Mathematical Models

Question 1

1. Describe the model on Decision Making for innovations and how to solve it.

Answer 1

• the agent decides on the starting time of an RnD project 𝑻𝟎 and the development strategy 𝑻𝑬 which maximizes the expected profits
• if 𝑻𝑬 is already set, the agent chooses the 𝑻𝟎 which maximises the ecpected benefit from the project
• important: costs go up with time (concave fct)
• Solving the model: Profit is the expected benefit from the project minus the total cost. –> integrate –> differentiate, set equal to zero –> solve for optimal 𝑻𝟎

Question 2

1. How does the introduction of competition affect the model on Decision Making for innovations?

Answer 2

• costs are lower when starting later, but competition pushes innovation to an earlier start
• imitators get a share of the market
• growth in competition can lead to earlier optimal point TE and an unprofitable development
• -> Comp works as driver for innovationm speed of development increases, but profit also decreases until development becomes unprofitable

Question 3

2. How can risk propensity be measured to compare agents?

Answer 3

• Arrow Pratt measure: negative quotient of the second order and the first order derivative of the utility function
• a positive coefficient stands for a risk averse agent

Question 4

2. How can the certainty Equivalent and the Risk premium be calculated and interpreted?

Answer 4

• CE is the secure payoff which as the same utility as the expected utility of the lottery
• RP is the difference between the CE and the expected value of the lottery –> Excess return an agent needs to take on the lottery
  RP>0 –> Risk-averse investor
  RP<0 –> Risk-seeking investor

Question 5

3. Describe the model on Firm formation based on risk aversion and how to solve it.

Answer 5

• agents are ordered by risk aversion (r) and become either entrepreneurs or workers
• become workers, if utility from certain wage is larger than utility from uncertain profit minus wage paid to workers (and other way around)
• market clearing: if the amount of workes who chose to be workers equals the amount of labour entrepreneurs need to maximise their utility
• Unique euilibrium: more risk averse people choose to be workers and less risk averse choose to be E. 0
• entrepreneur labour demand: integral from 0- 𝛼∗ [𝑙 (𝛼, 𝑤 ∗)] 𝑑𝛼
• workers labour demand: 1 − 𝛼∗
• -> have to be equal

Question 6

4. Describe the model on Moral hazard in teams and how to solve it.

Answer 6

• entrepreneurs maximise their private utility: share of monetary team outcome - private non-monetary costs
• solution: pareto-optimal allocation of work share of each worker, so that nobody can be better off from different allocation
• sharing rule: sum of all shares is the overall profit: the whole profit is allocated

Question 7

4. What is the difference between pareto-optimality and a Nash Equilibrium

Answer 7

Pareto-optimal: There is no player who´s payoff could be increased without decreasing the payoff of another player.
Nash Equilibrium: With players choosind strategies conditional on other players choices: no player can increase his payoff by deviating to another strategy

Question 8

4. What is the pareto-optimal alocation of Holmströms model on moral hazard in teams? Also: show if there is a pareto optimal NE

Answer 8

• Pareto-optimal solution:
• max: overall team outcome (as sum of S=x) minus sum of individual costs of each member
  –> PO if d/da(x(a)) - v’(a) = 0
  -test if allocation is a Nash-Equilibrium: is there any share, where some player can increase his payoff:
  S(𝑥, (𝑎∗)) − 𝑣𝑖 (𝑎𝑖∗) ≥ 𝑠𝑖 (𝑥( 𝑎1∗, … , 𝑎𝑖, … , 𝑎𝑛∗)) − 𝑣𝑖(𝑎𝑖)
  –> NE if si’(xa)d/da(x(a)) - v’(a) = 0
  –> for a PO NE the two FOCs lead to si’(xa)=1 , contradiction to by budget constraint implied sum of si’(xa)=1 –> there is no pareto optimal NE
  –> weeken the budget constraint so that not all output is distributed –> than there is a PO NE: output is distributed in share Bi if the team output is greater than a threshold, and not distributed otherwise

Question 9

How can Holmströms model on team production be critizised?

Answer 9

• not an enforceable model, as there will always be incentives to renegotiate after failure
• less input workers will recieve more in relation
• Incentive scheme leads to subsidizing of „weak“ team members at the cost of „strong“ ones. –> adverse selection
• Revelation of private costs is necessary which is difficult to
  accomplish!
  – Costs would be overstated, which is rational from an individual perspective but prevents cooperation.