10. Consider a government with preferences represented by the loss function L = 72 + au2 where it is the inflation rate, u is the unemployment rate, and a is a positive con- stant indicating the strength of the government's preference for low and stable unemployment relative to low and stable inflation. The expectations-augmented Phillips curve linking inflation and unemployment is: U = Un - b(7 - 7) where " denotes inflation expectations, Un is the natural rate of unemployment, and b is a positive constant. Expectations of inflation are assumed to be rational. Assume that monetary policy is able to control the inflation rate it directly. Suppose initially that the government chooses a monetary policy without being restricted by any past commitments (the government acts with discretion). This means inflation expectations ," are taken as given when monetary policy is cho- sen. (a) [3 marks] Derive the first-order condition for the inflation rate ? that mini- mizes the loss function subject to the Phillips curve for given ?". As expectations are formed rationally, everyone anticipates the actions of the cen- tral bank determined in part (a). (b) [3 marks] Find the equilibrium unemployment and inflation rates in terms of a, b, and Un. (c) [4 marks] Suppose political pressure forces the government to shift its focus away from inflation towards unemployment. Interpreting this as an increase in a in the loss function, what are the effects on the equilibrium values of u and ? Explain your findings with reference to the policy ineffectiveness proposition. The government announces it will follow a rule that strictly targets inflation . (d) [4 marks] Assuming that everyone believes the rule will be followed, find the optimal rule for the inflation rate that minimizes the loss function. What is the equilibrium unemployment rate when the government follows the rule? (e) [6 marks] Once everyone chooses inflation expectations consistent with your answer to part (d), suppose the government is now free to choose inflation taking a" as given as it did in part (a). Solve for the resulting values of u and ". By comparing your answer to part (d), explain why the optimal rule for inflation is time inconsistent.Question 2 (28 marks) Suppose the following is the market condition for the market of cupcake. Qd = 900 - 20P, where Qd is the quantity demanded (in units) and P is the price (in dollar term). Qs = -600 + 30P, where Qs is the quantity supplied (in units). (a) Define market equilibrium. Calculate the equilibrium price and equilibrium quantity of the cupcake market. Show your workings. Round your answer in integers if applicable. (4 marks) Calculate the consumer surplus, the producer surplus and total surplus for the market of cupcake. Show your workings. Round your answer in integers if applicable. Is the market condition efficient at the equilibrium? Explain briefly with data support. (8 marks) U Suppose that all cake shops agree to charge $40 a cupcake. What is the quantity planned to exchange in the cupcake market? Calculate the NEW consumer surplus, producer surplus and total surplus respectively. Round your answer in integers if applicable. (7 marks) When comparing with the case in part (b), is the total surplus maximized in part (c) if all cake shops agree to charge $40 a cupcake? Is the market efficient under the condition of part (c)? Explain briefly with data support. (5 marks) Based on your calculation, can you explain why all cupcake shops choose to charge $40 a cupcake on their side? Which party or parties would bear the negative impact? Explain briefly with relevant data. (4 marks) Recommendation: Draw ONE diagram with all relevant data for your reference when you attempt parts (a) to (e). Include this diagram in your answer script though no mark is allocated to this diagram. It helps the marker to understand how you derive all the working steps. - END OF PAPER - Page 4 of 4Problem 3 [24 marks] A competitive firm uses two inputs, capital (k) and labour (), to produce one output, ()). The price of capital, WA, is $1 per unit and the price of labor, w, is $1 per unit. The firm operates in competitive markets for outputs and inputs, so takes the prices as given. The production function is f (k, [) = 340.2510.25. The maximum amount of output produced for a given amount of inputs is y = f (k, !) units. a) Use the method of Lagrange to find the conditional factor demands for cost minimization. [8 marks] b) Find the firm's cost function. [2 marks] c) Would you call this a short-run cost function or a long-run cost function? Explain why. [1 mark] d) Write the equations for the firm's average cost function and marginal cost function. [2 marks] e) Draw the firm's total cost function, average cost function, and marginal cost function on a diagram. Clearly label the axes, the curves, and any key points on the graph (eg., axis intercepts, curve intersections, and minimums) with the numbers specifying the exact prices and quantities at these points. What are the coordinates of the points where the average cost curve and marginal cost curve intersect with the total cost curve? [6 marks] () Does your graph indicate increasing, decreasing, or constant returns to scale? Explain. [1 mark] Hint: Think about the relationship between the total cost function and returns to scale. g) Show the firm's long-run supply function on your diagram and write a supply function for the firm. [2 marks] h) Using your supply function, find the profit maximising quantity if the price of output P = 4. What price would be needed for the firm to supply 18 units of output? [2 marks]14:06 8/16: MI