DEPARTMENT OF ECONOMICS & FINANCE

ANSWER ALL QUESTIONS IN ASMUCH DETAIL AS POSSIBLE. SHOW YOUR DETAILED WORK. 1. Consider a world with no income and sales taxes. The utility function of a represen- tative consumer is given by u = px1x2. Her budget constraint is given by 7x1 + 10x2 = 3000. Solve for the optimal consumption bundle that the consumer will choose. Now suppose the government is debating between a 9:75% ‡at sales tax on both the commodities or a 5% income tax. The government is going to have either the sales tax or the income tax but NOT both. Solve for the optimal consumption bundles under sales tax and under income tax regimes separately. Which regime (sales tax or income tax) raises higher revenue for the government? Calculate the precise revenue collection in each scenario. Which regime (sales tax or income tax) results in higher consumption for our repre- sentative consumer? Which regime (sales tax or income tax) results in higher utility for our representative consumer? If there are one million consumers like our representative consumer, how much of the two commodities will be demanded in the economy? (Compute the total demand with NO taxes and then with sales and income taxes. Now compares these three sets of numbers and re‡ect on their di¤erences.) 2. Consider three consumption bundles X; Y;Z. Prove that if X is preferred to Y and Y is preferred to Z then X is preferred to Z: Use the insight from this proof to show that two indi¤erence curves may not intersect one another. 3. Suppose two commodities are perfect complements and the consumer’s utility function is given by U = minfX1;X2g. Both the commodities are priced the same way and they cost $20 per unit. The consumer makes $2000. Write down the budget constraint of the consumer. How much amount for each of the commodities will the consumer consume and why? What will be the utility level of the of the consumer if she is maximizing her utility subject to the budget constraint? 4. Argue in EACH case below as what returns to scale the production function exhibits (Show reason behind your conclusion): f(x1; x2) = x1 + x2; f(x1; x2) = (x1 + x2)2; f(x1; x2) = (x1 + x2)1=2 5. Critically assess if the following statements are true or false: (You MUST provide 3 ? 4 lines’of explanations for EACH answer and draw graphs wherever you can. Providing crisp mathematical explanation wherever possible will be good.) i) MC intersects the AV C where AV C is minimum. ii) MC intersects the ATC where AV C is minimum. iii) Break-even point refers to the point where MR = MC 6. The inverse demand curve faced by a monopolist is given by P = 2000?10Q. Monop- olist’s marginal cost is constant is given by c = 25. Assume that the monopolist maximizes its pro…ts. What is the amount that the monopolist would produce and what will be the market price? Compute the total pro…t of the monopolist and also the value of the consumer surplus. 7. Consider Question #6 and assume that the market is served by two …rms playing a Cournot game. Their constant marginals costs of production are given by c1 = 20 and c2 = 25. Assume that both …rms are pro…t maximizers. Compute the output produced by each of the …rms. Also compute the aggregate supply in the market. What will be the prevailing price in the market? Compute the pro…t for each of the …rms. Compare the consumer surpluses between Question #6 and Question #7. 8. Pro…t function of a representative …rm in a competitive market is given by = PQ ? wL ? rK ? F. Here, P is the market price, Q is the output produced (sold), w is the wage rate, L is the labor employed, K is the capital employed, r is the rental rate and F is the …xed cost that does not depend on the amount of quantity produced. The …rm uses a production function Q = AKL. Algebraically solve for the pro…t maximizing labor (L) and capital (K) that the …rm should be using. What is the total output produced by the …rm? What is the pro…t earned by the …rm? Now assume that w = 15; = 0:4; = 0:5; A = 10; and P = 15. Graph the pro…t maximizing labor (L) and capital (K) amounts when r increases continuously from 1% to 5%. (Use Excel to simulate as smooth an increase as possible. You may get quite a smooth path if you plot the changes in r at one …fth of one percent intervals). Suppose this economy has 100 such …rms. How will the total employment change as the rate of interest inches up from 1% to 5%? 9. Find the best strategies for EACH player and carefully …nd the Nash equilibrium of EACH of the following games: (Be careful about identifying pure strategy Nash equilibrium (equilibria). If no pure strategy Nash equilibrium exists then check for the mixed strategy Nash equilibrium) (i) Prisoner 2 ! Confess Don’t Confess Prisoner 1# Confess ?18;?18 ?2;?20 Don’t Confess ?20;?2 ?4;?4 (ii) Country 2 ! Trade Don’t Trade Country 1# Trade 1000; 1000 600; 500 Don’t Trade 500; 600 500; 500 (iii) Intel ! cooperate Don’t cooperate Apple# cooperate 20; 20 13; 17 Don’t cooperate 17; 13 18; 18 (iv) Tax Payer Joe ! Cheat Don’t Cheat IRS# Audit (with Probability 0:5) 450;?500 150;?200 Don’t Audit (with Probability 0:5) 0; 0 200;?200 (v) Applicant Russell ! Negotiate Don’t Negotiate Company X# O¤er a job 50; 40 60; 30 Don’t O¤er a job 0;?10 0; 0 (vi) Player B ! Go to Opera Go to Movie Player A# Go to Opera 1;?1 ?1; 1 Go to Movie ?1; 1 1;?1 10. Prove that all Gi¤en goods are inferior goods but all inferior goods are NOT Gi¤en goods. (A graphical proof will be su¢ cient). 11. When prices are (P1; P2) = (2; 4) a consumer demands (x1; x2) = (4; 2): When the prices are (4; 2) the consumer demands (2; 4). Argue if the behavior of the consumer is consistent with the utility maximizing behavior. 12. The total cost of a …rm is given by C = 2 + 3Q + 4Q2 where Q is the total amount of output. What is the cost of producing zero units? What do you call this amount? What is the total cost of producing four units? What is the marginal cost of producing the …fth unit? What is the marginal cost of producing the sixth unit? Draw a graph depicting the following as the output rises from 0 to 100: MC, AVC, ATC, AFC. (Tip: Use Microsoft Excel) Looking at your answers, what can you say about the change in marginal cost as the output level keeps increasing? 13. Suppose that the inverse market demand is given by P = 200?5Q. Assume now that the market is served by two …rms (Cournot Model). Marginal cost of production of …rm 1, c1 is 5 and marginal cost of production of …rm 2, c2 is 7. In other words, …rm 1 is the low cost …rm and …rm 2 is the high cost …rm. Derive the following for this market: (i) equilibrium price (P) (ii) equilibrium output of the …rst …rm (q1) (iii) equilibrium output of the second …rm (q2). (iv) equilibrium total supply in the market (Q = q1 + q2) (v) equilibrium output of the …rst …rm (1 ) (vi) equilibrium output of the …rst …rm (2 ) 14. Consider the demand function of Question #13. Suppose the two …rms are competing in a leader-follower fashion (Stackelberg Model) where Firm 1 is the Leader and Firm 2 is the follower. Marginal cost of production of …rm 1, c1 is 5 and marginal cost of production of …rm 2, c2 is 7. In other words, …rm 1 is the low cost …rm and …rm 2 is the high cost …rm. Derive the following for this market: (i) equilibrium price (P) (ii) equilibrium output of the …rst …rm (q1) (iii) equilibrium output of the second …rm (q2). (iv) equilibrium total supply in the market (Q = q1 + q2) (v) equilibrium output of the …rst …rm (1 ) (vi) equilibrium output of the …rst …rm (2)