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Consider the standard Solow growth model. Let productiv'rty be denoted by A and let the production function be Y=AF{K,N]. For simplicity, assume that there is no population growth {n20 so that N'zN]. Let depreciation be denoted by d and the savings rate by s. a. Derive the per capita capital accumulation equation for k=KIN and the steady state level of capital per worker, kss. Please show the details of your derivation to earn points. Draw the Solow model graph showing the savings line and the depreciation line and marking the steady state level of k. h. Now, consider the \"AK model\" with production function Y=AK, where A is the exogenous productivity level. Again, assume that there is no population growth in the economy. Follow the similar steps in part a. to derive the capital accumulation equation in terms of capital per capita k. c. Based on the equations you derived in part b, draw a graph similar to that of the Solow growth model to show the steady state of the model in part la. If you cannot nd such a steady state, please explain why. d. Recall that Solow growth model implies convergence in capital per capita. Does the model in part b has this feature? Exercises Line and Equation of a Line/Solving Two Equations 1. Consider a line given by the equation y = x + 2. a) Does this line intersect with a line going through (1, -3) with a slope of 3? If yes, solve for the intersection point. If no, why? b) Does this line intersect with a line going through (0, 4) and (2, 6)? If yes, solve for the intersection point. If no, why? c) Does this line intersect with a line given by the equation 4x - 4y = -8? 2. A policeman has $100 to spend by either buying x donuts ($5/donut) or y cups of coffee ($10/cup). Assuming he spends all $100, write the equation for the line that shows all combinations of donuts and coffee he can consume. (This is called "Budget Line.") Give your answer in both forms of Ax + By = C and y = mx + c. What are the x- and y-intercepts? 3. A line is given by the equation y = -2x + 5. If it is shifted up by 2 units, what is the new equation of this line? Instead, if it is shifted to the left by 2 units, what is the new equation of this line? 4. Repeat question 3 when the equation of a line is given by y = 2x + 5. 5. Suppose y is the distance travelled in miles, and x is the time travelled in hours. If the relationship between distance and time is given by equation y = 3x, what is the unit of the slope? Percentages and Weighted Averages 6. Alice has received her salary raise from $10 to $12.5, what is the percentage change in her salary? What about Bryan who has received his salary cut from $12.5 to $10? 7. In ECON 666, homework, midterm 1, midterm 2 and final exams worth 10, 25, 25 and 40 percent of final grade, respectively. Each item worth 100 points. You have received full points on homework, 90 points for midterm 1, and 80 points for midterm 2. To receive an A, you need to score 85 points. What is the minimum you need to score in final exam to receive A? Area of Triangle, Rectangle, and Trapezoid 8. What is the area of a triangle with the base of 3 units and height of 6 units? 9. What is the area of a rectangle with the width of 4 units and height of 8 units?Problems 1. Suppose a monopolistic local utility company faces a demand curve given by P = 120 - 40. Total cost for this firm is given by TC = 400 + 40, and MC is fixed at $4 per unit. a. Does the technology of a firm represent economies of scale? b. What is the fixed cost? Does this indicate high barriers to entry? c. What is the socially optimal level of production and price? d. Suppose this industry operates as a monopoly. Find the equilibrium price and quantity. e. The government, bowing to public pressure to regulate monopolies, decides to force firms to charge their marginal cost just like they would in perfect competition. How much will the monopolist produce? What is the profit for this monopolist? Is it sustainable? f. Suppose the government instead chooses to force the monopolist to charge a price equal to their average total cost, this monopolist will supply 25 units. What will be their profits? 2. Plastic molding has both industrial and dental uses. Consider a monopolist producer of this good with constant marginal cost MC = 4. The demand curves for the two market segments are given below Dental users P - 100 - 20 Industry users: P - 50 - 0.5Q a. If a monopolist can practice third-degree price discrimination, what price will they set in the two markets? What is the consumer surplus for each market? b. Now suppose the monopolist cannot price discriminate Instead, they must charge a single price in both markets. What price will they charge? c. Is consumer surplus higher or lower without price discrimination? d. (True story) Facing a market like this, one supplier of the plastic molding methyl methacrylate considered mixing arsenic with the product sold to industrial users. You might think about why this could be advantageous to the seller. 3. Anna and Boris went to the state fair together, but now can't find each other. They'd like to meet up, but can go to only one of two events to find the other. They can go to the horse show or to the truck rally. Neither person will enjoy the event if alone, but Amma would prefer the horse show while Boris would prefer the truck rally. This is modeled as a game in the table below Boris Horse Show Truck Kally 2, 1 Anna Horse Show 0.0 Truck Kully 0 , 0 1, 2 a. Does Anna have a dominant strategy? Does Boris? b. What are the equilibria in this game