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Another important application of exponential models is the Cobb-Douglas produc- tion function, which has the form Q = BOLBKB where Q is the quantity produced, L is the amount of labor used, and K is the amount of capital. B, and B2 are the relative contributions of changes in labor and changes in capital to changes in quantity produced. In one special case the sum of the coefficients is restricted to 1, and we have constant returns to scale. In that case B, and B2 are the percent contributions of labor and capital to productivity increase. The estimation of the coefficients when their sum is equal to 1 is one example of re- stricted estimation in regression models. Equation 12.29 is modified by the restriction B + B2 = 1 and, therefore, substitution of the form X14 B2 = 1 - B is included, and the new estimation equation becomes log(Y) = log(Bo) + B, log(X) + (1 - B.) log(X2) + log( 8 ) log(Y) log( X2) = log(Bo) + Bi[log(X1) - log(X2)] + log(8) log Xa = log(Bo) + Bi log(%) + log(e) (12.30) Thus, we see that the B, coefficient is obtained by regressing log(Y/X2) on log( X1/X2). Then, B2 is computed by subtracting B, from 1.0. All quality computer-based statistical packages can easily compute the required trans- formations of the data for logarithmic models. In the following example, we used Minitab, but similar results could be obtained using many other packages. Example 12.12 Production Function for Minong Boat Works (Exponential Model Estimation) The Minong Boat Works began producing small fishing boats in the early 1970s for northern Wisconsin fishermen. The owners developed a low-cost production method for producing quality boats. As a result, they have experienced increased demand over the years. The production method uses a workstation with a set of jigs and power tools that can be operated by a varying number workers. Over the years the number of workstations (units of capital) has grown from 1 to 20 to meet the demand for boats. At the same time the workforce has grown from 2 to 25 full-time workers. The owners are now considering expanding their sales to potential markets in Michigan and Minne- sota. Therefore, they need to decide how much to increase the number of workstations and number of workers to achieve various levels of increased production