Interpretation of categorial predictor in poisson regression I have performed a Poisson regression where my outcome/dependent variable is a count variable of how many technical devices someone ones (ranges from 1 to 9) and I have a bunch of predictor/independent variables, e.g. sociodemographic variables. I'm particularly interested in the interpretation of the variable Age which I'm not treating as a continuous variable but I put them into age groups and use this variable as a categorical (factor in R terminology) predictor.
I've read a few posts here and on the internet but am still struggling quite a bit how to interpret the coefficients. I've read that for Poisson regression the coefficients are multiplicative, but don't really have a clue what that means, especially since the results for a categorical variable always have to be interpreted in comparison to the reference category.
I'd appreciate if you could help me shedding some light on it.
Here is a plot with the descriptive results showing a clear relationship (older people own less technical devices):

And here is the regression output. I formatted it with the tidy package in R and I omitted all other predictors, so only showing the intercept and age. In addition to the raw coefficients I calculated the exponential versions (exp_estimate + confidence intervals).
# A tibble: 6 x 10
  term        estimate std.error statistic p.value conf.low conf.high exp_estimate conf.low.exp conf.high.exp
  <chr>          <dbl>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>        <dbl>        <dbl>         <dbl>
1 (Intercept)     0.14      0.2       0.72    0.47    -0.25      0.54         1.16         0.78          1.71
2 Age18-24       -0.03      0.01     -2.08    0.04    -0.05      0            0.97         0.95          1   
3 Age25-34       -0.07      0.02     -4.31    0       -0.1      -0.04         0.94         0.91          0.96
4 Age35-44       -0.04      0.02     -2.33    0.02    -0.07     -0.01         0.96         0.93          0.99
5 Age45-54       -0.1       0.02     -5.63    0       -0.13     -0.06         0.91         0.88          0.94
6 Age55-65       -0.09      0.02     -4.99    0       -0.13     -0.05         0.91         0.88          0.95

Age group 16-17 is my reference category and thus not showing in the output.
 A: A Poisson regression model models your observations as Poisson distributed,
$$ y_i \sim \text{Pois}(\lambda_i), $$
where each observation's Poisson parameter $\lambda_i$ depends on the covariates via an exponential link function:
$$ \lambda_i = \exp(\beta_0+x_{i1}\beta_1+\dots+x_{ip}\beta_p) =
\exp(\beta_0)\times \exp(x_{i1}\beta_1)\times\dots\times\exp(x_{ip}\beta_p).$$
Now, your predictors $x_{ij}$ are either 0 or 1 through your dummy coding, so your Poisson parameter for the $i$-th observation is just the product of (the exponential) of the appropriate parameters,
$$ \lambda_i = \exp(\beta_0)\prod_{j\in J_i}\exp(\beta_j) $$
for an index set $J_i$ that indicates which dummies are "active" for the $i$-th observation.
Now, your categorical predictor is a dummy-coded discretization of age. So precisely one of your dummies is active - or none, if age falls into the reference category. So
$$ \lambda_i=\begin{cases}
  \exp(\beta_0), & 16\leq\text{age}_i\leq 17 \\
  \exp(\beta_0)\exp(\beta_1), & 18\leq\text{age}_i\leq 24 \\
  \exp(\beta_0)\exp(\beta_2), & 25\leq\text{age}_i\leq 34 \\
  ...
\end{cases} $$
Thus, your Poisson parameter - that is, the mean amount of items the $i$-th participant owns - is $\exp(\beta_0)$ if that participant's age is in the reference category. (Disregarding other model parameters.) If they are in the $j$-th non-reference age category, this mean is $\exp(\beta_0)\exp(\beta_j)$. And this in turn is just $\exp(\beta_j)$ times the number of items someone in the reference category owns. This is the interpretation you are looking for.

Incidentally (but importantly!), don't discretize age. Your model amounts to assuming everyone at age 16-17 has the same mean, which differs from the mean of everyone at age 18-24. That is, there is a sharp step on your 18th birthday (do you get a ton of stuff as presents?), but then nothing happens until your 25th birthday (when you suddenly get another ton of stuff?), and then again nothing happens until you turn 35.
Actually, it's worse: your negative estimates mean that you throw stuff away on your 18th birthday (namely, 3% of your items, since $\exp(-0.03)\approx 97\%$) and on your 25th birthday (4% of what you had left: for the last seven years you lived with $\exp(-0.03)\approx 97\%$ of what you had at ages 16-17, and when you turn 25, you suddenly only own $\exp(-0.07)\approx 93\%$, and $\exp(-.07)/\exp(-0.03)\approx 96\%$). Then you get some stuff on your 35th birthday  and again throw stuff out on your 45th birthday.
Better: use age as a numercial covariate. If you are concerned about nonlinearities (which you would be right to be!), use splines to transform age. You currently expend five degrees of freedom in fitting an ecologically invalid model. Better to expend the same amount of dfs (or fewer!) in a spline model. Yes, I admit that the coefficients will be harder to interpret with splines. But at least the model will make sense. You can still compare $\hat{\lambda}_{\text{age} = 18}$ to $\hat{\lambda}_{\text{age} = 35}$, or plot $\hat{\lambda}_{\text{age}}$ against age (holding other predictors at a constant value). 
