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Part 2 Consider the endogenous growth model in which time can be used for work (it) or human capital accumulation (1 15). Output is produced according to Y = zuH, where z is total factor productivity and H is the current stock of human capital. The future human capital stock is given by H ' = M] u)H, where 5 represents eiciency in producing new human capital. The initial (current) human capital stock is H = 100. The economy is characterized by the following parameters: b = 6.1, 2 = 15, and u = 0.83. (a) Calculate future human capital (H '), and the growth rate of human capital. (b) Calculate current output (Y), future output (1\"), and the growth rate of output. How does it compare to the growth rate of human capital? (c) Suppose the government cuts technology funding for education, and as a result b declines to b = 6.0. Calculate the growth rate of output following this change. ((1) Are the growth rates you found in parts a and c the same or different? Explain intuitively why this policy change did or did not a'ect the economy's growth rate. (e) Explain how the decrease in b will affect (if at all) the current levels of human capital, output, and consumption: H, Y, and C. If there is no effect, explain why. (f) Explain how the decrease in b will affect (if at all) the future levels of human capital, output, and consumption: H', Y' , and C' . If there is no effect, explain why. (g) Explain how the decrease in b will affect the growth rates of human capital, output, and consumption. If there is no effect, explain why. (h) Draw a graph of logoutput, ln'), over time for this economy and illustrate how the decrease in b affects current and future output. Exercise # 1. Anna consumes two goods: milk (measured in gallons) and a composite good (measured in dollars). Let Xm represent the gallons of milk that Anna consumes in a given month and let Xc represent her expenditures on the composite good in a given month. Anna's preferences over consumption bundles (Xm,Xc) are summarized by the utility function: U (Xm,Xc) = Xm XE . Anna's monthly income is $400. Let Pm denote the dollar price of a gallon of milk. (a) [10 pts.] Suppose that Pm = $1. What is Anna's optimal consumption bundle? Show your work. Illustrate your answer with a neat and clear diagram showing Anna's budget line and indifference curves. Label the points at which the budget line intersects the axes and identify the optimal bundle. (b) [10 pts.] Suppose now that the local grocery store where Anna regularly shops decides to introduce a discount on milk. Specifically, for each gallon of milk that Anna buys, the grocery store reduces its price from $1 per gallon to $0.50 per gallon, up to a maximum number of 50 gallons of milk per month. If Anna buys more than 50 gallons she has to pay the regular price on every gallon beyond the 50-th. In a neat and clear diagram, graph Anna's budget line. Label the points at which the budget line intersects the axes and determine the coordinates of the kink point. (c) [15 pts.] Suppose now that the price of milk is again Pm = $1 (there are no discounts anymore). Due to a shortage of milk, the price of milk increases from $1 to $2. Describe how to compute the extra income that must be given to Anna in order to compensate her for this increase in the price of milk (i.e., the compensating variation) [Here you are not asked to compute this amount. Simply show which steps you would take to compute it.] Exercise # 2. John has the following demand function for beer Xb = m - 2pb + Pw where X denotes the gallons of beer he demands per month, Po is the dollar price of a gallon of beer, Pw is the dollar price of a bottle of wine, and m denotes John's income. (a) [5 pts.] Is beer an ordinary good in this case? Motivate your answer. [Notice: no credit will be given to yeso type of answers. In order to get credit you need to explain your answer.] (b) [5 pts.] Is beer a substitute for wine in this case? Motivate your answer. [Notice: no credit will be given to yeso type of answers. In order to get credit you need to explain your answer.] (c) [5 pts.] Suppose that the price of a bottle of wine is Pw = $10, and the price of a gallon of beer is po = $15. What is the relative price of a gallon of beer in terms of bottles of wine? (d) [10 pts.] Suppose that m = $100 and that Pw = $10. Compute the loss in John's consumer surplus that occurs when the price of a gallon of beer increases from $15 to $20. Support your analysis with a graph representing John's demand curve and his loss in consumer's surplus. [Remember that to draw a demand curve you need to place p. on the y-axis and x; on the x-axis.] (e) [10 pts.] From point (d) you can see that the loss in consumer's surplus can be decom- posed into two subregions, whose shapes are respectively rectangular and triangular. How can you interpret each of these two subregions? Exercise # 3. Consider the following statements and say whether they are true or false and why. To get credit you should provide a clear justification for your answers. (a) [10 pts.] If two goods are perfect complements the consumer will be just as well off facing a quantity tax as an income tax. (b) [5 pts.] If the price of one good increases the demand for that good always decreases. (c) [5 pts.] The marginal rate of substitution measures the rate at which the market is willing to substitute one good for the other. (e) [5 pts.] An indifference curve represents the collection of all bundles that a consumer can buy. (f) [5 pts.] By definition, a lump sum subsidy to a consumer does not affect his/her consumption behavior.Problem 4.9. Consider an overlapping generations economy where individuals in generation t maximize utility U = In(cit) + #In(c2+1) They work one unit when young, earn a wage wt, save an amount at, and earn interest re+1 on their savings. Ouput is produced with a Cobb-Douglas technology Y = KOL,"" and there is 100% depreciation. The number of individuals in generation t is Le. The size Le of generations t grows at a fixed rate n > 0. The governmnet operates a pay-as-you go social security system as follows: Each period, the young pay a tax Tit = Tut, where 0 0. Suppose Ty = 0 and Ty is set so that the budget equation is satisfied. Show how ke+1 depends on k, and d. Derive k*. Show that a higher d implies a lower &*