Output matrix field data
Assuming you have an entry to work with, something like this should work:
{% for block in entry.myMatrixFieldHandle if block.type == &quot;employment&quot; %}
    &lt;select&gt;
        {% for option in block.myDropDownFieldHandle.options %}        
            &lt;option value=&quot;{{ option.value }}&quot;&gt;{{ option.label }}&lt;/option&gt;
        {% endfor %}
    &lt;/select&gt;
{% endfor %}

Otherwise you may need to get the field through the fieldLayout or by it's handle.
Update
craft.matrix is not exposed to twig like it is in php. To get the dropdown field options through the matrix field directly you'll have to go the long way around (matrix &gt; blocktypes &gt; blocktype &gt; fields &gt; field &gt; options)
{% set matrixField = craft.fields.getFieldByHandle('employmentMatrix') %}
{% set blockTypes = matrixField.getFieldType.getSettings.getBlockTypes %}
{% for blockType in blockTypes if blockType.handle == &quot;myBlockTypeHandle&quot; %}
    {% for field in blockType.getFields() %}
        {% if field.handle == &quot;myDropDownFieldHandle&quot; %}
            &lt;select&gt;
                {% for option in field.settings.options %}
                    &lt;option value=&quot;{{ option.value }}&quot;&gt;{{ option.label }}&lt;/option&gt;
                {% endfor %}
            &lt;/select&gt;
        {% endif %}
    {% endfor %}
{% endfor %}

UPDATE: This is for craft 2. For craft 3 see Nathan S answer below.