Question

Numerical Example: Static GE - the social planner Given: household's utility: U(C,) 15 In 16 In C production function: Y 10vKN total time endowment h -9, government spending G 22.5 and capital stock K - Follow these steps: 9 o write down the planner's resource constraint o solve the planner's problem o find the endogenous variables, including C, e, N, Y

Answer 1

a) Planner's resource constraint:

Ct + It ≤ Yt.

its per capita term equaiton : ct + it ≤ yt

c)Endogenous varaibles: Let us assume  Lt denote the number of households (and the size of the labor force) in period t, Kt aggregate capital stock in the beginning of period t, Yt aggregate output in period t, Ct aggregate consumption in period t, and It aggregate investment in period t. The corresponding lower-case variables represent per-capita measures: kt = Kt/Lt, yt = Yt/Lt, it = It/Lt, and ct = Ct/Lt.

B) A feasible allocation is any sequence {ct, kt}∞ t=0 ∈ ¡ R2 + ¢∞ that satisfies the resource constraint kt+1 ≤ f(kt) + (1 − δ − n)kt − ct. consumption is, by assumption, a fixed fraction (1 − s) of output: Ct = (1 − s)Yt .consumption is given by ct = (1 − s)f(kt).

An “optimal” centralized allocation is any feasible allocation that satisfies the resource constraint with equality and ct = (1 − s)f(kt).we derive the fundamental equation of the Solow model: kt+1 − kt = sf(kt) − (δ + n)ktGiven any initial point k0 > 0, the dynamics of the dictatorial economy are given by the path {kt}∞ t=0 such that kt+1 = G(kt), (2.13) for all t ≥ 0, where G(k) ≡ sf(k) + (1 − δ − n)k. Equivalently, the growth rate of capital is given by γt ≡ kt+1 − kt kt = γ(kt), (2.14) where γ(k) ≡ sφ(k) − (δ + n), φ(k) ≡ f(k)/k.Suppose δ+n ∈ (0, 1) and s ∈ (0, 1). A steady state (c∗, k∗) ∈ (0, ∞)2 for the dictatorial economy exists and is unique. k∗ and y∗ increase with s and decrease with δ and n, whereas c∗ is non-monotonic with s and decreases with δ and n. Finally, y∗/k∗ = (δ+n)/s.