How do you calculate Golden Rule level?

The Golden Rule level of capital (k*) represents the steady-state capital stock that maximizes consumption. It is calculated where the marginal product of capital equals the depreciation rate.

Understanding the Golden Rule Level

The Golden Rule is a concept in economics that identifies the optimal level of capital accumulation that maximizes sustainable consumption in the long run. It helps determine how much of an economy's output should be saved and invested versus consumed.

Formula and Calculation

Based on the provided reference, the Golden Rule level of capital (k) is found where consumption (c) is maximized in the steady state. This occurs when:

*c = f(k*) − δk***

Where:

• **c*** is the consumption per capita in the steady state.
• *f(k)* is the output per capita as a function of the capital stock per capita (k).
• δ is the depreciation rate of capital.
• **k*** is the capital stock per capita in the steady state.

In the steady state, investment (i) equals depreciation (δk) because the change in the capital stock (Δk) is zero. Therefore:

i* = δk*

To find the Golden Rule level of capital (k*), you need to find the capital stock level where the marginal product of capital (MPK) equals the depreciation rate (δ):

MPK = δ

The marginal product of capital (MPK) is the additional output generated by adding one more unit of capital. Once you know the production function (f(k)), you can find the MPK (which is the derivative of f(k) with respect to k).

Steps to Calculate the Golden Rule Level

1. Determine the Production Function: Identify the economy's production function, f(k). A common example is the Cobb-Douglas production function: f(k) = k^α , where α is the capital share of output.

2. Calculate the Marginal Product of Capital (MPK): Take the derivative of the production function with respect to k: MPK = f'(k). For the Cobb-Douglas example: MPK = αk^α-1.

3. Set MPK equal to the Depreciation Rate: MPK = δ.

4. *Solve for k:* Solve the equation from step 3 for k. This value of k is the Golden Rule level of capital. Using the Cobb-Douglas example: αk^α-1 = δ => k = (α/δ)^(1/(1-α)).

Example

Let's assume a Cobb-Douglas production function: f(k) = k^0.3, and a depreciation rate (δ) of 0.1.

1. MPK = 0.3k^-0.7
2. 0.3k^-0.7 = 0.1
3. *k = (0.3/0.1)^(1/(1-0.3)) = (3)^(1/0.7) ≈ 4.35**

Therefore, the Golden Rule level of capital in this example is approximately 4.35. This level of capital stock maximizes consumption in the long run.