University:
University of Michigan
Course:
ECON 613 | Advanced Topic in Mathematics for Economists
Academic year:

2024
Views:

20

Pages:

2
Author:

Alvinska H.

Interceptions of Straight Lines: Understanding Changes in Constants in X and Y Relationships Name: Alvinska Hebat Grade: 910-Math Constant Change in Linear Functions A. Constant Change in Linear Function if the direction coefficient is positive: When Y = f(X) = a + mX where Y, X = variables; a = constant; m = direction coefficient. If there is a change in the constant "a" by Δa, the above equation becomes: Y' = a + mX + Δa = (a + Δa) +mX so that the intercept (constant) of the new function now becomes: a + Δa. Or from the picture beside: a2 = a0 + Δa where Δa > 0 a1 = a0 - Δa where Δa < 0 If Δa > 0 then the new constant becomes larger than before so that in the figure the line Y2 will be higher (above) the line Y0 by the vertical distance of the magnitude of Δa. If Δa < 0 then the new constant becomes smaller than the original so that in the figure the line Y1 will be lower (below) the line Y0 by a vertical distance of the magnitude of Δa. B. Constant Change in Linear Function if the directional coefficient is negative: When Y = f(X) = a - mX where Y, X = variables; a = constant; m = direction coefficient. If there is a change in the constant "a" by Δa, the above equation becomes: Y' = a - mX + Δa = (a + Δa) mX so that the intercept (constant) of the new function now becomes: a + Δa. Or from the picture beside: a2 = a0 + Δa where Δa > 0 a1 = a0 - Δa where Δa < 0 If Δa > 0 then the new constant becomes larger than before so that in the figure the line Y2 will be higher (above) the line Y0 by the vertical distance of the magnitude of Δa. If Δa < 0 then the constant is now smaller than before so that in the figure the Y0 line will be lower (below) the Y0 line by a vertical distance of the magnitude of Δa. Conclusion: Since the only thing that changes is the constant (intercept), the new line will be parallel to the original line with a vertical distance of Δa Examples of Application in Economic Concepts: 1. Effects of Sales Tax Generally, when a commodity is subject to sales tax, the price received by consumers will be more expensive (higher) than the price before the commodity was taxed. If a producer's supply function is P=f(Q), then if the government imposes a sales tax of Rp T per unit of the good offered by the producer, the producer's supply function (price function) becomes: P =f(Q) + T where P = price/unit; Q = quantity offered and T = sales tax per unit of goods If the producer is not subject to sales tax, the market equilibrium point occurs at Eo with quantities and prices equal to Qo and Po. } If the producer is subject to a sales tax of Rp T/unit then the equilibrium point is now at E1 with quantities and prices now equal to Q1 and P1. } The sales tax has increased the selling price from Po to P1 so that the equilibrium Q decreases from Qo to Q1.