Math based proof for currency changes (microeconomics)

So a US consumer picks their non-car bundle of goods, $x$, with prices $p$ denominated in dollars. The price of an American car is $a$ dollars, the price of a Japanese car is $y$ yen, and the price of a dollar in yen is the exchange rate, $r$ (how many yen does a dollar buy?). If the consumer buys an American car, $z = 1$, yielding utility $u_a$ at a price of $a$, and if the consumer buys a Japanese car, $z=0$, yielding utility $u_j$ at a price of $j$, $z in [0,1]$.

Then the consumer is solving $$ max_{xin mathbb{R}_N,zin [0,1]} u(x) + u_a z + u_j (1-z) $$ subject to $p'x + z a + (1-z) r j le I$ where $I$ is the consumer's wealth.

The Lagrangian is begin{eqnarray*} mathcal{L} &=& u(x) + u_a z + u_j (1-z) - lambda (p'x + za + (1-z)rj - I) &=& [u(x)-lambda (p'x-I)] + z(u_a-lambda a - u_j + lambda rj) -lambda rj + +u_j end{eqnarray*} The part in $[...]$ is the standard consumer problem: pick a basket of goods $x$ with income $I$ and prices $p$. The $z$ must be in $[0,1]$, with 1 meaning "buy American" and 0 meaning "buy Japanese". So if $u_a - lambda a > u_j - lambda rj$, the consumer buys American, and otherwise Japanese.

Now, if we increase $r$, what happens to the consumer's maximized utility? The envelope theorem tells us that if we want to see how welfare varies with a parameter, we partially differentiate with respect to that, then evaluate the Lagrangian at the optimum. Let's do that: $$ dfrac{partial mathcal{L}}{partial r} = (1-z) lambda j $$ So if the consumer is buying American, $z^*=1$, and this equals zero: the dollar becoming strong doesn't affect his welfare, because he is buying the American car anyway. But if $z^*=0$, he is buying Japanese, and a stronger dollar makes it easier to pay the $j$ price denominated in yen.

If you think about the Japanese consumer, everything is the same, except $r$ is going down and not up: he is worse off if he is buying an American car, but the same if buying a Japanese car.

Now, how is $r$ determined in equilibrium? The above problem gives you a demand curve for yen by American consumers: the ones who want to buy Japanese cars, i.e., they have a high $u_j$ relative to $u_a$. A similar problem gives demand for American cars by Japanese consumers, i.e. those Japanese households for whom $u_a$ is large relative to $u_j$. As the price $r$ varies, people will opt in or out of the market for the cars made by the other country. Equilibrium in the currency market occurs where the flow of dollars from Americans to Japanese and the flow of yen from Japanese to Americans is equal, just like your regular Supply-And-Demand diagram. This means that if a dollar buys fewer yen in equilibrium, it must also mean that yen buy more dollars in equilibrium, and vice versa.