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# Microeconomics 312: Exam 2 & 3 Review Economics 312

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This 12 page Study Guide was uploaded by Paul Hickey on Tuesday January 5, 2016. The Study Guide belongs to Economics 312 at Arizona State University taught by Brian Goegan in Summer 2015. Since its upload, it has received 93 views. For similar materials see Microeconomics in Economcs at Arizona State University.

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Date Created: 01/05/16

Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 18: Exam 2 Review We will use the following production function for this review: 1/2 1/4 ???? = ???? ???? 1. Derive the firm’s cost function. The cost function is found by solving the firm’s cost-minimization problem: min???????? + ???????? ????.????. ???? = ???? 1/???? 1/4 Step 1: Substitute the constraint into the cost function. 4 4 ???? =???? → min???????? + ???????? ???? 2 ????2 Step 2: Get the First-Order Condition ???? 2???????? 4 = ???? − 3 = 0 ???????? ???? Step 3: Solve for k. 4 4 1/3 2???????? 3 2???????? 2???? 4/3 ???? = ???? 3 → ???? = ???? → ???? = ( ???? ) ???? Step 4: Get l. ????4 ????4 ???? 2/3 4/3 ???? = 2 = 1/3 2 = ( ) ???? ???? 2???? 4/3 2???? [( ???? ) ???? ] Step 5: Plug back into the cost function and simplify. 1/3 2???? 4/3 ???? 2/3 4/3 ???? ????,????,???? = ????( ???? ) ???? + ????( 2???? ) ???? 2???????? 3 1/3 ???????? 3/2 2/3 ???? ????,????,???? = ( ) ???? 4/3+ ( ) ????4/3 ???? 2???? 2 1/3 4/3 0.5 2/3 4/3 ???? ????,????,???? = 2???????? ) ???? + 0.5???????? ) ???? 2 1/3 0.5 2/3 4/3 ???? ????,????,???? = [ 2???????? ) + 0.5???????? ) ] ???? For ease, let’s assume w and r are set to whatever they need to be to make it: 3 ???? ???? = ???? 4/3 4 2. Break it down into variable costs, fixed costs, average costs, and marginal costs. Variable Costs These are anything with a y in it. Which is all of it. Variable costs are: 3 ???? ???? = ???? 4/3 ???? 4 Fixed Costs These are anything with a y in it. So a +5 or +20 would be fixed costs. Here there are none. However, an excellent challenge to prepare you for the exam would be to try and figure out a production function that would give fixed costs. Cause the one on the exam does. Average Costs Average costs are: ???? ????) 3????4/3 3 ???????? ???? = = 4 = ???? 1/3 ???? ???? 4 Marginal Costs To get these, we take the derivative of the cost function with respect to y. ???????? ????) 1/3 ???????? ???? = ???????? = ???? This also represents the minimum price at which the firm is willing to supply the y unit of a th good. That is, if the firm has made 26 units, and we want them to make a 27 , at a minimum we will have to pay them $3, because their cost to make the 27 unit is= 3. Of course, that means that ???? = ????3is the inverse supply curve, since supply represents the minimum price at which the seller will sell that many units of the good. Thus, we get the supply curve by rearranging and solving for y. ???? = ???? 3 3. What happens if this industry is dominated by a monopoly? The monopolist’s profit maximization problem takes demand into account. They want to: max???? ???? ???? − ????(????) ???? 1 Step 1: Enter in the inverse demand function and cost function . Let’s say demand is: ???? = 100 − ????. Then: max 100 − ???? ???? − ???? 1 2 ???? 2 Step 2: Get the first-order condition. ???????? = 100 − 2???? − ???? = 0 ???????? If we wave our math wand: ???? ≈ 33.33 Step 3: Get other relevant variables. ???? = 100 − ???? = 100 − 33.33 = 66.67 ???? ???? = ???????? = 33.33 ∗ 66.67 = 2222.22 1 2 ???? ???? = ????2= 555.56 ???? ????) 555.56 ???????? ???? = = = 16.67 ???? 33.33 ???? = ???? ???? − ???? ???? = 2222.22 − 555.56 = 1666.67 Step 4: Graph it. I’ll do this on the board. 1I am switching this up because the math gets needlessly hard. 2Hope you didn’t miss class! 4. Now let’s try Stackelberg! This model has two firms, where firm 1 is the leader, and firm 2 is the follower. Step 1: Set up the follower’s problem. max???? = 100 − ???? − ???? ???? − ???? 1 2 1 2 2 2 2 Step 2: Get the first-order condition and so2ve for y . ???????? ???????? = 100 − ???? 1 2???? −2???? = 2 2 100 − ???? 1 ????2= 3 Step 3: Enter this into the leader’s problem. 100 − ???? 1 max???? = (100 − ???? − 1)???? − ???? 2 1 3 1 2 1 Step 4: Get the first-order condition and solve. ???????? 100 2 ???????? = 100 − 2???? 1 3 − 3 1 ???? =10 1 ???? = 18.18 1 Step 5: Get other variables of interest. 100 − 18.18 ???? 2 = 27.27 3 ???? = 100 − 18.18 − 27.27 = 54.55 1 ????1= 54.55 ∗ 18.18 − 18.18 = 826.46 2 1 ???? = 54.55 ∗ 27.27 − 27.27 = 1115.75 2 2 Neat! The follower is winning in this case! 5. Ready! Set! Cournot! Cournot has both firms choosing output simultaneously. Step 1: Set up the profit function for both firms. ???? = 100 − ???? − ???? ???? − ???? 1 2 1 1 2 1 2 1 1 ????2= 100 − ???? −1???? ???? 2 ????2 2 2 Step 2: Get first-order conditions for both. ???????? 1= 100 − 2???? − ???? − ???? = 0 → ???? = 100 − ???? 2 ???????? 1 1 2 1 1 3 ???????? 2 100 − ????1 = 100 − ???? 1 2???? 2 ???? =10 → ????2= ???????? 2 3 Step 3: Use the system of equations to solve. 100 − ???? 100 − 1 100 100 1 ???? 1 3 = − − ???? 1 3 3 9 9 8 200 200 ???? = → ???? = = 25 9 1 9 1 8 100 − 25 ????2= 3 = 25 Step 4: Get other relevant variables. ???? = 100 − 25 − 25 = 50 1 ????1= 50 ∗ 25 − 25 = 937.5 2 ???? = 50 ∗ 25 − 25 = 937.5 2 2 Huzzah! 6. Yes We Bertrand! Bertrand has each firm setting the price simultaneously, with the promise that they can meet market demand at that price. Competitively, firms set price equal to marginal cost. Step 1: Get Market Supply Each firm will supply ???? = ????, as this is where p is equal to the marginal cost. Market supply adds these together for all firms, so: ???? ???? = ???? 1 ???? =2???? + ???? = 2???? Step 2: Set this equal to market demand and solve. Well, we have inverse market demand, so let’s reverse to inverse market supply: 1 ???? = 2???? → ???? = 2???? Now set it equal to demand: 1 ???? = 100 − ???? 2 ???? = 66.67 Step 3: Get the market price. 1 1 ???? = ???? = 66.67 = 33.33 2 2 Step 4: Get other relevant variables. ????1= ???? = 33.33 ????2= ???? = 33.33 1 ????1= 33.33 ∗ 33.33 − 33.33 = 555.56 2 1 2 ????2= 33.33 ∗ 33.33 − 23.33 = 555.56 Note also that for both firms, the marginal cost is y, which is equal to the price. That’s a confirmation that we are at the competitive equilibrium. 7. Tonight, we dine in CARTEL! Cartels maximize the profits of the industry. Step 1: Set up the industry profit equation. 1 1 ???? = 100 − ???? −1???? 2)???? 1 ???? 2) − ???? − ???? 2 2 1 2 2 Step 2: Take the first-order conditions. ???????? = ???? + ???? )(−1 + 100 − ???? − ???? )1 − ???? = 0 ???????? 1 1 2 1 2 1 100 − 2????2 → ???? 1 3 ???????? = ???? +1???? 2)(−1 + 100 − ???? − 1 2)1 − ???? 2 0 ???????? 2 100 − 2???? → ???? 2 1 3 Step 3: Use the system of equations to solve. 100 − 2 100 − 2????1 ????1= 3 = 20 3 ???? = 100 − 2 ∗ 20= 20 2 3 Step 4: You know the drill. ???? = 100 − 20 − 20 = 60 ???? = 60 ∗ 20 − 20 = 1000 1 2 1 2 ????2= 60 ∗ 20 − 22 = 1000 Note that out monopolist did $1,666.67 in profits, but this cartel is doing $2,000. Why? 8. Compare the Models Specifically, calculate the marginal cost of the last unit produced in each model and then compare it to the price charged. Model MC P Diff. Monopolist 33.33 66.67 33.33 Stackelberg 27.27 54.55 27.28 Cournot 25 50 25 Bertrand 33.33 33.33 0 Cartel 20 60 40 Why does this matter? What questions could I possibly ask you where you would need to do this? 9. Some other questions. 1 Try this all again, but use the cost function: 2 ???? = ???? + 100. Solve the Cournot for n firms with this new cost function. Use your solution to the Cournot model with n firms to determine the maximum number of firms this market could sustain.

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