Euler Equation

Macro

The Euler Equation is the central first-order condition in a dynamic optimization problem, describing the optimal trade-off between consumption today and consumption tomorrow.

For example, the Euler Equation for the Growth Model is:

$$u^{\prime }(c_t)=\beta u^{\prime }(c_{t+1})(f^{\prime }(k_{t+1})+1-\delta )$$

This is from the EC417 Advanced Macroeconomics notes (lecture 2)

• The left side of the equation, you could understand as the marginal cost of saving. Because once you choose to save one more unit instead of consuming it, you lose this much utility $u^{\prime }(c_t)$.
• The right hand side is the marginal benefit of saving. Because if you save one more unit, then tomorrow you will have $(f^{\prime }(k_{t+1})+1-\delta )$ more units of consumption, and the utility you get from that is $\beta u^{\prime }(c_{t+1})(f^{\prime }(k_{t+1})+1-\delta )$. (you need to take the depreciation into consideration).