Derivations for Week 7

The Euler Equation

🔍 Step-by-step differentiation explicitly:

Step 1: State the household’s maximization problem

The household’s optimisation problem is explicitly given by:

\max_{c_1, c_2, d} U(c_1, c_2) = \sqrt{c_1} + \beta \sqrt{c_2}

• Here, the utility function explicitly shows diminishing marginal utility in both periods (square root utility) and includes discounting future utility by factor \beta.

Step 2: State the budget constraints

The household faces two budget constraints, explicitly:

• Period 1 budget constraint: c_1 + d = y

  • y is the endowment (income) in period 1.
  • c_1 is consumption in period 1.
  • d is the amount saved (deposited at the bank) in period 1, which will pay a return in period 2.

• Period 2 budget constraint: c_2 = R d + \pi

  • R is the gross interest rate on deposits.
  • \pi is lump-sum profit income received in period 2 from banks.

Note: Households take R and \pi as given. These will be endogenously determined later in equilibrium.

Step 3: Substitute the budget constraints into the utility

Substituting the constraints explicitly, the household’s problem reduces to choosing only the amount of deposits d:

From the first-period constraint: c_1 = y - d

From the second-period constraint: c_2 = R d + \pi

Thus, the simplified optimisation problem is:

\max_{d} \sqrt{y - d} + \beta \sqrt{R d + \pi}

Let’s carefully differentiate step-by-step and explicitly:

We have the simplified optimisation problem in terms of deposits d:

\max_{d}\left[ \sqrt{y - d} + \beta\sqrt{R d + \pi} \right]

We need to take the first derivative of this objective function with respect to d:

Step 4: Differentiate the first term \sqrt{y-d} .

The derivative of the first term with respect to d is found using the chain rule:

• Consider the term : (y - d)^{1/2}

• Apply the chain rule: \frac{d}{dd}(y - d)^{1/2} = \frac{1}{2}(y - d)^{-1/2}\cdot \frac{d}{dd}(y - d)

• Differentiating the inner function explicitly: \frac{d}{dd}(y - d) = -1

Thus, explicitly: = \frac{1}{2\sqrt{y - d}}\cdot(-1) = -\frac{1}{2\sqrt{y - d}}

Step 5: Differentiate the second term \beta\sqrt{Rd+\pi} .

We again apply the chain rule:

• Consider the second term explicitly: \beta (R d + \pi)^{1/2}

• Apply the chain rule explicitly: = \beta\cdot \frac{1}{2}(R d + \pi)^{-1/2}\cdot \frac{d}{dd}(R d + \pi)

  • Differentiating the inner function R d + \pi: \frac{d}{dd}(R d + \pi) = R

Thus, explicitly: = \beta \frac{1}{2\sqrt{R d + \pi}}\cdot R = \frac{\beta R}{2\sqrt{R d + \pi}}

Step 6: Combine both results clearly to get the full derivative

Combining results from steps 1 and 2 explicitly, we get the derivative :

\frac{d}{dd}\left[\sqrt{y - d} + \beta\sqrt{R d + \pi}\right] = -\frac{1}{2\sqrt{y - d}} + \frac{\beta R}{2\sqrt{R d + \pi}}

Step 7: Set the derivative explicitly equal to zero (First-order condition)

The first-order condition (FOC) sets this derivative equal to zero:

-\frac{1}{2\sqrt{y - d}} + \frac{\beta R}{2\sqrt{R d + \pi}} = 0

Multiplying explicitly by 2 simplifies the equation to:

-\frac{1}{\sqrt{y - d}} + \frac{\beta R}{\sqrt{R d + \pi}} = 0

Rearranging explicitly and substituting back definitions c_1 = y - d and c_2 = R d + \pi, we obtain the Euler equation:

\frac{1}{\sqrt{c_1}} = \frac{\beta R}{\sqrt{c_2}}

Final differentiated and derived result (Euler Equation):

Explicit Euler equation from clear step-by-step differentiation is:

\boxed{\frac{1}{\sqrt{c_1}} = \frac{\beta R}{\sqrt{c_2}} \quad \iff \quad c_2 = c_1(\beta R)^2}

📌 Economic Intuition:

• The Euler equation shows explicitly how the household optimally smooths consumption over time.
• It clearly balances the marginal utility of consuming today (1/\sqrt{c_1}) with the discounted marginal utility of consuming tomorrow (\beta R/\sqrt{c_2}).
• The higher the future return (R) or discount factor (\beta), the higher future consumption relative to current consumption (c_2 relative to c_1).

Slack vs. Binding Constraints

What does “slack” mean?

• A constraint is said to be “slack” if it holds as a strict inequality.
• Intuitively, this means the constraint is not currently limiting the decision-maker’s choices.
• In other words, you still have some “room to maneuver” without violating the constraint.

Mathematically:

(N+d)R^{k}-Rd > \theta R^{k}(N+d)

\text{LHS} > \text{RHS}

In our bank model, the constraint is slack if banks comfortably satisfy the no-default condition:

• Banks can safely increase deposits without the risk of default.

• The constraint isn’t restricting the banks’ optimal decisions at all.

What does “bind” mean?

• A constraint “binds” when it holds with equality.
• Intuitively, the constraint becomes a real limit or boundary.
• At this point, there’s no further room to maneuver without violating the constraint.

Mathematically:
\text{LHS} = \text{RHS}

In our model, the no-default constraint binds when banks have reached their maximum safe deposit-taking level:

• Banks can’t safely increase deposits any further.
• Any additional deposit-taking immediately triggers default incentives.

Summary of slack vs. binding constraints:

• Slack constraint: Not restrictive, decision-maker has flexibility.
• Binding constraint: Restrictive, the decision-maker is at a limit.

Thus, the terms “slack” and “bind” reflect the constraint’s status in influencing optimal choices.