Fix SymPy OverflowError in solow.md by using Rational

The SymPy solve() function was failing with 'mpz too large to convert
to float' when using Python float values (0.3, 0.5, 2.0) in symbolic
expressions with fractional exponents.

Using sympy.Rational for exact arithmetic avoids the large intermediate
mpz values that caused the overflow in factorint().

Author: mmcky

lectures/solow.md

@@ -510,21 +510,26 @@ plt.show()
 One can also try to solve this mathematically by differentiating $c^*(s)$ and solve for $\frac{d}{ds}c^*(s)=0$ using [sympy](https://www.sympy.org/en/index.html).
 
 ```{code-cell} ipython3
-from sympy import solve, Symbol
+from sympy import solve, Symbol, Rational
 ```
 
 ```{code-cell} ipython3
+# Use Rational for exact symbolic computation
+A_sym = Rational(2)
+alpha_sym = Rational(3, 10)
+delta_sym = Rational(1, 2)
+
 s_symbol = Symbol('s', real=True)
-k = ((s_symbol * A) / delta)**(1/(1 - alpha))
-c = (1 - s_symbol) * A * k ** alpha
+k = ((s_symbol * A_sym) / delta_sym)**(1/(1 - alpha_sym))
+c = (1 - s_symbol) * A_sym * k ** alpha_sym
 ```
 
 Let's differentiate $c$ and solve using [sympy.solve](https://docs.sympy.org/latest/modules/solvers/solvers.html#sympy.solvers.solvers.solve)
 
 ```{code-cell} ipython3
 # Solve using sympy
 s_star = solve(c.diff())[0]
-print(f"s_star = {s_star}")
+print(f"s_star = {float(s_star)}")
 ```
 
 Incidentally, the rate of savings which maximizes steady state level of per capita consumption is called the [Golden Rule savings rate](https://en.wikipedia.org/wiki/Golden_Rule_savings_rate).