No entiendo por qué en mi siguiente ejemplo, el cuadro combinado "Modelo de facturación" no muestra la propiedad BillingModel.BillingModelDescription cuando el cuadro de texto sí lo hace. Después de seleccionar un cliente, quiero que el cuadro combinado muestre la descripción del modelo de billng actual, pero permanece en blanco. El cuadro de texto vinculado a la misma cosa muestra la descripción. Tengo una colección de posibles modelos como ItemsSource, que funciona bien. ¿Cómo actualizo el cuadro combinado de modelo de facturación al seleccionar un cliente?

Aquí está el XAML:

<Window x:Class="WpfApplication7.MainWindow"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    Title="MainWindow" Height="350" Width="525">
<StackPanel>
    <StackPanel Orientation="Horizontal">
        <Label Content="Client"/>
    <ComboBox ItemsSource="{Binding AllClientData}" DisplayMemberPath="EmployerStr"
              SelectedItem="{Binding SelectedClient}"
              Width="300"/>
    </StackPanel>
    <StackPanel Orientation="Horizontal">
        <Label Content="Billing Model:"/>
    <ComboBox ItemsSource="{Binding AllBillingModels}" DisplayMemberPath="BillingModelDescription"
              SelectedItem="{Binding SelectedClient.BillingModel}"
              Width="300"/>
    </StackPanel>
    <StackPanel Orientation="Horizontal">
        <Label Content="Billing Model" />
    <TextBox Text="{Binding SelectedClient.BillingModel.BillingModelDescription}" Width="200"/>
    </StackPanel>
</StackPanel>

Y el código subyacente (esto es solo un ejemplo, estoy usando MVVM, etc. en la aplicación completa, pero esto sirve para mi propósito para ilustrar el problema):

public partial class MainWindow : Window, INotifyPropertyChanged
{
    public MainWindow()
    {
        AllClientData = new ObservableCollection<ClientRate>();
        AllBillingModels = new ObservableCollection<BillingModelType>();

        ClientRate uno = new ClientRate();
        uno.BillingModel = new BillingModelType();
        uno.BillingModel.BillingModelID = 3;
        uno.BillingModel.BillingModelDescription = "Free";
        uno.ID = 01;
        uno.EmployerName = "Employer1";

        ClientRate dos = new ClientRate();
        dos.BillingModel = new BillingModelType();
        dos.BillingModel.BillingModelID = 2;
        dos.BillingModel.BillingModelDescription = "Variable";
        dos.ID = 02;
        dos.EmployerName = "Employer2";

        ClientRate tre = new ClientRate();
        tre.BillingModel = new BillingModelType();
        tre.BillingModel.BillingModelID = 1;
        tre.BillingModel.BillingModelDescription = "Flat";
        tre.ID = 01;
        tre.EmployerName = "Employer3";

        AllClientData.Add(uno);
        AllClientData.Add(dos);
        AllClientData.Add(tre);

        BillingModelType one = new BillingModelType();
        one.BillingModelID = 1;
        one.BillingModelDescription = "Flat";

        BillingModelType two = new BillingModelType();
        two.BillingModelID = 2;
        two.BillingModelDescription = "Variable";

        BillingModelType three = new BillingModelType();
        three.BillingModelID = 3;
        three.BillingModelDescription = "Free";

        AllBillingModels.Add(one);
        AllBillingModels.Add(two);
        AllBillingModels.Add(three);

        InitializeComponent();
        this.DataContext = this;
    }

    private ObservableCollection<ClientRate> _allClientData;
    public ObservableCollection<ClientRate> AllClientData
    {
        get { return _allClientData; }
        set
        {
            if (_allClientData != value)
            {
                _allClientData = value;
                FirePropertyChanged("AllClientData");
            }
        }
    }

    private ClientRate _selectedClient;
    /// <summary>
    /// Gets/Sets Global SelectedClient object
    /// </summary>
    public ClientRate SelectedClient
    {
        get { return _selectedClient; }
        set
        {
            if (_selectedClient != value)
            {
                _selectedClient = value;
                FirePropertyChanged("SelectedClient");
            }
        }
    }

    //private BillingModelType _selectedBillingModel;
    //public BillingModelType SelectedBillingModel
    //{
    //    get
    //    {
    //        return _selectedBillingModel;
    //    }
    //    set
    //    {
    //        if (_selectedBillingModel != value)
    //        {
    //            _selectedBillingModel = value;
    //            FirePropertyChanged("SelectedBillingModel");
    //        }
    //    }
    //}

    private ObservableCollection<BillingModelType> _allBillingModels;
    /// <summary>
    /// Holds all possible billing model types 
    /// </summary>
    public ObservableCollection<BillingModelType> AllBillingModels
    {
        get { return _allBillingModels; }
        set
        {
            if (_allBillingModels != value)
            {
                _allBillingModels = value;
                FirePropertyChanged("AllBillingModels");
            }
        }
    }

    public event PropertyChangedEventHandler PropertyChanged;

    protected void FirePropertyChanged(string propertyName)
    {
        if (PropertyChanged != null)
            PropertyChanged(this, new PropertyChangedEventArgs(propertyName));

    }
}

public class BillingModelType
{
    /// <summary>
    /// Billing Model ID
    /// </summary>
    public int? BillingModelID { get; set; }
    /// <summary>
    /// Billing Model Description
    /// </summary>
    public string BillingModelDescription { get; set; }
}

public class ClientRate : INotifyPropertyChanged
{
    /// <summary>
    /// Employer name with Employer ID in parentheses
    /// </summary>
    public string EmployerStr { get { return EmployerName + " (" + ID + ")"; } }
    /// <summary>
    /// Employer ID
    /// </summary>
    public int? ID { get; set; }
    private string _EmployerName;
    /// <summary>
    /// Employer Official Name
    /// </summary>
    public string EmployerName
    {
        get { return _EmployerName; }
        set
        {
            if (_EmployerName != value)
            {
                _EmployerName = value;
                FirePropertyChanged("EmployerName");
            }
        }
    }

    private BillingModelType _billingModel;
    /// <summary>
    /// Rate Type  ID and Description
    /// </summary>
    public BillingModelType BillingModel
    {
        get { return _billingModel; }
        set
        {
            if (_billingModel != value)
            {
                _billingModel = value;
                FirePropertyChanged("BillingModel");
            }
        }
    }

    public event PropertyChangedEventHandler PropertyChanged;

    protected void FirePropertyChanged(string propertyName)
    {
        if (PropertyChanged != null)
            PropertyChanged(this, new PropertyChangedEventArgs(propertyName));

    }
}