The ^ symbol refers below to the percentage change in a variable. All prices initially equal 1.
The differential of the demand equation (40) can be expressed as:
where ϕ is the elasticity of substitution between the two products.
The differential of output in industry 1 is:
The change in output in industry 2 can similarly be written as:
We combine (Al), (A2), and (A3) noting that:
We now differentiate (29) and divide by Mc1kcl (that is, by Kcl) which yields:
The first four terms represent the changes in corporate capital in the two sectors. The next three terms are the changes in capital for noncorporate production in industry 1 due, respectively, to changes in capital in existing firms, changes due to new firms established by former corporate managers and workers, and by new firms established by entrepreneurs who previously had firms in industry 2.
We similarly differentiate equation (28) for a fixed stock of labor and divide by Mc1lc1 (which equals Lc1).
To solve the model, we solve equation (A1.4) for
which is now multiplied by
since ln1(A) = lc1, and kn1(A) = kc1. When equation (A1.4) is substituted into equation (A1.l’), the term
which is now multiplied by Kc1/K1 (recall that [Mc1kc1 = Kc1]) will cancel
by virtue of the fact that
By the same reasoning the term
will cancel the term
The same reasoning will also result in terms of canceling out when the new (A1.5) is substituted into (Al’).
The only integrals which do not cancel out through this process are those relating to the switching of existing entrepreneurs from one industry to another. These terms, which are each multiplied by
Note, however, that if we bring 111c2 with the first term inside the integral it becomes lc2B which is ln2(B)B, and if we bring kc2 in the second term into the integral it becomes kc2B which is kn2(B)B. Thus these parts of the integral will cancel, and similarly, parts of the second two terms will cancel. Moreover, since the number of entrepreneurs switching must net to zero, the terms with “1” cancel.
We are left with:
By using the relationship
We note that
We now substitute into (A1.l”) from equations (A1.6). In this substitution we use the facts that (α1/β1)ε1r = εkw and (α2/β2)η1r = ηkw.
The incidence formula (48) results from the above substitution and the relationships (A1.7) - (A1.11) given below:
which is the result of differentiating (43).
In (A8) - (A11) the term Den = θ1(1-β1) + θ2(1-β2).
For the CES function given in (50) it is easy to show that the factor demands for capital per manager and labor per manager satisfy:
In the case of the Cobb/Douglas/CES function capital per manager and labor per manager satisfy for i = 1,2:
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Dr. Jane G. Gravelle is with the Congressional Research Service; Professor Laurence Kotlikoff is with the Department of Economics, Boston University, and with the National Bureau of Economic Research. This research was conducted while Professor Kotlikoff was a Visiting Scholar at the International Monetary Fund. The authors wish to thank Alan Auerbach, Guillermo Calvo, Christophe Chamley, Arnold Harberger, Martin Feldstein, Michael Manove, Joe Pechman, Andy Postlewaite, Steven Salop, Vito Tanzi, Jack Taylor, Andrew Weiss and seminar participants at Boston University, Harvard, Pennsylvania, and the National Bureau of Economic Research for very helpful discussions. The authors are very grateful to the International Monetary Fund for supporting this research and to Ghazala Mansuri for independently checking our calculations. The views in this study do not necessarily represent those of the Congressional Research Service or the Library of Congress.
These tax rates represent marginal effective (personal plus corporate) tax rates on new investment. For the corporate sector the return (after all taxes) to stockholders and creditors is grossed up by personal taxes on dividends, interest, and capital gains (adjusted for the value of capital gains deferral and the taxation of the inflationary component of capital gains). The marginal tax rate methodology (see Gravelle (1982)) uses a discounted cash flow method to determine the pretax real return necessary to pay stockholders and creditors the after-tax return. A similar process is used to measure the noncorporate pre-tax return required to yield the same after-tax return. The differential between these two pre-tax returns is used to measure the corporate-noncorporate tax wedge. This wedge is given by 1 - (1-µ)/(1-t), where µ is the total effective tax rate on corporate capital, and t is the total effective tax rate on noncorporate capital.
After developing our model we became aware of Lucas’ (1978) paper that also models entrepreneurs as managers with differing abilities, but demonstrates how such a model can explain secular changes in firm size. Chamley (1983) is another example of an early analysis of differing entrepreneurial abilities and the choice of occupation.
Harberger and Shoven measure capital income as the sum of interest, profits, rents, and property taxes. These items, except for property taxes, appear on the proprietorship tax returns. Property taxes were imputed based on their fraction of capital income as reported in Rosenberg (1969). Labor payments as a share of total factor income were weighted by industry. Lessors of real estate were used to determine values of housing. For this industry there appears to be an error in the 1959 data (the first year for which the necessary detail is available). For this reason the ratio of labor income to total factor income for the next year available, 1962, was used.
The revenue base for measuring this tax rate is corporate profits plus interest. Note that the 0.45 tax rate, which does not consider personal taxation, is smaller than the 0.52 tax rate cited in Section II which corresponds to the 1957 differential tax on corporate versus noncorporate capital income taking into account both personal and corporate taxation. Hence, considering personal taxes would increase the estimates of excess burden in the MPM reported in Section VII.
K1/K2 = θ1l1 (1-t1)/θ2l2(1-t2).