The purpose of this paper is to investigate a relationship between a local government expenditure and a population size in Japan. The previous papers reported that a U shaped per capita expenditure curve is observed by a cross-sectional data. However, the these results might include omitted variable biases both the population size and the fixed effect for each municipality. Therefore, this paper tries to estimate the cubic function related to the population size. Also, in order to address the omitted variable biases related to the fixed effect, we estimate the fixed effect model using the panel data from FY1996 to FY2010. We found the first evidence of cubic function formed expenditure curve related to the population size. For the small municipalities, the form captures an increasing effect the population size on the local government expenditure. The medium-sized municipalities can have the decreasing effect. On the contrary, the large municipalities can have an increasing effect. In addition, we showed the relationship between the fixed effect for each municipality and the population size has the U shaped form. In other words, we could improve"the remaining omitted variable biases"about both the population size and the fixed effect on the previous papers.