Introduction to Modern Economic Growth (5) Show that unless the utility function is asymptotically equal to ( Ac(t)1−θ h(1 − l(t)) for θ 6= 1, 1−θ u (c (t) , − l (t)) = A log c (t) + Bh (1 − l (t)) for θ = 1, for some h(·) with h0 (·) > 0, there will not exist a balanced growth path with constant and equal rates of consumption and output growth, and a constant level of labor supply Exercise 8.18 Consider the standard neoclassical growth model with a representative household with preferences # " Z ∞ c (t)1−θ − U (0) = + G (t) , exp (−ρt) 1−θ where G (t) is a public good financed by government spending Assume that the production function is given by Y (t) = F [K(t), L (t)], which satisfies all the standard assumptions, and the budget set of the representative household is C (t) + I (t) ≤ Y (t), where I (t) is private investment Assume that G (t) is financed by taxes on investment In particular, the capital accumulation equation is K˙ (t) = (1 − τ (t)) I (t) − δK (t) , and the fraction τ (t) of the private investment I (t) is used to finance the public good, i.e., G (t) = τ (t) I (t) Take the sequence of tax rates [τ (t)]∞ t=0 as given (1) Define a competitive equilibrium (2) Set up the individual maximization problem and characterize consumption and investment behavior (3) Assuming that limt→∞ τ (t) = τ , characterize the steady state (4) What value of τ maximizes the level of utility at the steady state Starting away from the state state, is this also the tax rate that would maximize the initial utility level? Why or why not? Exercise 8.19 Consider the neoclassical growth model with a government that needs to finance a flow expenditure of G Suppose that government spending does not affect utility and that the government can finance this expenditure by using lump-sum taxes (that is, some amount T (t) imposed on each household at time t 413