# Why is there a stochastic discount factor in firms present value?

In a book I am reading, the problem of the firms is described as maximizing current market value of profits.

$$\max \sum_{k=0}^\infty \Delta_{t, t+k} \cdot "\text{profits in period k}"$$

So, basically, it's all profits, discounted back with discount factor $$\Delta_{t, t+k}$$. Except, the discount factor is defined in the book as $$\Delta_{t, t+k} = \beta^{k} U_{c, t+k}/U_{c, t}$$

where $$\beta$$ is a discount rate, and $$U_c$$ is the first derivative of the utility function of the consumers.

What I don't understand is, why are the derivatives of the utility function of consumers with respect to consumption involved in the discount factor? Why is it not just $$\beta$$? The book gives no explanation of where this discount factor came from.

• What makes you think this is stochastic in any way? – Giskard Dec 2 '18 at 17:13
• A stochastic element is an element that is randomly determined. Nothing in that discount factor is randomly determined. Your discount factor here is relatively standard. It is the discounted ration of marginal utility tomorrow/marginal utility today and note that $\beta \in (0,1)$ here shrinks to 0 as $k \to \infty$. So, I care less and less about things that happen further into the future whenever standing in period $t$. – 123 Dec 2 '18 at 18:01