Consider the following production function Yt ≡ zF (Kt, Nt):
Yt ≡ zF (Kt, Nt) = z [θ(αKt)γ + (1 − θ)(βNt)γ ] 1γ
where Yt is output at time t, the constant z measures total factor productivity (TFP), Kt is physical
capital, Nt is labor, 0 < θ < 1 and γ > 0 are parameters and 0 < α < 1, 0 < β < 1 represent the capital
and labor shares, respectively. Note that the subscript t denotes dependence with respect to time (i.e.,
those variables change over time).
a. Assume γ=1 and take Kt as given. Would this function make a good production function, in the
economic sense discussed in class? Argue, in favor or against, consider the properties of production
functions reviewed in class and provide an economic intuition (e.g., is this function increasing? is it
concave?).
b. Suppose z = 1, θ = 0.5, α = 0.66, β = 0.34, γ = 3 and Kt = 2. Given these parameter values, use
an Excel sheet to find the values of Yt for Nt = 0, 0.5, 1, 1.5, . . . , 20. Make sure to build in the cells
associated with the parameter values given above into the formula to compute Yt. What happens with
the marginal product of labor as Nt increases? Graph Yt as a function of Nt.
c. Leave the Excel graph in your spreadsheet and play with the parameter γ (since your formula to
compute Yt is linked to the cells containing the values of the parameters z, θ, etc, you can simply
change the value of γ in the cell associated and the graphed function should automatically update).
What happens with the shape of the function as you decrease the value of γ? What happens with the
shape of the function when γ = 1? And when γ → 0? What value (or range of values) of γ would you pick so that the production function Yt ≡ zF (Kt, Nt) makes economic sense? Graph Yt as a function of Nt for that value of γ.