4. Consider the economy in the example on page 7 of Notes 10. There are n firms in a competitive industry. The market demand function is given by p = 50 - 2Q. A firm's cost function is C (g) = q?. In class, we have calculated the total surplus is 51 = (n41)' 50." Now consider how taxation can change the total surplus. (a) [10 points] Suppose the government imposes a per-unit far t > 0 on consumers. That is, for each unit of goods purchased, the consumer pays p + t. As a result, the market demand function becomes p + = 50 - 2Q. The tax goes to the government and firms only receive p for each unit of output sold. So the profit of a firm is pq - q?. Derive the total surplus, denoted by $2. Is Sq > Si? (b) [10 points] Instead of the per-unit tax, suppose the government imposes a pro- portional for rate + > 0 on consumers. That is, the consumer pays a after- tax price of (1 + +) p for each unit of goods, and the market demand becomes (1 + 7)p = 50-2Q. Once again, all tax revenues go to the government. Derive the total surplus, denoted by S3. Is Sa > Si?1) Consider a one-period economy with a single representative consumer, a single representative firm and the government. The representative consumer derives utility from consumption c and leisure : u (c, I) = Inc The firm produces output Y using capital K and labor N according to Y = ZK NIG, where z is the total factor productivity and a is the Cobb-Douglas parameter. The firm maximizes profits it which are then transferred to the representative consumer. The government balances the budget using lump-sum taxes T on the representative consumer to finance government spending G. The hourly wage in this economy is w and the consumer has h hours (/=16) to divide between leisure and labor. (i) Write down the consumer's budget constraint and the firm's profits function. (ii) Assume that w = 10, z = 20, a = 0.3, and K = 1. Calculate the number of hours that the firm would like to hire and the profits of the representative firm. (ii) Assume that government spending G is 10, the representative consumer receives the profits that you calculated above, earns hourly wage w = 10, and has h = 16 hours to divide between leisure and labor. Calculate how many hours of work the representative consumer would like to supply in the market. (iv) In the economy described above, () the government budget is balanced, (ii) the representative consumer maximizes lifetime utility given the budget constraint and w, and (iii) the firm maximizes profits given the production function and w. Is this a competitive equilibrium? Why or why not?Consider a one-period economy with a single representative consumer, a sin- gle representative firm and the government. The representative consumer derives utility from consumption c and leisure /: u(c, l) = Inc+ In/ (1) The firm produces output Y using capital K and labor N according to Y = zK Nl- (2) where z is the total factor productivity and a is the Cobb-Douglas parameter. The firm maximizes profits , which are then transferred to the representative consumer. The government balances the budget using lump-sum taxes 7 on the repre- sentative consumer to finance government spending G. The hourly wage in this economy is w and the consumer has h hours to divide between leisure and labor. (i) Write down the consumer's budget constraint and the firm's profits function. (05 marks) (ii) Assume that w = 10, 2 = 20, a = 0.3, and K = 1. Calculate the number of hours that the firm would like to hire and the profits of the representative firm. (05 marks) (iii) Assume that government spending G is 10, the representative consumer receives the profits that you calculated above and earns hourly wage w = 10. Calculate how many hours of work the representative con- sumer would like to supply in the market. (10 marks) (iv) In the economy described above, (i) the government budget is balanced, (ii) the representative consumer maximizes lifetime utility given the budget constraint and w, and (iii) the firm maximizes profits given the production function and w. Is this a competitive equilibrium? Why or why not? (05 marks) Total: 25 marks
