Q.What do you understand by 'parametric shift of a line'? How does a line shift when its

A 'parametric shift of a line' refers to a change in the position of a line due not to a movement along the line, but to a change in one or more of its underlying parameters, such as its slope or its intercept. For a linear equation $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept, a parametric shift occurs when $m$ or $c$ (or both) change, causing the entire line to adopt a new position.

1. When its slope decreases: If the slope ($m$) of a line decreases, the line rotates clockwise around its y-intercept. If the original slope was positive, the line becomes flatter. If the original slope was negative, the line becomes steeper.
2. When its intercept increases: …

In economics, many relationships are represented by linear equations, such as demand curves, supply curves, or consumption functions. A 'parametric shift of a line' refers to a change in one of the constant values, or 'parameters', within the equation that defines the line. These parameters determine the line's position and orientation. When a parameter changes, the entire line shifts or rotates on the graph, rather than just moving along the existing line.

Consider the general equation of a straight line:

Here, $y$ and $x$ are the variables plotted on the axes. The parameters are:

• $m$: the slope of the line, which indicates its steepness and direction.
• $c$: the y-intercept, which is the point where the line crosses the y-axis (i.e., the value of $y$ when $x=0$).

A parametric shift means that either $m$ or $c$ (or both) change, leading to a new line. This is distinct from a movement along the line, which occurs when the values of $x$ and $y$ change while the parameters $m$ and $c$ remain constant. In economic contexts, parametric shifts often represent changes in underlying conditions or exogenous factors that are not explicitly plotted on the axes. For example, a change in income (a parameter) shifts the entire demand curve (a line relating price and quantity).

Let's examine how a line shifts under the specified conditions:

(i) Slope decreases

When the slope ($m$) of a line decreases, the line becomes less steep. The nature of this shift depends on the original slope:

• If the original slope ($m$) was positive: A decrease in $m$ means the line becomes flatter. It will rotate clockwise around its y-intercept ($c$). For instance, if a line initially had a slope of $2$ and it decreases to $1$, the line will become less steep, pivoting downwards around the point where it crosses the y-axis.
• If the original slope ($m$) was negative: A decrease in $m$ (meaning it becomes more negative, e.g., from $-1$ to $-2$) would make the line steeper. However, if 'decrease' implies moving closer to zero (e.g., from $-2$ to $-1$), the line becomes flatter. Assuming 'decrease' means the absolute value of the slope reduces (i.e., it becomes less steep), the line will rotate counter-clockwise around its y-intercept ($c$).
• If the original slope ($m$) was zero: A decrease in $m$ would make it negative, causing the horizontal line to tilt downwards from left to right, rotating clockwise around its y-intercept. …