Its economics kindly give correct solutions only

Application: Network externalities 1. Consider the following model of demand for a software product such as an operating system that we'll call OS." Consumers have heterogeneous preferences for OS. We'll index these preferences by x which is distributed uniformly from 0 to 1, (x ~ U[0,1]. People with a low x have a strong positive preference for OS whereas people with a high x have much less desire to use OS. Denote by n, 0 n 1 the total proportion of consumers using OS, and let p equal the purchase price. Finally, define the utility of a consumer as: U =(1-x)-p if shebuys OS otherwise Thus, the utility of each purchaser exhibits "network externalities" since it increases with the number of other purchasers. A. Write down the equation that shows the utility of a consumer, & , who is just indifferent between purchasing and not purchasing OS at a given price p. Call this equation (1). B. Notice that since & is indifferent between buying or not buying OS, this implies that everyone with x x is strictly happier not buying OS. Moreover, since x is uniformly distributed between 0 and 1, this implies that n in equation (1) is equal This model is due to Rohlfs, 1974 who developed it to describe the demand for phone services. 3 to x. Hence, rewrite equation (1) solely as a function of p and & and re-arrange so that you have p as a function of & and call this equation (2). Draw a figure that shows price as a function of & with re [0,1] on the X axis. What is unusual about this demand curve? C. Pick some point on the figure s, where i, , that is also consistent with this price? Intuitively, which of these points would you expect to be a more stable equilibrium and why? D. Now consider the production of OS. Since it's a software product, assume for simplicity that the marginal cost of producing the next copy is zero (for software, this is a reasonable approximation). What would be the price of OS in a competitive industry, and what fraction of all potential users n would purchase it at that price? E. Now imagine that OS is owned by a company called MacroSoft that has the sole rights to produce the software. Using equation (2), write down equation (3), the equation for MacroSoft's profits as a function of &, the number of users of OS. Draw a figure showing profits as a function of &. E. Solve MacroSoft's profit maximization problem. What is & , what are profits, and how does the number of users compare to your answer in (D)? F. Does this model suggest that MacroSoft is exploiting monopoly power to build its market share? If not, does it suggest that MacroSoft is behaving in the best interest of consumers? G. Imagine that MacroSoft is about to introduce a new software product that it believes is subject to the same type of network externalities as its OS. What would be the best pricing strategy? Specifically, should it choose a price that yields x users in long run equilibrium, or should it start at a higher price, lower price, etc.? Explain. (Make sure you consider the dynamics of how you get from x = 0 to * = i ). 2. Microsoft's (note: not MacroSoft's) detractors have raised the charge that Microsoft has slowly raised the price of Windows over time (a point that Bill Gates defends against in his article, "Compete, Don't Delete" found in your reading packet). Considering the model above, why is that charge relevant? After all, many companies raise their prices over time (e.g., Volkswagen's aren't as cheap as they used to be either)? 3. In Paul Krugman's article "Entertainment Values" (on your reading list), Krugman writes, "A world in which increasing returns are prevalent is one in which markets are likely to get it wrong." Can you think of one good example of a product or standard that the world (or a given country) has obviously "gotten wrong?" Can you name a product or standard that there is neither a wrong nor a right, but that it's nevertheless crucial that everyone use the same one?Problem 1. Utility maximization. (52 points) In this exercise, we consider a standard maximization problem with an unusual utility function. The utility function is u(x, y) = VI + vy. The price of good z is p, and the price of good y is py- We denote income by M, as usual, with M > 0. This function is well-defined for x > 0 and for y > 0. From now on, assume : > 0 and y > 0 unless otherwise stated. 1. Compute du/Or and Ou/02r. Is the utility function increasing in r? Is the utility function concave in x? (3 points) 2. The consumer maximizes utility subject to a budget constraint. Write down the maximization problem of the consumer with respect to a and y. Explain briefly why the budget constraint is satisfied with equality. (Hint: you can use the answer in point 1) (5 points) 3. Write down the Lagrangean function. (2 points) 4. Write down the first order conditions for this problem with respect to r, y, and A. (4 points) 5. Solve explicitly for r* and y* as a function of pr, py, and M. (8 points) 6. Do the solutions for a* and y* satisfy the positivity constraint, that is, a* > 0 and y* > 0?.(2 points) 7. Are these points maxima of the problem above? Check that the determinant of the bordered Hessian is positive at r*, y*, and A*. (8 points) 8. Use the expression for a* that you obtained in point 6. Differentiate it with respect to M, that is, compute or* /OM. Is this good a normal good? (4 points) 9. We are now interested in the sign of Or*/Op,- That is, we would like to know if the demand function is downward sloping. (at higher prices, a lower quantity of the good is demanded). Argue, using the answer that you gave in point 9 and something you learnt in class, that we know the sign of Or* /Op,. What is this sign? (8 points) 10. Can you guess the solutions for r* and y* for the following maximization problem? Develop your argument. (8 point) max (vi + vy) (1) s.t.pix + pyy = MShort problems. (26 points) In this part, you are required to provide short answers to the problems below. Provide the steps in the derivation of your answer. Short problem 1. (16 points) Consider the utility function if 05 x $ 10 U(x) = (20-x) if 10 U (5)? Is U (20) > U (5)? Are the preferences represented by this utility function monotonic (that is, if y 2 r, then y > x)? (5 points) 3. (Harder) Write down the set of preferences over the numbers between 0 and 20 that this utility function represents. Try to be precise, but you can help yourself using words. [Hint: for each r, y c [0, 20], when is r _ y?] (10 points) Short problem 2. (10 points) 1. Consider the implicit function g (r, y) = y - 1 - In(x * y) = 0 for r > 0, y > 0. (a) Use the implicit function theorem to write down dy/Or.(6 points) (b) Is this derivative well-defined for (x = 1, y = 1)? (4 points)