Problem Set 1 Introduction and Math Review

Problem Set 1. Introduction and Math Review EconS 526 1. Consider a monopolist whose total cost function is 𝐶 = 𝑔(𝑥) + 𝑓 where x is output, 𝑔(𝑥) is a function representing variable cost and f is a measure of fixed cost. The monopolist faces an inverse demand such that 𝑝 = ℎ(𝑥) where p is price and the function ℎ(𝑥) is characterized as hx<0 and hxx=0. Assume that the monopolist also faces a tax t for every unit of t sold. a. Set up the problem faced by a profit maximizing firm. b. Solve for the first and second order conditions that ensure profit is maximized. c. Derive a comparative static showing the effect of tax on optimal output. What is the only condition needed for the sign of the comparative static to hold? d. Derive a comparative static showing the effect of fixed cost on optimal output. What is the only condition needed for the sign of the comparative static to hold? a. max(ℎ(𝑥) − 𝑡)𝑥 − 𝑔(𝑥) + 𝑓 𝑥 b. First order condition, (ℎ(𝑥) − 𝑡) + 𝑥ℎ𝑥 − 𝑔𝑥 = 0 Second order condition, 2ℎ𝑥 + 𝑥ℎ𝑥𝑥 − 𝑔𝑥𝑥 < 0 c. Substitute x*(t) into FOC, (ℎ(𝑥 ∗ (𝑡)) − 𝑡) + 𝑥ℎ𝑥 (𝑥 ∗ (𝑡)) − 𝑔𝑥 (𝑥 ∗ (𝑡)) = 0 Take derivative with respect to t, (2ℎ𝑥 + 𝑥ℎ𝑥𝑥 − 𝑔𝑥𝑥 ) 𝑑𝑥 ∗ −1=0 𝑑𝑑 1 𝑑𝑥 ∗ = <0 (2ℎ𝑥 + 𝑥ℎ𝑥𝑥 − 𝑔𝑥𝑥 ) 𝑑𝑑 Only need the necessary condition for the second order condition to hold. d. 0 𝑑𝑥 ∗ = =0 (2ℎ𝑥 + 𝑥ℎ𝑥𝑥 − 𝑔𝑥𝑥 ) 𝑑𝑑 Therefore, fixed cost has no effect on output. 2. A monopolist producing a single output has two types of consumers. If it produces Xa units for consumer type a, then these consumers are willing to pay a price 100-Xa dollars per unit. If it produces Xb units for customer type b, then these customers are willing to pay a price of 1202Xb dollars per unit. The total cost of producing the good is 20X dollars where X is aggregate output produced. a. Determine the optimal output and prices in both types of consumers. b. Check the second order conditions to verify that the output is a maximum. a. max 𝜋 = (100 − 𝑥𝑎 )𝑥𝑎 + (120 − 2𝑥𝑏 )𝑥𝑏 − 20(𝑥𝑎 + 𝑥𝑏 ) 𝑥𝑎 ,𝑥𝑏 FOC, 𝜕𝜕 = 100 − 2𝑥𝑎 − 20 = 0 𝜕𝑥𝑎 𝜕𝜕 = 120 − 4𝑥𝑏 − 20 = 0 𝜕𝑥𝑏 𝑥𝑎 = 40; 𝑥𝑏 = 25. Price of a is 100-40=60. Price of b is 120-2(25) = 70. b. 𝜕2 𝜋 𝜕𝑥𝑎 𝜕𝑥𝑎 = −2; 𝜕2 𝜋 𝜕𝑥𝑏 𝜕𝑥𝑏 = −4; 𝜕2 𝜋 𝜕𝑥𝑎 𝜕𝑥𝑏 = 𝜕2 𝜋 𝜕𝑥𝑏 𝜕𝑥𝑎 =0 First leading principal is negative at -2. Second leading principal is positive at 8. 3. Under current tax law, some individuals can save up to $2000 a year in Individual Retirement Account (IRA), a savings vehicle that has an especially favorable tax treatment. Consider an individual at a specific point in time who has income Y, which she wants to spend on 𝛽 consumption C, IRA savings, S1, or ordinary savings S2. The utility function is 𝑢 = 𝑆1𝛼 𝑆2 𝐶 𝛿 . The budget constraint is 𝐶 + 𝑆1 + 𝑆2 = 𝑌 and the limit on IRA contribution is L. a. Set up the constrained optimization problem of the individual whose limit L is not binding. Solve for the first order conditions and second order conditions. b. Using the first order conditions, derive the demand functions for C, S1 and S2 for a consumer whose limit L is not binding. Derive the comparative static showing the effect of an increase in Y on C, S1 and S2. c. Set up the constrained optimization problem of the individual whose limit L is binding. Solve for the first order conditions. d. Derive the demand functions for C, S1 and S2 for a consumer whose limit L is binding. a. The problem becomes, 𝛽 max 𝑢 = 𝑆1𝛼 𝑆2 𝐶 𝛿 𝑠. 𝑡. 𝐶 + 𝑆1 + 𝑆2 = 𝑌 𝑆1 ,𝑆2 ,𝐶 The Lagrange is, 𝛽 𝐿 = 𝑆1𝛼 𝑆2 𝐶 𝛿 + 𝜇(𝑌 − 𝐶 − 𝑆1 − 𝑆2 ) The FOCs are, 𝜕𝜕 𝜕𝑆1 𝜕𝜕 𝛽 = 𝛼𝑆1𝛼−1 𝑆2 𝐶 𝛿 − 𝜇 = 0 𝛽−1 𝛿 = 𝛽𝑆1𝛼 𝑆2 𝐶 −𝜇 =0 𝜕𝑆2 𝜕𝜕 𝛽 = 𝛿𝑆1𝛼 𝑆2 𝐶 𝛿−1 − 𝜇 = 0 𝜕𝐶 𝜕𝜕 = 𝑌 − 𝐶 − 𝑆1 − 𝑆2 = 0 𝜕𝜇 The SOCs are, 𝜕2 𝐿 𝜕𝑆12 = (𝛼 − 1)𝛼𝑆1𝛼−2 𝑆2 𝐶 𝛿 < 0; 𝜕2 𝐿 𝜕𝐶 2 = (𝛿 − 1)𝛿𝑆1𝛼 𝑆2 𝐶 𝛿−2 < 0; 𝜕2 𝐿 𝜕𝑆22 𝜕2 𝐿 𝜕𝜇2 b. 𝛽 𝛽−2 𝛿 = (𝛽 − 1)𝛽𝑆1𝛼 𝑆2 𝛽 = 0; 𝜕2 𝐿 𝜕𝑆1 𝜕𝜕 = −1; 𝐶 < 0; 𝜕2 𝐿 𝜕𝑆2 𝜕𝜕 𝜕2 𝐿 𝜕𝑆1 𝜕𝑆2 𝜕2 𝐿 𝜕𝑆1 𝜕𝑆2 𝜕2 𝐿 𝜕𝑆1 𝜕𝐶 = −1; 𝛽−1 𝛿 = 𝛽𝛽𝑆1𝛼−1 𝑆2 𝛽−1 𝛿 = 𝛽𝛽𝑆1𝛼−1 𝑆2 𝛽 𝐶 ; 𝐶 ; = 𝛿𝛼𝑆1𝛼−1 𝑆2 𝐶 𝛿−1 ; 𝜕2 𝐿 𝜕𝐶𝜕𝜕 = −1 𝜕2 𝐿 𝜕𝑆1 𝜕𝐶 𝜕2 𝐿 𝜕𝑆2 𝜕𝐶 𝜕2 𝐿 𝜕𝑆2 𝜕𝐶 𝛽 = 𝛿𝛼𝑆1𝛼−1 𝑆2 𝐶 𝛿−1 ; 𝛽−1 𝛿−1 = 𝛿𝛽𝑆1𝛼 𝑆2 𝐶 𝛽−1 𝛿−1 = 𝛿𝛽𝑆1𝛼 𝑆2 𝐶 Simultaneously solving for the four equations yields, 𝑆1 = 𝛼𝑌 ;𝑆 𝛼+𝛽+𝛿 2 = 𝛽𝑌 ; 𝛼+𝛽+𝛿 𝐶= Using the optimal solutions, 𝜕𝑆1 𝜕𝜕 𝛿𝑌 𝛼+𝛽+𝛿 = 𝛼 𝛼+𝛽+𝛿 > 0; 𝜕𝑆2 𝜕𝜕 = 𝛽 𝛼+𝛽+𝛿 > 0; 𝜕𝜕 𝜕𝜕 = 𝛿 𝛼+𝛽+𝛿 >0 ; ; 𝜕2 𝐿 𝜕𝑆1 𝜕𝜕 𝜕2 𝐿 𝜕𝑆2 𝜕𝜕 𝜕2 𝐿 𝜕𝐶𝜕𝜕 = −1 = −1 = −1 c. The problem becomes, 𝛽 max 𝑢 = 𝐿𝛼 𝑆2 𝐶 𝛿 𝑠. 𝑡. 𝐶 + 𝐿 + 𝑆2 = 𝑌 𝑆2 ,𝐶 The Lagrange is, 𝛽 𝐿 = 𝐿𝛼 𝑆2 𝐶 𝛿 + 𝜇(𝑌 − 𝐶 − 𝐿 − 𝑆2 ) The FOCs are, 𝜕𝜕 𝛽−1 𝛿 = 𝛽𝐿𝛼 𝑆2 𝐶 −𝜇 =0 𝜕𝑆2 𝜕𝜕 𝛽 = 𝛿𝐿𝛼 𝑆2 𝐶 𝛿−1 − 𝜇 = 0 𝜕𝐶 𝜕𝜕 = 𝑌 − 𝐶 − 𝐿𝛼 − 𝑆2 = 0 𝜕𝜇 d. So now, 𝑆1 = 𝐿; 𝑆2 = 𝛽𝑌 ; 𝛽+𝛿 𝐶= 𝛿𝑌 𝛼+𝛽 4. Suppose that a firm has its policies determined by a manager whose objective function is to maximize sales, i.e. revenues, without letting profit drop below some fixed level, m. Let R(y,a) denote the firm’s revenue when the level of production is y and the advertising level is a. Let C(y) denote the cost of manufacturing y units of output and p is the per unit price of advertising. We assume that 𝜕𝜕 𝜕𝜕 > 0, 𝜕𝜕 𝜕𝜕 > 0 and 𝜕𝜕 𝜕𝜕 > 0. a. Set up the constrained optimization problem of the firm. b. Derive the first order conditions that maximize the firm’s objective and write the optimal solution of 𝑎∗ and y* as a function of parameters. c. What is the effect of an increase in the price of advertising on revenue? (Hint: use the envelope theorem) a. b. max 𝑅(𝑦, 𝑎) 𝑦,𝑎 𝑠. 𝑡. 𝑅(𝑦, 𝑎) − 𝐶(𝑦) − 𝑝𝑝 ≥ 𝑚 max 𝐿 = 𝑅(𝑦, 𝑎) + λ (𝑅(𝑦, 𝑎) − 𝐶(𝑦) − 𝑝𝑝 − 𝑚) 𝑦,𝑎 𝜕𝜕 𝜕𝜕 𝜕𝜕 = (1 + λ ) − =0 𝜕𝜕 𝜕𝜕 𝜕𝜕 𝜕𝜕 𝜕𝜕 = (1 + λ ) −𝑝=0 𝜕𝜕 𝜕𝜕 λ (𝑅(𝑦, 𝑎) − 𝐶(𝑦) − 𝑝𝑝 − 𝑚) = 0 Solving for y and a yields 𝑦 ∗ = 𝑦(𝑝, 𝑚) and 𝑎∗ = 𝑎(𝑝, 𝑚). c. 𝑑𝑑 𝜕𝜕 = = −𝑎λ 𝑑𝑑 𝜕𝜕 So revenue decreases by 𝑎λ for every unit increase in price of advertising. 5. Identify which of the following functions are homogeneous and if they are homogeneous, determine the degree of homogeneity. Provide a short proof for each answer. a. 3𝑥 5 𝑦 + 2𝑥 2 𝑦 4 − 3𝑥 3 𝑦 3 b. 𝑥 0.5 𝑦 −0.5 + 3𝑥𝑦 −1 + 7 c. 𝑥 0.75 𝑦 0.25 + 6𝑥 + 4 d. 3𝑥 5 𝑦 + 2𝑥 2 𝑦 4 − 3𝑥 3 𝑦 4 e. 𝑥 0.75 𝑦 0.25 + 6𝑥 f. a) b) c) d) e) f) 𝑥 2 −𝑦 2 𝑥 2 +𝑦 2 +3 HOD 6 HOD 0 Not homogeneous Not homogeneous HOD 1 HOD 0