Too much math not enough intuition: undergraduate microeconomics

In any undergraduate microeconomics course, students are always expected to solve the basic consumer problem:

Nick has an income level $M$ to spend on two goods, imaginatively named, $x_1$ and $x_2$. The price of these two goods is fixed at prices $P_1$ and $P_2$ and Nick's preferences over these two goods are described by a utility function $U(x_1,x_2)$. Further, assume Nick is an extraordinarily greedy guy (technical term: Nick has locally non-satiated preferences for these two goods) and is always happier when he consumes more of $x_1$ or $x_2$, so Nick will always spend all his income.

The basic problem is how should Nick allocate his income $M$ between these two goods?

Formally, what we're really looking for is the solution to the following maths problem:

subject to

The standard approach offered to undergraduates is to solve this problem by forming the Lagrangian and then computing the first-order conditions and solving them for $x_1$ and $x_2$.

This was the approach I used throughout my undergraduate studies but I would argue this approach is very long-winded and skips over some important economic intuition. This second approach was not introduced to me till my math prep course at the start of my postgraduate studies - even though its less complicated and much more intuitive!

Ask yourself the following question: What's the opportunity cost for Nick of consuming a bundle $(x_1,x_2)$?

Well, Nick could reduce his consumption of $x_1$ by 1 unit freeing up $P_1$ units of income and allocating that additional income to buying $P_1/P_2$ units of $x_2$. Given Nick's preferences, which are represented by $U(x_1,x_2)$, the above reallocation causes his level of utility to change by

Where, $MU_1$ and $MU_2$ describe Nick's marginal utility of $x_1$, and $x_2$ respectively. 

If 

Then, Nick prefers this bundle over the old bundle, hence the old bundle can't be his most preferred bundle.

Likewise,

If

which is equivalent to

Then Nick would be better off by reducing his consumption of $x_2$ by 1 unit and buying $P_2/P_1$ units of $x_1$. Once again, his old bundle can't be his most preferred bundle.

Thus it must be the case that

which is equivalent to

Which is the same optimality condition you would get by using the Lagrangian approach. But instead of just blindly applying the maths, we got the result by asking what Nick is sacrificing given he consumes an arbitrary bundle $(x_1,x_2)$

To solve for Nick's optimal bundle we would make use of the budget constraint, giving us two equations with two unknowns which can be solved by substitution.

We can use this approach when solving macroeconomic models too.

In most macroeconomic models, they often start by assuming a representative consumer and use the Euler condition to solve the model. But the above example can be easily adapted.  Suppose the discount rate in the economy is given by $\beta$, price of a zero-coupon bond at date t expiring at date t+1 is $Q_t$, $P_1 = P_t$ , $P_2 = P_{t+1}$ are the prices of the consumption good at dates t and t+1 respectively,  and $x_1 = C_t$ and $x_2 = C_{t+1}$ denoting levels of consumption in period t and period t+1 respectively.

Then,

for all t.

  

A final example is Hotelling's rule which states that for an exhaustible resource the real rate of interest must equal the expected price change in the natural resource.  This can be derived by asking yourself the following questions: 

What is the payoff of keeping the natural resource in storage? 

Answer: $E_tP_{t+1}$, the expected price obtained tomorrow, whose present value is $E_tP_{t+1} / (1+r)$

So what payoff am I sacrificing by waiting till tomorrow?

Answer: $P_t$, the price today.

Suppose $P_t$ was higher than $E_tP_{t+1} / (1+r)$, then owners of the exhaustible resource would sell as much as possible today rather than wait til tomorrow. But this would depress the price today and increase the scarcity of the resource tomorrow. Thus the expected price obtained tomorrow would rise. This process would continue till $P_t = E_tP_{t+1} / (1+r)$.

Which can be expressed as $\Delta P^e = r$. Where $\Delta P^e$ denotes the expected change in price.

Throughout this post, all we've been doing is using the idea of opportunity cost - something which is taught in every pre-university Economics course.