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We consider generalized production functions, introduced in Zellner and Revankar (1969), for output y. g(f) where g is a monotonic function and f is a homogeneous production function. For various choices of the scale elasticity or returns to scale as a function of output, differential equations are solved to determine the associated forms of the monotonic transformation, gW. Then by choice of the form off, the elasticity of substitution, constant or variable, is determined. In this way, we have produced and generalized a number of homothetic production functions, some already in the literature. Also, we have derived and studied their associated cost functions to determine how their shapes are affected by various choices of the scale elasticity and substitution elasticity functions. In general, we require that the returns to scale function be a monotonically decreasing function of output and that associated average cost functions be U-or L-shaped with a unique minimum. We also represent production functions in polar coordinates and show how this representation simplifies study of production functions properties. Using data for the U.S. transportation equipment industry, maximum likelihood and Bayesian methods are employed to estimate many different generalized production functions and their associated average cost functions. In accord with results in the literature, it is found that the scale elasticities decline with output and that average cost curves are U-or L-shaped with unique minima. *Research financed by the National Science Foundation and by income from the H.G.B. Alexander Endowment Fund, Graduate School of Business, University of Chicago. We thank two referees and the Co-Editor, John Geweke for helpful comments. ..