Question 1 Consider an economy with / consumers and L goods. For each consumer i e {1, .... / }, the consumption set is R4. Her utility function is given by u'(x') = -) are-Biz, K=1 where ox, ; > 0 for k e {1. ..., L}. As usual we denote r' = (r;, ...,;). Consider price vectors (p1, .... pz) and wealth levels (w], ...,w') for which the solution to the utility maximization problem is interior for every consumer i e {1, .... /}. a.) Derive the Walrasian demand function for good j by consumer i. (Be careful with your calculations. Double-check! It is easy to make mistakes.) b.) What is the slope of consumer i's Engel curve for good j at (p, wi)? c.) Find a condition as general as possible on parameters ox, By, i e {1,..., / }, k E {1. ..., L} guaranteeing the existence of a positive representative consumer. Do we need restrictions on parameters ox, ke {1, .... L}? d.) Consider now the special case with just a single consumer and two goods. The consumer's utility function is given by Derive the wealth-expansion path for a given price vector (P1, p2)- e.) In problem d.), when does the wealth-expansion path intersect the r,-axis and when does it intersect the ry-axis?EF5400 Advanced International Trade Practice Problem Set 1 1. Consider an economy in which there are two countries, i = 1,2, and a continuum of goods, indexed by ke [0, 1]. Goods are produced using labor: Mi (k ) = 4 (1) / m (k) . where a; (k) is the unit labor requirement for producing good & in country i, y (k) is the output of good & produced in country i and l (k) is the amount of labor used in production. Assume that the unit labor requirement functions are an (k) = call-*) 12 (k) = cox The representative consumer in each country has the utility function [ lage ( R ) LA. This consumer is endowed with 4 units of labor where h = 12 = I. (a) Define an equilibrium of the economy. Calculate expressions for all of the equilibrium prices, p.(k), and quantities, c(k), y:(k), and (k). Draw a graph that illustrates the pattern of specialization in production, consumption, and trade. (b) Suppose now each country imposes an ad valorem tariff ~ on imports from the other country. Repeat the analysis of part (a). (c) For the model in part (b), calculate GDP in each country. 2. Consider an economy in which there is a continuum of goods, s E [0, 09). which are ordered so that higher-indexed goods involve more advanced tech- nologies in the production process. An individual consumer has preferences given by u(c) = e- log (c(s, t ) + 1) ds dt.In a double decrement model with two independent decrements (causes), assume the following distributions for the exit times due to decrements 1 and 2 over one year from age x to age x +1: . Decrement 1: , px =1-tq, (uniform exit), 0 0 to derive an approximate formula for the expected present value EPV, of $1 under the policy payable to a policyholder currently aged x within one year at annual interest rate i in terms of x, qx, and 8 = log(1+ i) . Then calculate an approximate EPV of $10,000 payable to a policyholder of current age 55 within next two years, using qx in part (b) above, and i = 0.04. [5]Homework 2 (Due: In class, Tuesday, 09/24/19) 1. a. A zero-coupon bond pays $1000 in five years. i) What is the value of the bond today? The nominal interest rate i is 3%. ii) Price the bond if the interest rate is now 7%. ili) Describe the relationship between the interest rate and the bond price. b. A zero-coupon bond pays $1000 in one year and sells for $920 now. Calculate the yield to maturity of the bond. c. A zero-coupon bond pays $1000 in six years and sells for $750 now. Calculate the yield to maturity of the bond to the nearest basis point (one hundredth of one percent). d. A consol pays $20 each year forever. The consol is currently selling for $160. The consol begins payments next year. What is the yield to maturity of the consol? e. A one-period coupon bond has a face value of SFV and pays an annual coupon amount C. Show that if the bond is priced at par (that is, the bond price equals its face value), then the yield of the bond is the coupon rate c. f. An annual coupon bond has a maturity of twelve years, a face value of $1000, and a coupon rate of 4%. The current market price of the bond is $780. Use a grid search to find the yield to maturity of the bond to the nearest basis point (one hundredth of a percent). Other methods will not receive full credit. Use a spreadsheet program and print a copy of your work. g. A U.S. Treasury note with a face value of $1000 pays interest semiannually at matures in three years. If the coupon rate is 4% and the bond equivalent yield is 5%, find the price of the note. Obey all pricing conventions. h. i) At the grocery store, apples cost 96 cents and bananas cost 92 cents. If a shopper buys twenty apples and ten bananas, what is the total cost at checkout? ii) A one year zero coupon bond with a face value of $100 has a yield of 4.1667%. A similar bond with a maturity of two years has a yield of 4.2572%. If a bond manager holds a bond portfolio with twenty zero coupon bonds that mature in one year and ten zero coupon bonds that mature in two years, what is the value of the bond portfolio? iii) Compare part (i) to part (ii) in a sentence or two. i. Consider the prices for the following zero-coupon bonds: Bond Maturity Face Value Price Bond A 1 year $1 0.98 Bond B 2 year $1 0.94 Bond C 3 year $25 23 Calculate the price of an annual coupon bond that matures in 3 years with a coupon rate of 5% on a face value of $100