Step 1: Write the modulation index.
In amplitude modulation the modulation index is the ratio of the modulating to carrier peak voltages, \[ m = \frac{V_m}{V_c} \]
Step 2: Apply the percentage increases.
The modulating signal grows by $1\%$ and the carrier by $3\%$, so \[ V_m' = 1.01\,V_m, \qquad V_c' = 1.03\,V_c \]
Step 3: Write the new modulation index.
The new value is \[ m' = \frac{1.01\,V_m}{1.03\,V_c} = m\left(\frac{1.01}{1.03}\right) \]
Step 4: Use the small-change approximation.
For small percentage changes, the fractional change in a ratio is the difference of the fractional changes, \[ \frac{\Delta m}{m} \approx (1\%) - (3\%) \]
Step 5: Evaluate the change.
This gives \[ \frac{\Delta m}{m} \approx -2\% \]
Step 6: State the answer.
Hence the modulation index changes by \[ \boxed{-2\%} \]