The firm's production function is as follows: (0.3) T = Y W* Na=2*N) - W* Na where K sets at 1. Let the consumer's utility function be: (0.4) U = C*1- Y3 Think of the second term (-Y3) as a loss to the consumer because of pollution from the output. The more output is produced, the more the society is polluted and the greater is the harm to the consumer. 1. Write down the usual problem of the firm and solve for the optimal amount of labor demand (N.). Given w = 73, calculate the optimal numerical values of labor demand, output and profits (Hint: Try to go in this order). 2. Using the utility function above, write down the usual problem of the consumer and solve for the optimal level of consumption and leisure assuming T = 0 (Hint: keep in mind that the consumer is not choosing output Y but is taking it as given while optimizing). Given that h=1 and w = 73, calculate the optimal numerical values of leisure (1), labor supply (N,) and consumption (c) and utility (U) taking Y and re from part 1 (Hint: Try to go in this order). You do not need to fully compute U and leave it as a fraction. It can be negative. Don't be surprised! (What we care about is not its actual number, its order. For example, if U (C1, 1) = -0.1>U (C2, l2)=-0.2, then the consumer prefer (C1,4) to (C2, l2)). | 3. Do you observe that the labor supply and labor demand are equal in parts 1 and 2? In other words, does the wage w = V3 clear the labor market? Likewise, do you also observe that the output and consumption are the same under zero spending on public good? The firm's production function is as follows: (0.3) T = Y W* Na=2*N) - W* Na where K sets at 1. Let the consumer's utility function be: (0.4) U = C*1- Y3 Think of the second term (-Y3) as a loss to the consumer because of pollution from the output. The more output is produced, the more the society is polluted and the greater is the harm to the consumer. 1. Write down the usual problem of the firm and solve for the optimal amount of labor demand (N.). Given w = 73, calculate the optimal numerical values of labor demand, output and profits (Hint: Try to go in this order). 2. Using the utility function above, write down the usual problem of the consumer and solve for the optimal level of consumption and leisure assuming T = 0 (Hint: keep in mind that the consumer is not choosing output Y but is taking it as given while optimizing). Given that h=1 and w = 73, calculate the optimal numerical values of leisure (1), labor supply (N,) and consumption (c) and utility (U) taking Y and re from part 1 (Hint: Try to go in this order). You do not need to fully compute U and leave it as a fraction. It can be negative. Don't be surprised! (What we care about is not its actual number, its order. For example, if U (C1, 1) = -0.1>U (C2, l2)=-0.2, then the consumer prefer (C1,4) to (C2, l2)). | 3. Do you observe that the labor supply and labor demand are equal in parts 1 and 2? In other words, does the wage w = V3 clear the labor market? Likewise, do you also observe that the output and consumption are the same under zero spending on public good
