If the Lorenz Curve were used to graph the distribution of income in an economy where all households earn the exact same income, the graph would show a straight line and sloped diagonally upward.
This is because, in a situation of perfect income equality, the cumulative share of income received by the bottom x% of households would equal the cumulative share of households. Hence, the Lorenz Curve would coincide with the line of perfect equality, which is a straight diagonal line from the origin to the top right corner of the graph.