# A Competitive Firm A Competitive Firm Seeks To Maximize Where 946 8712 0 1 And Time 1946815

A competitive firm A competitive firm seeks to maximize

Where β ∈ (0, 1), and time-t revenue Rt is

Where pt is the price of output, and yt is the time-t output of the firm. Here .5d(yt+1 − yt)2 measures the firm’s cost of adjusting its rate of output. The firm starts with a given initial level of output y0. The price lies on the market demand curve

Where Yt is the market level of output, which the firm takes as exogenous, and which the firm believes follows the law of motion

With Y0 as a fixed initial condition.

a. Formulate the Bellman equation for the firm’s problem.

b. Formulate the firm’s problem as a discounted optimal linear regulator problem, being careful to describe all of the objects needed. What is the state for the firm’s problem?

c. Use the Matlab program olrp.m to solve the firm’s problem for the following parameter values: A0 = 100, A1 = .05, β = .95, d = 10, H0 = 95.5, and H1 = .95. Express the solution of the firm’s problem in the form

giving values for the hj’s.

d. If there were n identical competitive firms all behaving according to equation (5), what would equation (5) imply for the actual law of motion (4) for the market supply Y?

e. Formulate the Euler equation for the firm’s problem.