Please answer the attached problem question with a detailed step-by-step process.

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Question 3. Price Competition. (30 marks) Consider the price competition between two rms: Firm 1 and Firm 2. The demand for Firm 1's product can be approximated by the following linear function: Qlipl-le = a1 51P1+d1PL where P1 is the price of Firm l's product and P2 is the price of Firm 2's product. The demand for F irrn 2's product can be approximated by the following linear function: Qriplt P2] = 02 52?: + depr- Firm 1's total production cost is TC(Q1) = 1:10;, and Firm 2's total production cost is TCler = 72(92- Suppose we have the following historical sales data: Observations P1 P2 Q1 Q2 Data Point 1 1000 1000 1500 1500 Data Point 2 300 600 1300 2300 Data Point 3 600 800 2300 1300 (a) (3 marks) Determine the demand function for each rm. (1:) {3 marks) Suppose 71 = 1:2 = 500. Assume both rms simultaneously choose prices P1 and P3, respectively. Determine each nn's optimal pricing strategy. Calculate the equilibrium prots for each rm. Next consider the following three scenarios independently. Scenario 1. Government Tax. Suppose the government imposes a new tax. As a result, both 71 and 72 increase from 500 to 600 The demand functions remain unchanged. (c) (3 marks) Determine each firm's optimal pricing strategy. Calculate the equilibrium prots for each rm. (d) (3 marks) Based on your answers in (b)-(c), discuss how the tax affects (i) the product prices. (ii) the market share of each firm, and (iii) the prots of each rm. Scenario II. Research and Development. Suppose now each firm can make an investment in 85.50 to enhance its product features and create differentiation from its competitor's product. Denote by I: the amount of money Firm 1 spends on R351), and I; the amount of money Firm 2 spends on R&D. For simplicity. assume I; and I: admit the following two values: 30 and $100,000. Given (I:, h), the demand for Firm 1's product is determined by the following equation: 2.:1 h Q1(P1,P2,fl,.fl]=d1+ 5m 01P1+d]4p2, and the demand for Firm 2's product is I ! (92(1),, P2, at} = ea + 2 ' ' 472102 +dga. 5m The coefcients ((11,412, 51.62411. and d2] are the same as that obtained in {a}. and A\" = 1:2 = 500. Consider the following two-stage game. In the rst stage, the two firms simultaneously choose R&D spending 1': and 1'2. respectively. In the sewnd stage. after observing each other's investment in R3513 in the rst stage. both rms simultaneously choose prices P1 and P2, respectively. (e) (6 marks) Find the subgame perfect Nash equilibrium of this tnostage game. (f) (3 marks) Based on your answer in (e). discuss whether the R350 race on product differen- tiation leads to the Prisoner's dilemma between Firm 1 and Firm 2. Scenario III. Incorrect Demand Estimation. Suppose Firm 1 mistakenly believes that Firm 2's product is not a. close substitute of its product. Consequently, Firm 1 ignores the price effect stemming from P2 and thinks that the demand for its product can be approximated by the following linear function: Q1091}: a1 bipl- Moreover. Firm 1 uses Data Points 2 and 3 to estimate its demand. Similariy. Firm 2 makes the same mistake and thinks that the demand for its product can be approximated by the following linear function: C32(102} = '12 52102- Firm 2 also uses Data Points 2 and 3 to estimate its demand. (g) (3 marks) Determine the demand function for each rm. [11) (3 marks] Redo (b) based on the demand functions obtained in [g]. (i) (3 marks) Based on your answers in (b) and (h). discuss how the incorrect demand estimation aects each rm's pricing strategy and prots