Principal agent model in the goods market

Assume the principal-agent model described in chapter 10 (the Benetton case). The agent’s expected value is as follows: \(v(p,q)=u(p,q)T(q),\) where \(u(p,q)=p-\frac{\underline{u}}{(1-q)} \text{ and } T(q)=\frac{1}{t(q)}=\frac{1}{1-q}.\)

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  1. Sketch and explain the timeline of the game.
  2. See the interactive figure. What do the green curves show? What is the sign of the slope of the green curves? Explain.
  3. Calculate the slope of the green curves and show the sign algebraically.
  4. Assume that $\underline{u}=5$ and the price equals 30 ($p=30$). Use the slider to choose that price and value of $\underline{u}$. Drag the horizontal line called ``Agent’s response” to find agent’s best response quality for that price. What is the value of $q$ you observed?
  5. What happens if you decrease the price to 10 ($p=10$)? What is the new best response value of quality, $q$? Use the interactive figure and explain.
  6. What happens if you increase the price to 40 ($p=40$)? What is the new best response value of quality, $q$? Use the interactive figure and explain.
  7. Find the agent’s BRF algebraically.
  8. Assume that $\underline{u}=5$ as before and check the box on the interactive figure to see the agent’s BRF. Verify that the BRF passes through the points you found in parts (d) - (f). Explain why it must pass through those points.
  9. What is the objective of the principal?
  10. Check the second box on the interactive figure to see one of principal’s iso-cost curves. Then, use the slider to find principal’s optimal price offer, given the agent’s BRF. Notice that this point will be the equilibrium point. Hint: Remember what happens to agent’s BRF and principal’s iso-cost line at the equilibrium point.
  11. Find the Nash equilibrium algebraically. Assume again that $\underline{u}=5$. Verify that the equilibrium price and quality is the one you have observed in the interactive graph.
  12. Is the Nash equilibrium point Pareto efficient? Click the third box on the interactive graph to see one extra point on the graph. Use that extra point to illustrate your answer.
  13. (additional practice) Notice that until now we assumed $\underline{u}=5$. Use the slider on the interactive graph to change the value of $\underline{u}$ and see what happens to the model. Why is the equilibrium point you found above not still an equilibrium? What is the new equilibrium point and why? Explain the intuition.

 

Credits for the code tools belong to Chris Makler and credits for the initial graphs used belong to Bridget Diana and Chris Makler.