Economic Models with Parameters - Case Study Example

ECON3002- Economic Modeling

ECON3002- Economic Modeling

SECTION I

• OLS Assumptions on Regression Model Yt=a+bxt+et

OLS assumption states that the regression model should be linear in parameters, wherein the errors need to be statistically independent from each other and independent variable’s measurement needs to be done precisely to generate reliable result of the analysis (Kendall, 2007). The regression model Yt=a+bxt+et on OSL assumptions include implicit independent variables in which missing X variables would not be appropriate. The existence of implicit block effect has also caused more dependency on the Y variable. Lack independence of the Y value has made the estimation of analysis unreliable. Due to the presence of outliers, the values obtained are not identical. Hence, outliers increase the residual variances, which reduce the chance of discarding null hypothesis. The non-normality is hard to detect in the model, which adds violation to OLS assumptions. Pronouncement of heteroscedasticity has been missing in the model. Therefore, until heteroscedasticity of Y is not asserted the effectiveness would not be severe (Iowa State University Department of Economics, 2004).

The assumptions have provided misleading results of the analysis. The violated assumptions of the independent variables have resulted in the inappropriate linear regression. Violation of the normality has given the inefficient fit test of the linear regression (Iowa State University Department of Economics, 2004).

• Appropriate Model for Describing the Relationship between Yt And Xt

The general linear model is a simple and elegant model that concentrates on identifying the relationship between Yt and Xt. It is a procedure through which variable y is characterized by combination of variables x1, x2…..xp.

Or,

The general linear model is comprehensive in nature, which assists in interpreting in easier manner. The linear model y=xb helps in separating the error distribution from any kind of effect. The model is tractable due to the availability of the closed form parameters closed form estimates in which the variance and mean can be independently estimated. The interpretation of the estimations and test provides the similar results of the analysis with this model (Wickens, 2004).

• Computation of the elasticity coefficient of Yt and Xt

The following table (Table 1) illustrates the data based on which the elasticity coefficient of Yt and Xt has been calculated as shown in the Table 2:

Table 1: Data set for Calculating correlation between Xt and Yt

Table 2: Coefficient Calculation

Form the coefficient calculation, it is evident that there is a positive correlation between Xt and Yt, as the correlation value has been obtained 0.74, which is >0 and <1.

Section II

• Estimate the Cobb-Douglas Production Function Y ̂T=A (T) Kt^A ̂ Lt^B ̂ through Gretl Software

Figure 1: Calculation of the Cobb Douglas Production Function using Gretl Software

The possible technology constant growth rate has been obtained as 0.98.

Figure 2: Calculation of Constant Technology Growth Rate

• Plotting Isoquant curves on the (Kt, Lt) space for Period 30

The following diagram depicts the plotting of isoquant on two variables namely on Kt and Lt:

Figure 3: Plotting of Isoquant on (Kt, Lt)

• Solving Output Maximization Problem

The following table provides a clear understanding on the output of cost function based on the calculation provided:

Table 3: Calculation of Cost function

• Output Maximization Problem and the Dual Mirror Image Representation

The total variable cost function has a close relationship with the production function. The production function has the underlying dual cost function, which is expressed in physical terms and inversed to the production function. The cost function is identified as a mirror image of the fundamental production function. For instance, in case the production function increases at increasing rate, then the cost function will increase in terms of decreasing rate. Therefore, it is important to consider physical quantity of input for determining the co-efficient of the production function (University of Kentucky, 2003).

• Estimation of Restricted Cobb-Douglas Production Function and its Statistical Test

Figure 4: Restricted Production Function

Figure 5: Statistical Test

Section III

• Expected Shape For The Short Run Curves For The Total Function Cost, Average Function Cost And Marginal Cost

The short run cost curve STC(Q) helps in understanding the total cost production of Q units in which one unit is constant at certain level. For instance, if the amount of capital invested by the firm is constant at K, the short run total cost curve is actually the sum total of total fixed cost and the total variable cost curve. Therefore, STC (Q) = TFC (Q) + TVC (Q). The total variable cost comprises the cost of variable inputs, such as labor and other materials in the short run. Again, as the total fixed cost is equal to the expenses of the fixed capital services, TFC= rK, which does not change with the shifts in the output. Therefore, because total fixed cost does not have any impact with the output, it can be ascertained as independent of output.

Figure-6 Cost Function Costs

Source: (University of Porto, 2004)

The short run marginal cost (SMC) or SAC(Q) = [= [STC(Q)]/Q and SMC(Q) = (STC)/(Q). As can be inferred thus, the short run marginal cost is equal to the slope of the short run total cost curve. The figure below gives an illustration that the short average cost is obtained by adding the vertical average variable cost and the average fixed cost. Therefore, increase in the output (Q) decreases the average fixed cost. This implies that the rise in the output leads the fixed cost to decline even to zero, as average fixed cost keeps on decreasing whenever Q increases.

Figure-7 Marginal and Average Cost Curves

Source: (University of Porto, 2004)

• Regression Models For Estimating Total Cost, Average Cost And Marginal Cost

For estimating the short run cost function, the total fixed cost has to be identified. Total average cost function can be obtained by adding all the components of the fixed cost. For example, the real data of variable cost and the output has been obtained. The further step would be estimating the parameters of the functional form of the multiple regression technique to interpret the data. With the multiple regression technique, if the association between the cost and the output is linear, the function form will be TVC=b0+ b1Q. If b0 an b1 are properly estimated with the regression further, the average variable function will be AVC=b0/Q+b1 and the marginal cost function will be MC=b1. TVC, AVC and MC curves are consistent with this regression model, which is exhibited in the figure below (Petersen, Lewis, & Jain, 2006).

Figure-8 Total Variable, Marginal And The Average Cost Function

Source: (Petersen, Lewis, & Jain, 2006)

• Summary Of The Empirical Research

After the regression analysis, if the empirical data shows a U-shaped average cost curve, the linear function would fail to establish the relationship between cost and the output. The quadratic total variable function would be TVC = c0+ c1Q+ C2 Q2 and the cubic cost function would be TVC = d0 +d1Q+ d2 Q2+ d3 Q3.

These functions are forms of the hypothesized nonlinear association. Therefore, the shape of the cubic and the quadratic functions would depend on the estimated parameters using the least square regression analysis. The total cost increases at a growing rate, marginal cost is the function of the output and the average cost is calculated by total variable cost function by Q (Petersen, Lewis, & Jain, 2006).

The relationship between cost and output is understood with the help of cost curves. The shape of the cost curves is based on the nature of the cost function. The different cost function forms different types of cost curves. Therefore change in the level of the output effects the different curves of the cost function (Dwivedi, 2015).

SECTION IV

• Summary Statistics and Graphs for the Preliminary Data Analysis of Graddy (2006) Data

Figure 9: Summary Statistics Using Gretl

Figure 10: Graphs of Price and Quantity Analysis

Figure 11: Graphical Representation Using Other Variables

• Gretl Command Log Report and Gretl output for estimations

Figure 12: Output for Estimation

Figure 13: Gretl Command Log Report

• Strategy and Steps for Estimating the Graddy (2006) Model with a 2SLS Approach and the Role of Variable Stormy in the Estimation Process

The Grady model (2006) utilizes 2 SLS approach or Instrumental variable (IV) to determine supply and demand of the fish. IV has identified the weighted average of derived behavioral association of interest, wherein 2SLS approach has also contributed in formulating the important assumptions of demand and supply based on different prices. The essential strategy in the IV estimation is to search for a variable, which can generate exogenous variation in the interest interpreter. Thus, the application of IV helps in addressing the issues of demand and supply of the fish (Crosby, Dowsett, Gennetian & Huston, 2012).

There are two stages of 2SLS or the IV estimates. In the first stage, endogenous predictor regresses on the instrument to obtain the coefficients of price and fish quantity and in the second stage the actual values of all the variables get replaced with the expected values to adjust the standard errors of the analysis (Crosby, et al., 2012). The stormy variable is an important factor in the price fluctuation of fish market, which reduces the quantity and increases the price. The effect of stormy variable creates an impact on the supply, which results in quantity decrease and price increase of fish (Graddy, 2006).

• Comparing OLS and IV demand estimates

OLS is a type of analytical tool, which is used to generate impartial linear estimates. Hence, if the assumptions in OLS regression are violated, then the outcome of the analysis would also be biased. This is the common method applied in the Graddy (2006) model to observe the characteristic of variables available in the model. Instrumental variable (IV) is utilized to remove the endogeneity bias in the analysis. IV analysis is derived only from the variation attributable to exogenous instrument. IV helps in generating the consistent parameters estimates of the analysis, whereas OLS estimates lack consistency due to the endogeneity’s presence (Crosby, et al., 2012).

• Graddy (2006) Table 2 Analysis

Table 2 of the Graddy (2006) represents the estimations related to the elasticity of quantity demand and price. With the help of the OLS and IV as instrument, fish has been treated as a homogenous product. The first column of the table (Figure 12) shows the OLS with the log quantity as an independent variable and the second column shows the unchanged estimations with dummy variables to measure the weather on shore. In accordance, third column illustrates log price as dependent variable and the stormy weather as descriptive variable, whereas the fourth column descries about the pricing power of fish dealer in the market (Graddy, 2006).

Figure 14: Ordinary Least Squares and Instrumental Variable

Source: (Graddy, 2006)

• Demand for fish on a Monday with IV equation estimated change in Tuesday

The demand of fish on Monday has been calculated by using Gretl and provided in the below diagram:

Figure 15: Estimation of Demand on Monday