4. Consider a static economy with a representative consumer, a representative firm, and a gov- ernment. The consumer has preferences over private consumption c, work effort n, and public consumption g, described by u(c, n, g) = c v(n) + $(g), where v and are strictly increasing and strictly concave. The firm produces according to = zkan- --a, where a [0,1]. The consumer is endowed with ko > 0 units of capital. The level of public spending 9 is exogenous. The government can fund public spending by taxing total household income at the proportional rate 7. Write down the consumer's problem in a competitive equilibrium. (b) Derive and interpret the consumer's optimality condition. (c) Write down the firm's problem. Derive and interpret the firm's optimality conditions. (d) Write down the government's problem. (e) Formally define a competitive equilibrium. (f) Suppose the government had the option to tax capital and labor income at different proportional rates. What would be the best way to raise tax revenue? Would a lump- sum tax be better? Explain. 2 (g) Consider the parametric case v(n) = n and a = 0. Show that the government faces a Laffer curve. Explain why this is the case. (h) Suppose the government increases g, with the economy always on the increasing portion of the Laffer curve. Under the parametric case of the previous question, describe the equilibrium effects of the fiscal expansion (i.e. say whether each of the variables you identified in your definition of a competitive equilibrium increases, decreases, or stays constant). Please justify your answer. (i) What is the sign of the government spending multiplier? Explain the economic reasoning behind your finding. Explain carefully how your finding contrasts with the government spending multiplier we studied in class (please identify and explain the main economic mechanisms in each case).