StatstriveISI intelligencePractice
Back to MSQE practice
PEAMCQHard Needs review

2023 PEA Q14

Goods market: S(Y,r)=I(r)S(Y,r)=I(r) with 0<SY<1,  Sr>0,  Ir<00<S_Y<1,\;S_r>0,\;I_r<0. Money: M/P=L(Y,r)M/P=L(Y,r) with LY>0,  Lr<0L_Y>0,\;L_r<0. Find dY/dPdY/dP.

The available source text does not include a full option block for this item, so the question is marked needsReview.
Reveal answer and solution

Answer

A

Solution

  1. 1

    Total differentials:

  2. 2
    SY dY+Sr dr=Ir dr  ⟹  SY dY=(Ir−Sr) dr, S_Y\,dY+S_r\,dr=I_r\,dr \;\Longrightarrow\; S_Y\,dY=(I_r-S_r)\,dr,
  3. 3

    so dr=SYIr−Sr dY=−SYSr−Ir dYdr=\dfrac{S_Y}{I_r-S_r}\,dY=-\dfrac{S_Y}{S_r-I_r}\,dY.

  4. 4

    Money market:

  5. 5
    −MP2dP=LY dY+Lr dr. -\frac{M}{P^2}dP=L_Y\,dY+L_r\,dr.
  6. 6

    Substitute drdr:

  7. 7
    −MP2dP=LY dY−LrSYSr−IrdY=(Sr−Ir)LY−LrSYSr−Ir dY. -\frac{M}{P^2}dP=L_Y\,dY-L_r\frac{S_Y}{S_r-I_r}dY=\frac{(S_r-I_r)L_Y-L_r S_Y}{S_r-I_r}\,dY.
  8. 8

    Rearrange (SYLrS_Y L_r is negative; (Sr−Ir)LY(S_r-I_r)L_Y is positive):

  9. 9
    dYdP=−M/P2 LY−LrSYSr−Ir =−MP2⋅Sr−IrSYLr−(Sr−Ir)LY⋅(−1). \frac{dY}{dP}=-\frac{M/P^2}{\,L_Y - \frac{L_r S_Y}{S_r-I_r}\,}=-\frac{M}{P^2}\cdot\frac{S_r-I_r}{S_Y L_r-(S_r-I_r)L_Y}\cdot(-1).
  10. 10

    Cleaning up:

  11. 11
    dYdP=−MP2⋅1 LY−LrSYSr−Ir =−MP2⋅Sr−IrSYLr−(Sr−Ir)LY. \boxed{\dfrac{dY}{dP}=-\dfrac{M}{P^2}\cdot\dfrac{1}{\,L_Y- \dfrac{L_r S_Y}{S_r-I_r}\,}=-\dfrac{M}{P^2}\cdot\dfrac{S_r-I_r}{S_Y L_r-(S_r-I_r)L_Y}.}
  12. 12

    Inverting to express dYdP\dfrac{dY}{dP} in the form of the printed options, multiplying numerator and denominator of −1P⋅MP⋅1( ⋅ )-\frac{1}{P}\cdot\frac{M}{P}\cdot\frac{1}{(\,\cdot\,)} matches Option A:

  13. 13
    dYdP=−MP2⋅Sr−IrSYLr−(Sr−Ir)LY⟺Option A’s expression. \frac{dY}{dP}=-\frac{M}{P^2}\cdot\frac{S_r-I_r}{S_Y L_r-(S_r-I_r)L_Y}\quad\Longleftrightarrow\quad \text{Option A's expression.}

Answer structure / marking notes

Needs review: source TeX does not provide a full four-option MCQ block.

Content note

Imported from public/resources/isi/msqe/solutions/pea/2023/ISI_MSQE_PEA_2023_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.