Solve the following questions and draw well labeled graphs where need be

1. Let u(q1, 42) = Ing1 + 92 be the (direct) utility function, where q and oz are two goods. Denote p and pa as the prices of those two goods and let M be per period money income. Derive each of the following: (a) the ordinary or Marshallian demand functions of = di(p1, p2, M) for i = 1, 2; (b) the compensated or Hicksian demand functions of = hi(pi, pa, M) for i = 1,2 ; (c) the Indirect Utility Function UO = V(p1, p2, M); (d) the Expenditure Function E(p1, p2, (70). (e) Draw a diagram of the solution. There should be two graphs, one above the other; the first containing the indifference curves and budget constraint that character- ize the solution to the consumer's choice problem; the second characterizing the demand functions.3. A consumer has the following utility over childcare c (measured in hours) and food f (measured in units): u(c, f ) = cift. The price of childcare is p. = 2, the price of food is py = 4 and income is M = 20. (a) What is the consumer's ordinary or uncompensated demand for childcare and food?4. Hillary spends her money on two goods q and 92, with prices p, and pa. Her ordinary demand for good 1 is estimated to be; 41 = di(pi, ma, M) = M - - 1. P1 [You need the Slutsky decomposition (i.e., equation) for part of this question.] (a) What is her ordinary or Marshallian demand for good 2?5. You are going to spend $100 on either new CD's (good of) which cost $10 or old CD's (good 42) which cost only $5. You like a bit of old and new music, so your utility function has the Cobb-Douglas form: 0.5 0.5 41 42 Given this utility function, you initially purchase 5 new CD's and 10 old CD's. You may want to verify this.] (a) If the price of old CD's falls to $2, how many do you purchase of each?6. The government wants to raise a specific amount of revenue with either (1) an excise tax t or (2) a lump-sum tax 7. Assuming that preferences for the taxed good and all other goods are strictly convex, and assuming that the lump-sum tax must raise the same exact amount of revenue as the per unit excise tax, which tax do people prefer? Draw a carefully labelled diagram to illustrate your answer.Problem 1. Consumer's surplus Mattias has quasilinear preferences and his demand function for books is B = 15 - n.5p. a} Write the inverse demand function h} Mattias is currently oonsu ming 1i] books at a price of 1D kr. How much money would he be willing to pay to have this amount, rather than no books at all?r What is his level of consumer's surplus? Problem 1. Consumer's surplus I11 [LEI Suppose Birgitta has the utility function U = v.1 x2 . She has an income of 1m and P1: 1 and P2: 1. Calculate compensating and equivalent variation when the price of 3:1 increases to 2. Also. try to estimate the change in consumer's surplus measured by the area below the demand function. Problem 3. Consumer's surplus Explain the concept of "consumer surplus" in words and illustrate by a diagram. Problem 4. Consumer's surplus The inverse demand curve {the demand curve but with p instead of :1 on the left hand side] is given by p{q]=l-1q. The consumer consumes five units of the good to}. a]: How much money would you have to pay to compensate her for reducing her consumption to zero? (The consumer is not paying anything for the goods} b] Suppose now that the consumer is buying the goods at a price of ED per unit. If you now require her to reduce her purchases to zero. how much does she need to get compensated? Hint: The number you will nd is the net consumer's surplus. Problem 5. Consumer's surplus New housing is planned in Karlstad but the location where it is to be built is used as a popular recreation area for people in neighbouring pans of the city. In order to decide whether to build or not, the city authorities want to make a survey to measure the decrease in welfare due the loss of this recreation area. They are told by an economist that two measures are possiblel compensating variation {ch and equivalent variation {EU}. a] How should they formulate the question if they want to measure the compensating variation? b] How should they phrase it if they want to measure the equivalent variation? Problem 3. Market demand Find the price elasticity of demand for the following demand functions. a) D(p)=30-6p b) D(p)=60-p c) D(p)=a-bp d) D(p)=40p2 e) D(p)=Apt f) D(p)=(p+3)-2 Problem 1. Equilibrium Suppose we have the following demand and supply equations D(p) = 200 - p S(p) = 150 + p a. What is the equilibrium price and quantity? b. The government decides to restrict the industry to selling only 160 units by imposing a maximum price and rationing the good. What maximum price should the government impose? c. The government doesn't want the firms in the industry to receive more than the minimum price that it would take to have them supply 160 units of the good. Therefore, they issue 160 ration coupons. If the ration coupons were freely bought and sold on the open market, what would be the equilibrium price of these coupons? d. Calculate the dead-weight loss from restricting the supply of the goods. Will the dead-weight loss increase or decrease if the government would not allow the coupons to be sold on the open market? Problem 2. Equilibrium The demand curve is qD = 100 - 5p and the supply curve is q5 = 5p. a. A quantity tax of 2 kr per unit is placed on the good. Calculate the dead-weight loss of the tax. b. A value (ad valorem) tax of 20 % is placed on the good. Calculate the dead-weight loss of the tax. Problem 3. Equilibrium Assume that both demand and supply for a good are linear functions of its price: D(p) = a + bp, a > 0, b0 a) Draw curves that fit this description in a diagram. b) Assume that a tax t per unit has to be paid by the consumer. Show the effects on demand, supply, equilibrium price, quantity consumed and consumer and producer welfare in your diagram. c) Assume instead that an equally large tax has to be paid by the producer. What are the effects now on demand, supply, equilibrium price, quantity consumed and consumer and producer welfare. (Use a diagram to illustrate.) Problem 1. Intertemporal choice Suppose that a consumer has an endowment of 200.000 kr each period (period 1 and 2). He can borrow money at an interest rate of 200%, and he can lend money at a rate of 0%. a. Illustrate his budget set. b. The consumer is offered an investment that will change his endowment to m = 300.000 and m =