Accuracy of Numerical Solution to Dynamic Programming Models

Dynamic programming models with continuous state and control variables are solved approximately using numerical methods in most applications. We develop a method for measuring the accuracy of numerical solution of stochastic dynamic programming models. Using this method, we compare the accuracy of various interpolation schemes. As expected, the results show that the accuracy improves as number of nodes is increased. Comparison of Chebyshev and linear spline indicates that the linear spline may give higher maximum absolute error than Chebyshev, however, the overall performance of spline interpolation is better than Chebyshev interpolation for non-smooth functions. Two-stage grid search method of optimization is developed and examined with accuracy analysis. The results show that this method is more efficient and accurate. Accuracy is also examined by allocating a different number of nodes for each dimension. The results show that a change in node configuration may yield a more efficient and accurate solution.Research Methods/ Statistical Methods,

Households’ Willingness to Pay for Improved Tap Water Services in Karachi, Pakistan

Access to safe water is a basic need for human survival and health. Water is one of the most important commodities for households, who use it for drinking and many other important household activities including cooking a meal, washing dishes, bathing, laundry, cleaning, and watering the home gardens. Households need safe water and its availability on regular basis. Irregular and uncertain access to safe water affects not only these activities directly but also households’ health and workforce productivity indirectly. Thus, households give great importance to water, its quality and regular supply

Impact of Nature-based Tourism on Earnings of Local People: Evidence from Keenjhar Lake in Pakistan

Nature-based tourism is the fastest growing part of tourism [Kuenzi and McNeely (2008)]. Wetland areas including lakes are an important source of nature-based tourism as tourists like their scenic views and enjoy doing different activities including swimming, canoeing, diving and learning about nature [van der Duim and Henkens (2007)]. Wetlands are amongst the most important ecosystems on Earth and provide numerous goods and services including recreational services [Mitsch and Gosselink (2007)]. Increasing demand for nature-based tourism has raised the importance of wetland

Measuring Production Function and Technical Efficiency of Onion, Tomato, and Chillies Farms in Sindh, Pakistan

Pakistan is blessed with vast agricultural resources on account of its fertile land, well-irrigated plains, huge irrigation system and infrastructure, variety of weathers, and centuries old experiences of farming. Agriculture is the single largest sector of the economy which contributes 20.9 percent in GDP and employees 43.4 percent of total work force. The estimated GDP of agricultural crops at current factor cost is Rs 1,608,522 million with major crops contributing Rs 579996 million and minor crops valued at Rs 191,835 million for the year 2006-07 [Pakistan (2007)]. The horticulture crops (fruits, vegetables and condiments) alone contribute Rs 116.645 billion, equivalent to US$2 billion, which is 26 percent of the total value of all crops and 81.8 percent of the total value of minor crops. Pakistan annually produces about 12.0 million tons of fruits and vegetables. Fruit and vegetable export trade in Pakistan amounts to US$ 134 million (2003-04), of which fruits account for US$102.7 million (76.6 percent), vegetables US$ 25.7 million (19.2 percent) and fruit and vegetable preparations (mostly juices) US$ 5.6 million which is 4.2 percent [Pakistan (2004)]

Testing Onion Market Integration in Pakistan

This paper analyses spatial market integration using monthly wholesale real price of onion in four regional markets located in each of the four provinces of Pakistan. Unit root test indicates that the price series in each location are stationary, and the series are represented as autoregressive model for eachlocation. The error correction model results show that the regional markets of onion have strong price linkages, and thus are spatially integrated.

Testing Onion Market Integration in Pakistan

Spatial market integration of agricultural products has been widely used to indicate overall market performance [Faminow and Benson (1990)]. In spatially integrated markets, competition among arbitragers will ensure that a unique equilibrium is achieved where local prices in regional markets differ by no more than transportation and transaction costs. Information of spatial market integration, thus, provides indication of competitiveness, the effectiveness of arbitrage, and the efficiency of pricing [Sexton, et al. (1991)]. If price changes in one market are fully reflected in alternative market, these markets are said to be spatially integrated [Goodwin and Schroeder (1991)]. Prices in spatially integrated markets are determined simultaneously in various locations, and information of any change in price in one market is transmitted to other markets [Gonzalez-Rivera and Helfand (2001)]. Markets that are not integrated may convey inaccurate price signal that might distort producers marketing decisions and contribute to inefficient product movement [Goodwin and Schroeder (1991)], and traders may exploit the market and benefit at the cost of producers and consumers. In more integrated markets, farmers specialise in production activities in which they are comparatively proficient, consumers pay lower prices for purchased goods, and society is better able to reap increasing returns from technological innovations and economies of scale [Vollrath (2003)]

Accuracy of Numerical Solution to Dynamic Programming Models

Dynamic programming models with continuous state and control variables are solved approximately using numerical methods in most applications. We develop a method for measuring the accuracy of numerical solution of stochastic dynamic programming models. Using this method, we compare the accuracy of various interpolation schemes. As expected, the results show that the accuracy improves as number of nodes is increased. Comparison of Chebyshev and linear spline indicates that the linear spline may give higher maximum absolute error than Chebyshev, however, the overall performance of spline interpolation is better than Chebyshev interpolation for non-smooth functions. Two-stage grid search method of optimization is developed and examined with accuracy analysis. The results show that this method is more efficient and accurate. Accuracy is also examined by allocating a different number of nodes for each dimension. The results show that a change in node configuration may yield a more efficient and accurate solution

Measuring Spatial Integration in Tomato and Onion Markets of Pakistan: An Application of Error Correction Model in the Presence of Stationarity

This paper empirically evaluates the degree of integration among spatially separated markets using monthly wholesale real price series of tomato and onion from regional markets located in each of the four provinces of Pakistan. Unit root test indicates that the price series are stationary, and are represented as autoregressive model for each region. Results of the error correction model show that the markets of both tomato and onion are spatially integrated, however, the overall degree of integration in onion markets is relatively stronger than in tomato markets. This difference may be attributed to relatively lower degree of perishability in onion as compared to tomato

Minnesota Agricultural Economist 687

Agricultural Finance,