The Fisher Model

Lorenzo Naranjo

Fall 2026

Utility Functions

Consumption

The theory of finance is concerned with how investors allocate resources over time.
Investors must decide today how much to save, how much to consume, and how to invest their savings.
We would like to determine the best way to maximize the benefit that investors derive from their consumption today and tomorrow.
We denote this consumption bundle by \{C_{0}, C_{1}\}.

Utility

In economics, a very convenient way to rank consumption bundles is to use a utility function.
The idea is to assign a real number to each consumption bundle.
That is, in these notebooks a utility function is a functional U: \mathbb{R}^{2} \rightarrow \mathbb{R}.
A consumption bundle is then preferred to another if the utility number is larger.

Example

Consider the following function U(C_{0}, C_{1}) = \ln(C_{0}) + \ln(C_{1}).
We can compute U(3, 2) = 1.79 and U(2.5, 2.5) = 1.83
The agent prefers consuming 2.5 units today and tomorrow over consuming three units today and two units tomorrow, which we write as (2.5, 2.5) \succsim (3, 2).
Note that U(2, 1) = U(1, 2) = 0.69, which we denote by (1, 2) \sim (2, 1).
Therefore, the consumer is indifferent to some consumption bundles.
The set of all consumption bundles that provide the same utility is called an indifference curve.

The Notion of Order

The value of the utility function is irrelevant since applying any increasing function to a utility function will not change the rankings of consumption bundles.
Since U(C_{0}, C_{1}) = \ln(C_{0}) + \ln(C_{1}) = \ln(C_{0} C_{1}), the new utility function V(C_{0}, C_{1}) = C_{0}C_{1} generates the same rankings of consumption bundles.

Indifference Curves

Figure 1: The figure shows three indifference curves.

Properties of Utility Functions

The figure displays something that we expect to find in real life: utility should increase with consumption.
In the previous example, we found that (1, 2) \sim (2, 1), but of course we would expect (1, 2) \precsim (1, 3).
In other words, the marginal utility of consumption must be positive for consumption in both periods, i.e., \frac{\partial U}{\partial C_{i}} > 0.
Marginal utility should also decrease with consumption, i.e., \dfrac{\partial^{2} U}{\partial C_{i}^{2}} < 0, since each additional unit of consumption can only increase utility at a lower rate.

Some Utility Functions

In finance, it is common to use separable utility functions of the form U(C_{0}, C_{1}) = u(C_{0}) + \beta u(C_{1}).
The choice u(C) = \begin{cases} \frac{C^{1 - \gamma} - 1}{1 - \gamma}, & \text{if}\ \gamma \geq 0, \gamma \neq 1 \\ \ln(C), & \text{if}\ \gamma = 1 \end{cases} is called power utility if \gamma \neq 1 and log utility if \gamma = 1.
Another common choice for u(C) is u(C) = - e^{-a C}, usually called exponential utility.

Marginal Rate of Substitution

We can compute the utility differential as dU = \frac{\partial U}{\partial C_{0}} dC_{0} + \frac{\partial U}{\partial C_{1}} dC_{1}.
Since an indifference curve keeps the utility level constant, for all points in the indifference curve, we have that dU = 0, implying that \frac{dC_{1}}{dC_{0}} = - \frac{\frac{\partial U}{\partial C_{0}}}{\frac{\partial U}{\partial C_{1}}}. \tag{1}
The absolute value of the derivative of C_{1} with respect to C_{0} is called the marginal rate of substitution (MRS) between C_{1} and C_{0}.
The MRS compares how important it is to consume tomorrow versus today at any given point.

Production Functions

The Idea

An investor must first decide how much to consume today and how much to save for the next period.
Two factors determine this decision.
- On the one hand, the MRS determines how future consumption feels compared to current consumption.
- On the other hand, the ability to transform current consumption into future consumption is essential in deciding how much to consume today versus tomorrow.
We model the ability to convert current consumption into future consumption through a production function.

Economic Setup

All consumers start with a certain level of wealth, W, measured in terms of current consumption.
Consumers can then decide how much to consume today, given by C_{0}^{*}, and how much to invest.
An investment of K = W - C_{0}^{*} will generate C_{1}^{*} = f(K) of consumption tomorrow.

A Portfolio of Projects

The production function combines all available investment projects and ranks them from best to worse in terms of return.
Of course, if you have little to invest you want to use it in projects that have the best profitability.
For example, consider the following portfolio of investment opportunities, ranked by internal rate of return (IRR).

Project	Maximum Investment	IRR
I	1	500%
II	3	300%
III	5	100%
IV	11	0%

A Production Function

Assuming that we can invest fractions of today’s consumption, we can then generate the following production function f(K).

Figure 2: The figure shows a piece-wise linear production function.

Properties of Production Functions

The production function we just built is continuous in its range of definition.
It is also increasing in K as long as we assume limited liability.
Note that Project IV has a net return of 0%, which means that each unit of consumption invested generates one unit of consumption tomorrow.
- The worst possible scenario under limited liability is that the IRR of the project is -100%, at which point the production function would be flat.
The production function in Figure 2 is also convex, which is a consequence of investing in the projects with better profitability first.

Smoothness Assumptions

Typically, we assume that the production function is smooth such that f'(K) > 0 and f''(K) < 0, which yields a continuous, increasing, and convex function.
In the following, it is useful to express the function in terms of K = W - C_{0}, so that C_{1} = f(W - C_{0}).
- If the consumer decides to invest nothing and consume everything today, we have that K = 0 and C_{0}^{*} = W.
- If, on the other hand, the consumer decides to consume nothing today and invest everything, then we have that K = W and C_{0}^{*} = 0.

Investment Opportunity Set

Figure 3: The figure shows the investment opportunity set available to an investor.

Maximizing Utility

The Investor’s Problem

Consider now an investor with utility function U(C_{0}, C_{1}) and initial wealth W.
The investor has the ability to invest K = W - C_{0} into a production function that yields next period C_{1} = f(K).
We can write the investor’s problem as follows \begin{aligned} \max_{\{C_{0}, C_{1}\}} & U(C_{0}, C_{1}) \\ \text{s.t. } & C_{1} = f(W - C_{0}) \end{aligned}
To solve the previous optimization problem, we can write the Lagrangian as \mathcal{L} = U(C_{0}, C_{1}) - \lambda (C_{1} - f(W - C_{0})).

The Solution

The first-order conditions (FOC) are \begin{aligned} \frac{\partial \mathcal{L}}{\partial C_{0}} & = \frac{\partial U}{\partial C_{0}} - \lambda f'(W - C_{0}) = 0, \\ \frac{\partial \mathcal{L}}{\partial C_{1}} & = \frac{\partial U}{\partial C_{1}} - \lambda = 0, \\ \frac{\partial \mathcal{L}}{\partial \lambda} & = C_{1} - f(W - C_{0}) = 0. \end{aligned}
At the optimum, the marginal rate of substitution (MRS) must be equal to the marginal rate of transformation (MRT) between C_{1} and C_{0}.
The last FOC says that whatever the investor does not consume today is invested and can be consumed tomorrow to yield C_{1} = f(W - C_{0}).

Optimal Choice

Figure 4: The figure shows the optimal consumption choice given a production function and initial wealth W.

Example

Consider an investor with utility U(C_{0}, C_{1}) = \ln(C_{0}) + \ln(C_{1}).
The investor has initial wealth W and can invest K = W - C_{0} in a technology that produces f(K) = \sqrt{K} next period.
The investor maximizes her utility if her consumption (C_{0}^{*}, C_{1}^{*}) satisfies \text{MRS} = \frac{C_{1}}{C_{0}} = \frac{1}{2 \sqrt{W - C_{0}}} = \text{MRT}.
Since C_{1} = \sqrt{W - C_{0}}, we have that \frac{\sqrt{W - C_{0}}}{C_{0}} = \frac{1}{2 \sqrt{W - C_{0}}}, which implies that C_{0} = \frac{2}{3} W and C_{1} = \sqrt{\frac{1}{3} W}.

The Role of Capital Markets

The Production Decision

Investors can do better than autarky if they organize their economy differently.
Let’s delegate the production decision to a manager with access to a technology f(K).
Furthermore, assume that consumers have access to capital markets where they can borrow or lend at an interest rate r.
The manager is given a certain amount of wealth W, and must decide how much to sell today, investing the rest in the technology for future production, which we denote by (Q_{0}, Q_{1}), respectively.

The Manager’s Problem

Shareholders expect the manager to choose (Q_{0}, Q_{1}) to maximize the value of the firm V = Q_{0} + \frac{Q_{1}}{1 + r}.
The manager faces the budget constraint that he can only invest what the firm does not sell today, i.e., K = W - Q_{0}.
The problem that the manager must solve is given by \max_{\{Q_{0}\}} Q_{0} + \frac{f(W - Q_{0})}{1 + r}.
The FOC is \text{MRT} = f'(W - Q_{0}^{*}) = 1 + r.

Optimal Production Choice

Figure 5: The figure shows the optimal production policy for the firm given a production function and initial wealth W.

Maximizing Firm Value

The intercept of the CML with the x-axis determines the firm value.
By choosing the tangency point between the two lines, the manager maximizes the firm’s value by selecting the intercept that is furthest to the right.
To increase the firm’s size, the manager would need a more significant initial investment of W.
The difference between V and W is the net present value (NPV) created using the technology.
By investing an initial capital of W, shareholders now have an asset worth more than the initial investment.
The manager should then accept all projects with positive NPVs!

Example

Consider the same production function as the previous example, i.e., f(K) = \sqrt{K}.
The market interest rate is r.
The policy (Q_{0}^{*}, Q_{1}^{*}) that maximizes firm-value is such that \begin{aligned} \frac{1}{2 \sqrt{W - Q_{0}^{*}}} & = 1 + r, \\ Q_{1}^{*} & = \sqrt{W - Q_{0}^{*}}. \end{aligned}
Thus, Q_{1}^{*} = \frac{1}{2 (1 + r)} and Q_{0}^{*} = W - \frac{1}{4 (1 + r)^2}.
The value of the firm is then V = W - \frac{1}{4 (1 + r)^2} + \frac{1}{2 (1 + r)^{2}} = W + \frac{1}{4 (1 + r)^{2}} > W.

The Consumption Decision

In the model, shareholders agree on how the firm should maximize its value, regardless of their utility for today’s and future consumption.
An investor with initial wealth W can create the previous firm, hire a manager, and incorporate the firm.
The firm will then produce Q_{0}^{*} today, invest K = W - Q_{0}^{*} and produce f(K) = Q_{1}^{*} for consumption next period.
The investor could sell the firm for V, which can be used to consume C_{0} today and invest the rest to consume C_{1} = (V - C_{0}) (1 + r) next period.

The Investor’s Problem

The investor’s problem is \begin{aligned} \max_{\{C_{0}, C_{1}\}} & U(C_{0}, C_{1}), \\ \text{s.t. } & C_{1} = (V - C_{0}) (1 + r). \end{aligned}
The Lagrangian of the problem and the FOCs are \begin{aligned} \mathcal{L} & = U(C_{0}, C_{1}) - \lambda (C_{1} - (V - C_{0}) (1 + r)), \\ \frac{\partial \mathcal{L}}{\partial C_{0}} & = \frac{\partial U}{\partial C_{0}} - \lambda (1 + r) = 0, \\ \frac{\partial \mathcal{L}}{\partial C_{1}} & = \frac{\partial U}{\partial C_{1}} - \lambda = 0, \\ \frac{\partial \mathcal{L}}{\partial \lambda} & = C_{1} - (V - C_{0}) (1 + r) = 0. \end{aligned}

Interpretation of the Solution

The first two FOCs imply that the MRS for the consumer is equal to the rate of return of the CML,i.e., \text{MRS} = \frac{\frac{\partial U}{\partial C_{0}}}{\frac{\partial U}{\partial C_{1}}} = 1 + r.
The last FOC says that the present value of today’s and future consumption must equal V, i.e., V = C_{0} + \frac{C_{1}}{1 + r}.

Optimal Consumption With Capital Markets

Figure 6: The figure shows the optimal production policy for the firm and optimal consumption decisions for two investors given a production function and initial wealth W.

Example

Consider an investor with initial wealth W who owns the technology function of the previous example.
By producing Q_{0}^{*} = W - \frac{1}{4 (1 + r)^2} and Q_{1}^{*} = \frac{1}{2 (1 + r)}, she maximizes the firm value at V = W + \frac{1}{4 (1 + r)^{2}}.
Assume that the investor has a utility function of the form U(C_{0}, C_{1}) = \ln(C_{0}) + \beta \ln(C_{1}).
At the optimum, \dfrac{C_{1}}{\beta C_{0}} = 1 + r.
Since C_{1} = (1 + r) (V - C_{0}), we obtain C_{0}^{*} = \dfrac{1}{1 + \beta} V and C_{1}^{*} = (1 + r) \dfrac{\beta}{1 + \beta} V.

Fisher Separation

The previous analysis suggests that we can separate the firm’s investment decision from the investment decision faced by the consumer.
This separation result is known as Fisher Separation Theorem after economist Irvin Fisher.
Well-functioning capital markets play a crucial aspect in creating this separation.
All consumers are better off when firms maximize their values by undertaking positive NPV projects.
It is the role of the firm’s manager to ensure that companies maximize their values to shareholders.