Question

Economic Mobility The income of a child (low, medium, or high) generally depends on the income of the child's parents. The matrix \(P\), given by \(P=\left[\begin{array}{lll}0.4 & 0.2 & 0.1 \\ 0.5 & 0.6 & 0.5 \\ 0.1 & 0.2 & 0.4\end{array}\right] \begin{array}{ll}\mathrm{L} \\ \mathrm{H}\end{array} \quad\) Child's income is called a left stochastic transition matrix. For example, the entry \(P_{21}=0.5\) means that \(50 \%\) of the children of low- income parents will transition to the medium level of income. The diagonal entry \(P_{i i}\) represents the percent of children who remain in the same income level as their parents. Assuming that the transition matrix is valid from one generation to the next, compute and interpret \(P^{2}\).

Short answer

The matrix \(P^2\) is \[\left[\begin{array}{lll} 0.25 & 0.22 & 0.19 \ 0.61 & 0.65 & 0.55 \ 0.14 & 0.13 & 0.26 \end{array}\right].\It shows transition probabilities over two generations.

Step-by-step solution

Understand the Given Matrix

The given transition matrix, P, represents the probabilities of children transitioning between different income levels (low, medium, and high) based on their parents' income levels. P is a left stochastic matrix with the sum of each row equal to 1.

Recall Matrix Multiplication Rules

When multiplying two matrices, the element in the resulting matrix at position \(i, j\) is computed as the dot product of the i-th row of the first matrix and the j-th column of the second matrix.

Square the Matrix P (Calculate P^2)

Compute \(P^2\), which is the matrix P multiplied by itself. This involves taking the dot product of the rows and columns as described in Step 2: \[ P^2 = P \times P = \left[\begin{array}{lll} (0.4)(0.4) + (0.2)(0.5) + (0.1)(0.1) & (0.4)(0.2) + (0.2)(0.6) + (0.1)(0.2) & (0.4)(0.1) + (0.2)(0.5) + (0.1)(0.4) \ (0.5)(0.4) + (0.6)(0.5) + (0.5)(0.1) & (0.5)(0.2) + (0.6)(0.6) + (0.5)(0.2) & (0.5)(0.1) + (0.6)(0.5) + (0.5)(0.4) \ (0.1)(0.4) + (0.2)(0.5) + (0.4)(0.1) & (0.1)(0.2) + (0.2)(0.6) + (0.4)(0.2) & (0.1)(0.1) + (0.2)(0.5) + (0.4)(0.4) \end{array}\right] \ = \left[\begin{array}{lll} 0.25 & 0.22 & 0.19 \ 0.61 & 0.65 & 0.55 \ 0.14 & 0.13 & 0.26 \end{array}\right] \]

Interpret the Squared Matrix

Each element of the matrix \(P^2\) represents the probability of transitioning from one income level to another over two generations. For example, the entry \(P^2_{21} = 0.61\) indicates that there is a 61% chance that a child of low-income parents will have medium income two generations later.

Key concepts

Stochastic Matrices

Stochastic matrices are a mathematical construct used to model systems that undergo transitions from one state to another. In the context of economic mobility, stochastic matrices capture the probabilities of income transitions from one generation to the next.
Each row of a stochastic matrix represents a different starting state (e.g., low, medium, or high income), and each column represents an ending state. The elements of the matrix are probabilities:

• They must be non-negative
• Each row must sum to 1, indicating total probability

In our economic mobility matrix, the element at row i, column j tells us the probability of moving from income level i to income level j. For example, if P is the income transition matrix, then P_{21} = 0.5 implies that there's a 50% probability that children of low-income parents (state 1) transition to medium-income (state 2). This type of matrix ensures that we account for all possible outcomes within the model.

Income Transition Probabilities

Income transition probabilities represent the likelihood that individuals will move between different income levels from one generation to the next. We use these probabilities to study and predict patterns of economic mobility within populations.
These probabilities are captured within the elements of a stochastic matrix. For example, if an entry in the matrix is 0.2, it signifies a 20% chance of transitioning from one specific income bracket to another. These probabilities help us analyze:

• Intergenerational mobility
• Likelihood of income persistency
• Spatial and temporal income transitions

Understanding income transition probabilities is vital for creating models that can evaluate social policies and economic programs. For instance, the entry P_{ii} in our matrix shows the probability that the child remains in the same income bracket as their parent, giving insights into the degree of income immobility.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices combine to produce a third matrix. When applied to stochastic matrices, it can reveal deeper insights into multi-step processes, such as economic transitions over multiple generations.
To multiply two matrices, the element in the resulting matrix at position (i, j) is the sum of the products of corresponding elements from the i-th row of the first matrix and the j-th column of the second matrix. For instance, to compute the element at the first row and first column of the resulting matrix, you'd perform the following calculations:

• Multiply each element of the first row of the first matrix by the corresponding element of the first column of the second matrix
• Sum these products

When applied to our economic mobility matrix P, multiplying P by itself (P^2) gives us probabilities of income transitions over two generations rather than one. For example, P^2_{21} is the probability that children of low-income parents will transition to medium-income levels in two successive generations. This extended horizon gives more comprehensive insights into long-term economic mobility trends.