Problem 1. Consider a closed market economy with L utility maximizing households and
M profit maximizing firms,
a) Find the capital intensity k; = Ki/L; chosen by firm i (expressed as a function of r, o and G). b) Find the equilibrium level of aggregate output at some arbitrary point in time. Hint: You may use that ki = k = K/L for all i (why?) to find an expression for Y in which (*) can be inserted to find G. NB! You should end up with Y expressed as a function of only predetermined variables, exogenous variables and parameters. c) Find the equilibrium real rate of interest and the equilibrium real wage at some arbitrary point in time. d) Suppose the households, all alike, have an intertemporal utility function with infinite horizon and a constant rate of time preference p > 0. The instantaneous utility function has elasticity of marginal utility equal to a constant 0 > 0. Derive the Keynes-Ramsey rule, given the described taxation system. e) Write down the transversality condition in a form comparable to the No-Ponzi-Game condition of the household. Comment on the discount rate. f) Find the growth rate of k = K/L and y = Y/L in this economy (an informal argu- ment, based on your general knowledge, is acceptable). In case, you need to introduce a restriction on some parameters, do it. g) Comment on the solution to f). h) Suppose lumpsum taxation is not feasible. Hence, let 7 = 0 for all t 2 0. Is it possible to fix + at a level (constant over time) such that the government budget is still balanced in equilibrium for all t > 0? i) If the welfare of the representative household is the criterion, what proposal to the gov- ernment do you have w.r.t. the size of g and the form of taxation (given that taxation must be non-lumpsum)? Hint: There may be a problem with the tax candidate that first comes to one's mind.Problem 1. Consider a closed market economy with L utility maximizing households and M prot maximizing rms, operating under perfect competition (L and Mr are constant, but \"large\"). There is also a government, supplying a nonrival productive service G per time unit. Each household supplies inelastically one unit of labour per time unit. Aggregate output is Y per time unit, and output is used for private consumption, C E CL , the public productive service G and investment I in (physical) capital, i.e., Y = C + G + I. The stock of capital K changes according to K = I 5K, where (5 2 U is the rate of physical decay of capital. Variables are dated implicitly; t denotes time. The initial value Kw) > U is given. The capital stock in society is owned, directly or indirectly, by the households. There is perfect competition at the labour market, and the equilibrium real wage is called to. There is a perfect market for loans at the real rate of interest 3", and there is no uncertainty. A dot over a variable denotes the time derivative . The government chooses C so that g = a v) where the constant .5 is exogenous and is such that positive growth in the economy occurs in equilibrium. The government budget is always balanced. The service G is the only public expenditure, and the tax revenue is [re + w) + T5] L = G, (CBC) where a- is per capita nancial wealth, and \"i" and T5 denote the income tax rate and a lumpsum tax, respectively. The tax rate T is a given constant, whereas Tg is adjusted when needed for (CBC) to be satised. The production function for rm 3' is 1;: AKf'(GL,)1\
