MacroEcon, Stock Prices, Recurrence Relations (and Linear Algebra?)

[bpr]

Homework Statement

This is a question from an upper level econ course that is giving me quite a bit of trouble. Fluency in linear algebra is assumed for the course. I'm taking a linear algebra course for the first time this semester so I'm still scrambling to learn the basics. If anyone has a good resource to learn linear algebra I would appreciate that as much as any help on this particular problem. So... {subscript} The price of a stock obeys the equations: (1) P{t} = y{t} +β* P{t+1}.
It is "guessed" that equation (1) implies P{t} = Hx{t} + cλ^t. A, G, H are matrices, c and λ are scalars. β is the discount factor.
Solve for H, c, and lambda - are they unique?

Homework Equations

Equations x{t+1} = Ax{t}, y{t} = Gx{t} are also given.

The Attempt at a Solution

I found P{t+1} and substituted in the other equations to get: Hx{t} +cλ^t = Gx{t} + β(HAx{t} + cλ^(t+1)) assuming cλ^t = βcλ^(t+1) (therefore beta =1/lambda) H = G(I-βA)^-1
I'm told this is incorrect but it was my best guess so far.. My next guess would be to put P{t+1} into a geometric series which would solve to a form that looks like cλ^t and then putting the rest into a matrix form. But I feel unfortunately out of my element. Any and all help is greatly appreciated, thank you in advance!

 

[mighty2000]

Hello,
Thank you for reaching out for assistance with this problem. Linear algebra can be a challenging subject, especially when it is new to you. It is great that you are seeking additional resources to help you learn the basics. One good resource for learning linear algebra is the textbook "Linear Algebra and Its Applications" by David C. Lay. This textbook is often used in introductory linear algebra courses and provides clear explanations and examples.

Now, let's take a look at the problem you are trying to solve. It appears that you are on the right track with your attempt at a solution. However, there are a few things to keep in mind. First, when solving for H, c, and lambda, you are trying to find a solution that satisfies all of the given equations. This means that you should not make any assumptions about the relationship between c and lambda. Instead, you should treat them as separate unknowns and solve for each of them individually.

Additionally, when substituting P{t+1} into the other equations, be sure to use the correct form for P{t+1}. In this case, P{t+1} = Hx{t+1} + cλ^(t+1). Also, when substituting P{t+1} into the equation for P{t}, keep in mind that y{t} = Gx{t+1} (not Gx{t}). This will give you the equation Hx{t} + cλ^t = Gx{t+1} + β(HAx{t+1} + cλ^(t+1)).

From here, you can continue with your approach of solving for H using the equation for P{t}. Then, you can solve for c and lambda by plugging in the values you found for H and using the equation for P{t+1}. Keep in mind that you may need to use the matrix inverse to solve for H, and you can use the geometric series to simplify the equation for P{t+1}.

In terms of uniqueness, it is possible that there may be more than one set of values for H, c, and lambda that satisfy the given equations. However, if you follow the steps outlined above, you should be able to find a unique solution.

I hope this helps guide you in the right direction. If you have any further questions or need clarification, please don't hesitate to ask. Good