The following is known as the transcendental production function (TPF), a generalization of the well-known Cobb€“Douglas production function:

Y_i = β₁ L^β2 k^β3 e^β4L+β5K

where Y = output, L = labor input, and K = capital input.
After taking logarithms and adding the stochastic disturbance term, we obtain the stochastic TPF as

ln Y_i = β₀ + β₂ ln L_i + β₃ ln K_i + β₄L_i + β₅K_i + u_i

where β₀ = ln β₁.

a. What are the properties of this function?
b. For the TPF to reduce to the Cobb€“Douglas production function, what must be the values of β₄ and β₅?

c. If you had the data, how would you go about finding out whether the TPF reduces to the Cobb€“Douglas production function? What testing procedure would you use?
d. See if the TPF fits the data given in the following table. Show your calculations.

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Fixed Capital Employmentt Year GDP* 1955 8310 114043 182113 1956 120410 8529 193749 1957 129187 8738 205192 8952 1958 134705 215130 9171 1959 139960 225021 9569 1960 150511 237026 1961 157897 9527 248897 1962 165286 9662 260661 1963 178491 10334 275466 10981 295378 1964 199457 1965 212323 11746 315715 1966 226977 11521 337642 1967 241194 11540 363599 1968 260881 12066 391847 1969 277498 12297 422382 1970 296530 12955 455049 1971 306712 13338 484677 1972 329030 13738 520553 1973 354057 15924 561531 609825 1974 374977 14154