The economy is populated by 100 agents. Each agent has to divide I unit of time between work and leisure given the wage rate w paid on the labor market. In addition to the salary, each agent also receives a dividend of 7. The total profit of the firm is II. The agent's utility function depends on consumption c and leisure , and it is assumed to satisfy u(c, () = 0 In(c) + (1 - 0) In((), where 0 E (0, 1). On the other side of the market, there are firms that hire workers and produce output. The representative firm operates with a Cobb-Douglas production technology Y = zK Ni, where z denotes the total factor productivity, and K is a fixed amount of capital. Each of the firm's employees receives wage w, i.e., the total labor cost of the firm is equal to wNV. The firm's profit will be distributed back equally to the shareholder (the agent here) as dividend income. Suppose that the government does not incur expenditure, so G = 0. (a) Suppose initially 0 = 4, 2 = 1, K = 1600. 1. Write down the equation to show the relation between each individual's dividend income o and the firm's total profit II. 2. Write down the consumer's budget constraint and maximization problem. Plot her budget constraint and solve the optimal consumption allocation function c(w*) and labor supply Ns(w*).3. Write down the maximization problem of the firm. Find the labor demand No(w*). 4. Write down the government's budget constraint. How much does the agent need to pay as the lump sum tax *?5. Write down the market clear conditions in the labor and goods market. Find the equilibrium price w*, allocations c*, N*.(b) Now, suppose there is a positive shock to TFP (i.e., technological improvement) and z increases to 2, keeping 0 = , K = 1600 unchanged. What are the effects of positive TFP shock on equilibrium allocations and prices? Explain both mathematically and intuitively. (c) Now, suppose there is a positive shock to 0 (i.e., change in preference) and 0 increases to -, keeping z = 1, K = 1600 unchanged. What are the effects of such shock on equilibrium allocations and prices? Explain both mathematically and intuitively.(d) Now, suppose the economy has a larger supply of capital K and K increases to 8100, keeping 0 = 3, 2 = 1. What are the effects of positive capital shock on equilibrium allocations and prices? Explain both mathematically and intuitively. (e) Now, suppose that Congress, concerned about the welfare of the working class, passes a law setting the minimum wage that is 10 percent above the equilibrium wage you just derived. Assuming that Congress cannot dictate how many workers are hired at the mandated wage, what are the law's effects? Specifically, calculate what happens to the real wage, the employment, the output, and the total amount earned by workers