Provide, step by step solutions to the given questions.

Firms in the perfectly competitive constant cost widget industry have access to technology that allows them to produce widgets at a cost of C(q) = 300 + 10q + 3q2, with sunk costs of 225. Demand for widgets is Q = 1300 - 10P. a) The industry is currently in a long-run competitive equilibrium. What is the market price and quantity that each firm produces? How many firms are there currently operating in the industry. b) Illustrate the current state of the industry on carefully drawn and labeled graphs of the typical firm's marginal and average costs and the market. c) What is the short-run market supply curve for the industry? d) Suppose that demand for widgets changes to Q = 1540 - 10P. What will happen to the price in the short-run? What will happen to the price in the long-run?(i) Describe briefly the Vasicek one-factor model of interest rates and its key statistical properties. [4] (ii) According to a particular parameterisation of this model, the instantaneous forward rate applicable at a fixed time T >t implied by the market prices at time t is found to be: f(t,T)=near +(1-e-")+k(1-e-")e-ar where r = T-t and a > 0. Show that, if a humped curve is required for f(t,T), the parameter values must satisfy the condition k> |0 - 4. [5] Here a "humped curve" means one where the value of the function for some intermediate values of T exceeds the values for both : =0 and r = co. In other words, there will be a maximum value for some positive value of r . (iii) Describe briefly the main advantages and limitations of the Vasicek model. [4] [Total 13] Claims occur according to a compound Poisson process at a rate of % claims per year. Individual claim amounts, X , have probability function: P(X =50) =0.8 P(X =100) =0.2 The insurer charges a premium at the beginning of each year using a 20% loading factor. The insurer's surplus at time t is U(t). Find P[U(2) qH, 2. both projects have positive NPV if the entrepreneur works, but negative if she shirks PHR > qHR > I > PLR + B> qLR +b 3. pledgeable income is higher for project 2 but lower than the cost of investment PH ( R - Ap) B ) qHR > I and Show that monitoring is useful if and only if B-b C