Multiplicative Model in Commercial Bank

Complexity and Versatility Multiplicative Model for Nonlinear Cost Functions If we want to allow the effect of each independent variable to be a function of the level of itself and all other independent variables, then we should attempt to estimate the model in multiplicative form: Cost = B0X1^B1 X2^B2 X3^B3 With this model, the effect of a change in any independent variable, say X, is a function of the level of X, as well as the levels of X, and X3. The estimated coefficient, B2, represents the percentage change in the dependent variable caused by a given percentage change in the independent variable, X2. Students of economics will notice that 2 represents the elasticity of cost with respect to the variable X2. The multiplicative model assumes a constant elasticity B; for each of the independent variables. The multiplicative model is most useful when the dependent variable is naturally written as the product of independent variables. For example, total labor expense is the product of total labor-hours and the average wage rate. Interest expense is the product of loans outstanding and the interest rate. In both cases, we would expect the estimated exponential coefficients, B, and B2, of the independent variables to be close to 1. It would be a mistake to estimate either of these relationships as additive in the independent variables, since the cost category is a function of the product, not the sum, of the independent variables. The multiplicative model is easily estimated using standard OLS packages by taking the logarithm of the left - and right-hand side of the regression equation to yield: log Y, = log Bo + B1 log x1t + B2 log X2t + B3 log X3t This equation is now linear in the sum of the logarithms of the independent variables. Therefore, we transform the original data by computing the logarithm of the dependent and independent variables and running a standard regression routine on these "logged" values. The estimated coefficients of the logged variables represent the constant elasticities, B, in the multiplicative model. A particularly ambitious application of the multiplicative regression model was used to estimate the cost functions of commercial banks.' Data from nearly one thousand commercial banks were collected and used to estimate total bank costs as a function of the banks' many different activities and characteristics. Because of the complexity of a bank's operations and the expectation that the cost for any particular service would depend on the current activity levels of all the services, a multiplicative model containing twelve independent variables was used: Variable Number Variable Description 1 Average number of accounts per office 2 Average dollar size of a demand deposit account 3 4 Average number of debits per account Average number of deposits per account Average number of non-home-bank 5 (transit) checks deposited per account 6 Average number of official checks issued per account 7 Average number of checks cashed per account Average number of transit checks 8 9 cashed per account Number of offices operated by the bank Ratio of number of regular checking 10 accounts to the sum of regular and special accounts 11 Average annual wage rate per employee Ratio of dollar volume of demand 12 deposits to sum of demand and time