QFTVG20 ) = i??0 + i??2ln ( PFTVG ) + i??3ln ( PTEA ) + i??9ln ( PALLOTH ) + i??10ln ( INCOME ) + i??aD1 + i??bD2 + i??cD3 + et

This shows that the silent person variables are undistinguished at 5 % significance degree. To take the silent person variables, I run an F-test to look into for the combined significance.

H0: i??a = i??b = i??c = 0

Hour angle: H0 is non true

F = ( SSRR-SSUR ) /r ~ F ( R, n-k )

( SSUR ) /n-k

F = ( 16.7224302 – 16.3332741 ) /3 = 0.8890750414

16.3332741/ ( 120 – 8 )

At 5 % significance degree, the critical value for F ( 3,112 ) is 2.68. Since 0.889 & A ; lt ; 2.68 I fail to reject the void hypothesis. From this, it can be seen that at the 5 % significance degree, there is no grounds of seasonality. I can now take the seasonal silent person variables.

Homogeneity

A demand map is homogeneous if when both monetary values and income are doubled, the optimum measures demanded do non alter.

H0: i??2 + i??3 + i??9 + i??10 = 0

Hour angle: i??2 + i??3 + i??9 + i??10 ? 0

If H0 is true, the equation can be rearranged as:

i??10 = – i??2 – i??3 – i??9

The arrested development theoretical account therefore becomes:

ln ( QFTVG20 ) = i??0 + i??2ln ( PFTVG ) + i??3ln ( PTEA ) + i??9ln ( PALLOTH ) + ( – i??2 – i??3 – i??9 ) ln ( INCOME )

From logarithmic regulations, the equation can be written as:

Ln ( QFTVG20 ) = i??0 + i??2ln ( PFTVG/INCOME ) + i??3ln ( PTEA/INCOME ) + i??9ln ( PALLOTH/INCOME )

F = ( SSRR-SSUR ) /r

( SSUR ) /n-k

F = ( 17.3810772 – 16.7224302 ) /1 = 4.529509413

16.7224302/ ( 120-5 )

The critical value for F ( 1,115 ) is 3.92. Since 4.5295 & A ; gt ; 3.92 I reject the void hypothesis and conclude that demand is non homogeneous, it exhibits heterogeneousness. Laitinen has undertaken a survey which concludes that the trial of homogeneousness is ‘seriously biased ‘ towards rejecting the void hypothesis. This leads me to believe that my consequence is acceptable and could be due to this, or the money semblance, where consumers mistake alterations in nominal values to be alterations in existent values.

Slutsky

The Slutsky equation shows how a monetary value alteration can take to an income consequence and a permutation consequence.

To cipher the monetary value snap of demand I multiply through by P/Q and multiply the last term by I/I giving:

This means:

Price snap of demand = permutation consequence – ( income snap x fraction of income spent )

From table 10 it can be seen that the income snap of demand is -0.470995 and monetary value snap of demand of fruit and veggies is -0.626791. The fraction of income spent on fruit and veggies is 3 % .

Income consequence = -0.470995 x 0.03 = -0.01412985

Substitution consequence = -0.626791 – -0.01412985 = -0.61266115

Since income snap of demand is negative, this means that fruit and veggies are inferior goods. The permutation consequence must ever be negative.

Interpretation Of The Preferred Model

Having identified that there are no structural interruptions in the theoretical account and that there is no grounds of seasonality, I can run a 3rd arrested development with all the undistinguished variables removed. The demand map is determined by:

ln ( QFTVG20 ) = i??0 + i??2ln ( PFTVG ) + i??3ln ( PTEA ) + i??9ln ( PALLOTH ) + i??10ln ( INCOME ) + et

The restricted arrested development theoretical account gives the undermentioned consequences to the aforementioned diagnostic trials:

* Significant at 5 % significance levelThe restricted log-log theoretical account base on ballss every trial carried out and passes the F trial and White ‘s Test more satisfactorily than the unrestricted log-log theoretical account.

I will now run farther t-tests and see whether the staying variables are still important. The consequences are shown in the tabular array below.

The tabular array shows that all the staying parametric quantities ( except the invariable ) are important at a 5 % significance degree.

Regression equation for the preferable theoretical account

ln ( QFTVG20 ) = 0.814700 – 0.626791ln ( PFTVG ) – 0.579563ln ( PTEA ) + 2.80783ln ( PALLOTH ) – 0.470995ln ( INCOME )

The equation suggests that fruit and veggies are inferior goods as the coefficient for income is negative. This means that as income additions, the demand for fruit and veggies lessening.

Interpretation of Elasticities

Changeless – represents the value that is predicted for the dependent variable when all the independent variables are equal to zero.

LPFTVG – A 1 % addition in monetary value will take to a 0.626791 % autumn in quantity demand of fruit and veggies. The mean own-price snap for fresh fruit from 10 surveies combined by Durham and Eales is -0.6 which is really near to the snap I have found.

LPTEA – A 1 % addition in monetary value of tea will take to a autumn in demand of FTVG20 of 0.579563 % . This could be due to fruit and tea being consumed together, for illustration, as portion of breakfast.

LPALLOTH – a 1 % addition in the monetary value of all other goods will do a 2.80783 % addition in demand for fruit and veggies

LINCOME – A 1 % addition in income means the demand for fruit and veggies will fall by 0.470995 % . From this I can reason that fruit and veggies are inferior goods. Purcell and Raunikar found that at lower incomes, fruit and veggies are normal goods but at higher incomes they are inferior goods. They besides found that green veggies are inferior goods for all degrees of income from 1958-62. Their consequences correspond to a recent survey ( 2007 ) by Ruel, Minot and Smith, who found that in 10 ( comparatively hapless ) African states the mean income-elasticity of demand for fruit and veggies was 0.766, i.e. fruit and veggies are normal goods for low-income states.

Decision

In this undertaking I have estimated a demand map for fruit and veggies ( 20 ) in Ruritania. Through utilizing diagnostic trials and arrested development analysis I have found it to be a log-log theoretical account. I was able to take undistinguished variables go forthing independent variables of monetary value of fruit and veggies, tea, all other goods and income. I so tested the information for seasonality and structural interruptions and found no grounds of seasonality or structural interruptions between 1981 and 2010. I found the informations to be heterogenous and justified this with mention to Laitinen ‘s research. Using Slutsky ‘s equation, I found that fruit and veggies are inferior goods.

To better the theoretical account I could divide the demand for fruit and veggies to see whether they both remain inferior goods. It would besides be interesting to see socioeconomic factors, such as those studied by Nayga. Additionally, since a big proportion of demand for fruit is made up of the demand for juice, it would useful to see the demand of whole fruit and veggies instead than that pressed into juice. These factors combined may better the theoretical account so that a proportion of the staying 6.6 % of the information tantrums my arrested development theoretical account.