4. [20 points] So far we have assumed that these taxes are just wasted. Now let us assume that government spending is actually part of the production function. 1" = AKC'LJQG1__J'9. Also, assume that the government runs a balanced budget and uses only one tax, which is levied on output (at rate 1" per unit output). Also assume o: E {0,1}, ,6 E {0,1} and o:+,6 c: 1. (a) Write down the government budget constraint if there is a balanced budget every period. (b) Show that output is increasing in government expenditure but at a decreasing rate. Provide an intuitive explanation of this property. Do you think this is realistic? Why or why not? (c) Write down the rms prot maximization problem and obtain the F00 for K and L. {Note that rms do not choose G). (d) Assume for the rest of the exercise that ,5 = 0 (Note that 2:\" = l). Rewrite the F00 with respect to capital. Use the government budget constraint to eliminate G from the F00. Then derive r as a function of 1', or, A and 5. (e) Derive the Euler equation and replace rr' with the expression found in 3d. (f) \"That is the relationship between the growth rate in consumption and the tax rate? Find 7* the rate of taxation that maximizes consumption growth. Provide a graphical representation of the growth rate of consumption as a function of the tax rate