Q1. Assume the economy has only 1 representative agent. She has to divide 1 unit of time between work and leisure given the wage rate 10 paid on the labor market. In addition to the salary, he or she also receives dividend a income of 7r. Suppose that the government does not incur expenditure, so G = O. The agent's utility function depends on consumption (2 and leisure E, and it is assumed to satisfy u(c, 13) = 0.51n(c) + 0.5ln(13). On the other side of the market, there are rms who hire workers and produce output. The representative rm operates with a Cobb Douglas production technology Y = 2K0'5N0'5, where z = 1 denotes the total factor productivity, and K = 16 is a xed amount of capital. Each of the rm's employees receive wage w, i.e. the total labor cost of the rm is equal to wN. The prot of the rm will be distributed back to the shareholder (the agent here) as dividend income. (a) Write down government's budget constraint. How much does the agent need to pay as the lump sum tax 1""? (b) Write down consumer's budget constraint and maximization problem. Solve the optimal consumption allocation function c(w*) and labor supply N 8(10'\"). (c) Write down the maximization problem of the rm. Find the labor demand N d(w*). (d) Write down the market clear conditions in labor and goods market. Find the equilibrium price w*, allocations 6*, N *. (e) (Analyzing the effects of COVID19) Suppose now the total factor produc tivity decreases from 2 = 1 to z = 0.5. i. What are the effects of this negative TFP shock on the equilibrium allocations and prices