Download presentation
Presentation is loading. Please wait.
1
microeconomics spring 2016 the analytics of
final solutions a monopolistic bank ………….1 software upgrades ………….2 competition in prices ………….4 price customization ………….5 electricity markets ……..….8 spring 2016 microeconomics the analytics of constrained optimal decisions
2
marginal cost/revenue
microeconomics final solutions the analytics of constrained optimal decisions a monopolistic bank marginal cost/revenue ► Supply function (with Deposits as “quantity” and interest rate on Deposits as “price”): D = krD or equivalently rD = D/k ► Demand function (with Loans as “quantity” and interest rate on Loans as “price”): L = A – BrL or equivalently rL = A/B – L/B ► The supply curve is basically the marginal cost of the bank thus: MC(D) = D/k ► For a demand function P = a – bQ the marginal revenue is MR = a – bQ, thus MR(L) = A/B – L/B monopolist solutions ► The monopolist will choose it’s optimal “quantity” such that MC = MR. In the bank’s case there might be certain restrictions on how deposits are made available for loans, in particular one restriction is that only fraction f of deposits can be loaned out: L = fD. ► Thus the monopolist will solve: MR(L) = MR(D) A/B – L/B = D/k A/B – fD/B = D/k D = (kA)/(B + 2fk) L = fD L = fD L = fD L = f(kA)/(B + 2fk) Remark: for f = 1 we get the solution for the case in which all deposits are made available for loans. 2016 Kellogg School of Management final solutions page | 1
3
microeconomics profit maximization software upgrades final
solutions the analytics of constrained optimal decisions software upgrades profit maximization ► The monopolist maximizes its profit by setting the quantity such that the resulting marginal revenue equals the corresponding marginal cost: MR = MC. Remember that for a demand P = a – b∙Q the marginal revenue is MR = a – 2b∙Q monopoly price P(Q) ►Marginal revenue is: MR = 100 – 2Q ► Marginal cost is: MC = 0 ► Profit maximization gives: 100 – 2Q = 0 with QM = 50 ►The market price is obtained from the demand function for the quantity QM as: PM = 100 – QM = 50 monopoly quantity (PM ) MR(Q) MC(Q) (QM ) ► Graphically we look for the intersection of the marginal revenue and marginal cost curves – this will give the quantity QM that maximizes monopolist’s profit. The price charged by the monopolist is found on the demand curve for the quantity QM. 2016 Kellogg School of Management final solutions page | 2
4
microeconomics profit maximization software upgrades final
solutions the analytics of constrained optimal decisions software upgrades profit maximization ► The amount received next year is $40 which is valued at 0.9∙$40 = $36 this year. ► There is an additional revenue of $36 to the MR calculated above for each additional unit sold thus the marginal revenue is now MRnew(Q) = 136 – 2Q ► Graphically only the new marginal revenue line is relevant; we can drop the initial demand and initial marginal revenue. ► With a marginal cost of zero the profit maximization gives: 136 – 2Q = 0 with QM = 68 ►The market price is obtained from the demand function for the quantity QM as: PM = 100 – QM = 32 136 MRnew(Q) monopoly quantity MC(Q) 68 (QM ) ► Graphically we look for the intersection of the marginal revenue and marginal cost curves – this will give the quantity QM that maximizes monopolist’s profit. The price charged by the monopolist is found on the demand curve for the quantity QM. 2016 Kellogg School of Management final solutions page | 3
5
microeconomics bertrand solution myopic solution competition in prices
final solutions the analytics of constrained optimal decisions competition in prices bertrand solution ► Using the Excel application it is immediate to find: P1 = P2 = 40 ► The reaction functions are now: firm 1: P1 = ·P2 firm 2: P2 = P1 The solution is P1 = 40 and P2 = 40 (same as in Task 1 but for completely different reason) ► We get the same answer because the reaction function for the second firm is P2 = P1 and so it will produce in equilibrium the same price for both firms. If we pick a difference reaction function, say a “discount” reaction function, e.g. P2 = P1 – 1, the results in Task 1 and Task 2 will no longer be the same. myopic solution 2016 Kellogg School of Management final solutions page | 4
6
separate markets and pooled solution
microeconomics final solutions the analytics of constrained optimal decisions price customization separate markets and pooled solution Type 1 Type 2 Pooled Demand: Q1 = 5000 – 4P1 Inverse demand: P1 = 1250 –Q1/4 Marginal rev.: MR1 = 1250 – Q1/2 Marginal cost: MC1 = 200 Optimal decision: MR1 = MC1 1250 – Q1/2 = 200 → Q1 = 2100 P1 = 1250 – 2100/4 = 725 Demand: Q2 = 4000 – 16P2 Inverse demand: P2 = 250 – Q2/16 Marginal rev.: MR2 = 250 – Q2/8 Marginal cost: MC2 = 200 Optimal decision: MR2 = MC2 250 – Q2/8 = 200 → Q2 = 400 P2 = 4000 – 400/16 = 225 Demand: Q = Q1 + Q2 = 9000 – 20P Inverse demand: P = 450 – Q/20 Marginal rev.: MR = 50 – Q/10 Marginal cost: MC = 200 Optimal decision: MR = MC 450 – Q/10 = 200 → Q = 2500 P = 450 – 2500/20 = 325 2016 Kellogg School of Management final solutions page | 5
7
limited supply solution
microeconomics final solutions the analytics of constrained optimal decisions price customization limited supply solution Type 1 Type 2 ► How many units? We have to find the Q1 for which MR1 = 250: 1250 – Q1/2 = 250 Q1 = 2000 Since we have 2200 units available it follows that some units will also be sent to Type 2 store. How many units? 1250 MR1 250 250 MR2 2500 2000 2000 All these units offer a higher marginal revenue if sent to Type 1 store than to Type 2 store. For each unit you have to decide to which type of store to send it… ► What’s the decision criteria? Send it to store type that offers the highest MR for that particular unit… From the diagrams it obvious that you’ll start sending units to Type 1 stores (since you get MR higher than 250 for at least the first few units) ► But once the marginal revenue from sending units to Type 1 stores hits 250 (or slightly below) you should start sending the units to Type 2 store… Again revert to send units to Type 1 store as soon as MR from Type 2 store is below MR from sending to Type 1 store 2016 Kellogg School of Management final solutions page | 6
8
limited supply solution
microeconomics final solutions the analytics of constrained optimal decisions price customization limited supply solution Type 1 Type 2 ► The idea is that as soon as the monopolist realizes that it will send units to both markets it must be the case that at the optimum (the last unit it sends) must satisfy MR1 = MR2 That is 1250 – Q1/2 = 250 – Q2/8 ► But there is a constraint on the total quantity available to distribute between the two stores: Q1 + Q2 = 2200 1250 MR1 250 230 230 MR2 2500 2000 2040 160 2200 ► We get a system of two equations with two unknowns: 1250 – Q1/2 = 250 – Q2/8 Q1 + Q2 = 2200 ► Conclusion: Q1 = 2040 units to Type 1 store and Q2 = 160 units to Type 2 store, in total all 2200 units available for distribution. Notice that the marginal cost plays no role here (sunk cost by now). ► For Type 1 store P1 = 1250 – Q1/4 = 1250 – 2040/4 = 740 ► For Type 2 store P2 = 250 – Q2/16 = 250 – 160/16 = 240 2016 Kellogg School of Management final solutions page | 7
9
microeconomics cournot solution electricity markets final
solutions the analytics of constrained optimal decisions electricity markets cournot solution time line market demand for electricity is initially P = 120 – Q ● where Q is the total electricity supplied to the market by the players Q = Q1 + Q2 + QG Government commits to supply electricity QG with resulting market demand P = [120 – QG] – (Q1 + Q2) ● this initial commitment of electricity by the government basically “reduces” the market size left for the two players the two players compete now in a Cournot model given a demand P = [120 – QG] – (Q1 + Q2) ● the solution to this game follows the same logic as the solution without the Government ► For firm 1, residual demand is P1(Q1) = [120 – QG – Q2] – Q1 from which we derive the marginal revenue MR1(Q1) = [120 – QG – Q2] – 2Q1 ► Since marginal cost is zero we get, from MR1 = MC1 the reaction function for firm 1 as Q1 = 60 – 0.5QG – 0.5Q2 ► For firm 2, residual demand is P2(Q2) = [120 – QG – Q1] – Q2 from which we derive the marginal revenue MR2(Q2) = [120 – QG – Q1] – 2Q2 ► Since marginal cost is zero we get, from MR2 = MC2 the reaction function for bank 2 as F2 = 60 – 0.5QG – 0.5Q1 2016 Kellogg School of Management final solutions page | 8
10
microeconomics cournot solution electricity markets final
solutions the analytics of constrained optimal decisions electricity markets cournot solution ► We have to solve now the system Q1 = 60 – 0.5QG – 0.5Q2 Q2 = 60 – 0.5QG – 0.5Q1 ► The solution is found in the usual way (plug the first equation into the second, solve for Q2, then use this back into the first equation to find Q1) however the algebra is slightly more cumbersome since we have to carry over the extra term for QG. Anyway: Q1(QG) = 40 – QG/3 Q2(QG) = 40 – QG/3 ► The resulting price is thus P(QG) = 120 – (Q1 + Q2 + QG) = 120 – (40 – QG/ – QG/3 + QG) = 40 – QG/6 ► The Government wants a target P* thus solves 40 – QG/3 = P* with solution QG = 120 – 3∙P* ► For P* = 20 we get QG = 120 – 3∙20 = 60 and Q1 = 40 – 60/3 = 20, F2 = 40 – 60/3 = 20 2016 Kellogg School of Management final solutions page | 9
11
microeconomics cartel solution electricity markets final
solutions the analytics of constrained optimal decisions electricity markets cartel solution time line market demand for electricity is initially P = 120 – Q ● where Q is the total electricity supplied to the market by the players Q = QC + QG Government commits to supply electricity QG with resulting market demand P = [120 – QG] – QC ● this initial commitment of electricity by the government basically “reduces” the market size left for the cartel the cartel maximizes its profit given a demand P = [120 – QG] – QC ● the solution to this game follows the same logic as the solution without the Government ► For the cartel with a demand is P(QC) = [120 – QG] – QC the marginal revenue is MRC(QC) = [120 – QG] – 2QC ► Since marginal cost is zero we get, from MRC = MCC the profit maximizing quantity for the cartel is QC = 60 – 0.5QG 2016 Kellogg School of Management final solutions page | 10
12
P(QG) = 120 – (QC + QG) = 120 – (60 – QG/2 + QG) = 60 – QG/2
microeconomics final solutions the analytics of constrained optimal decisions electricity markets cartel solution ► The resulting price is thus P(QG) = 120 – (QC + QG) = 120 – (60 – QG/2 + QG) = 60 – QG/2 ► The Government wants a target P* thus solves 60 – QG/2 = P* with solution QG = 120 – 2∙P* ► For P* = 20 we get QG = 120 – 2∙20 = 80 and Q1 = Q2 = QC/2 = (60 – 80/2)/2 = 10 2016 Kellogg School of Management final solutions page | 11
Similar presentations
