ECON 211 2022-23 T1 ASSIGNMENT #3

Q1: (10 points) DEMAND CURVES and ENGLE CURVES: Consider the following consumer’s     problem: U(X,Y) = X1/2  Y1/2 .  Suppose that she has income (M) and faces prices Px and Py .

a) (1 point) Sketch the budget set. What is the slope of the Budget Line? What are maximal possible consumptions of X and Y? (note: these will depend on what values of I, Px and Py are given).

b) (1 point) Show that the MRSXY = Y/X.

Note:    by defn MRSab = MUa/MUb

STEP 1:                           MUx  = ½  X- 1/2 Y1/2

STEP 2:                           MUy  = ½   X1/2 Y- 1/2

STEP 3:                           MRS = MUx /MUy  = (½  X- 1/2 Y1/2)/( ½   X1/2 Y- 1/2 )

STEP 4:  simplify to get:           MRS = Y/X

Interpretation: we are willing to exchange 1 unit of Y to get 1 unit of X.

c) (2 point) Show that the consumer will spend ½ of her income on Good X and ½ on Good Y.

STEP 2: by substitution:                         Y/X = Px/Py

STEP 3:  isolate Y                                     Y = Px X/Py       or         Y = (Px /Py) X

STEP 4: substitute eq 1 into eq 2

STEP 5: simplify

Px X + (Px/Py X) Py = m

2X Px = m    implies X= ½ m /Px

And Y = ½ M/Py

d) (2 point) Now, sketch the Demand Curve for X and then for good Y. Show that both are downward sloping.

X= ½ m /Px:  so as Px rises, X falls.  Both are right-angle Hyperbolas and are downward sloping.  Eg, as Px rises, X falls since each share of income is now on a higher priced good.

e) (2 point) Sketch the Engel Curve for X and then for good Y. Show that both are upward sloping.

From the above X = (½ 1/Px) M.  so is increasing in M at a rate 1/2Px.  Similar for Y.

f) (2 point) Suppose income (I) is $1000, Px = $6 and Py = $4. Find the optimal consumption bundle. From above X= ½ m /Px = 500/6 = 83.3

And Y = ½ M/Py = 500/4 = 125

Note: you can use excel for this but using the demand curve is better (as you see in Q2)

Q2: (10 points) SUBSTITUTION and INCOME EFFECTS: Suppose we are given the following       utility function for a consumer:  U(X,Y) = X1/2y1/2   : Suppose also that her income (I) is $1000, Px = $6 and Py = $4.

a) (1 point) Find the consumer’s optimal choice given the prices and income above. What is the utility she derives from this income?

From above     X= ½ m /Px       = 500/6             = 83.3

And                   Y = ½ M/Py       = 500/4             = 125

Hence U = (83.31/2)(1251/2) = 102.0621

b) (1 point) Find the new optimum if Py falls to $3.

X= ½ m /Px       = 500/6             = 83.3                no change

Y = ½ M/Py       = 500/3             = 166.6             so Y rises given the lower price.

New U = (83.31/2)(1661/2) = 117.8511                which is higher than before.

c) (3 point) Show that the income required to just make the previous utility from (a) attainable with Px = $6 and Py = $3 is $866.03. Show and explain the process you use to get this result.      (Eg. you have the answer so just show the steps to get there.)

STEP 1: from above we have X= ½ m /Px  and Y = ½ M/Py

STEP 2:  substitute into the utility function U(X,Y) = X1/2y1/2   = ( ½ m /PX )1/2  ( ½ m /PY      )1/2 STEP 3:  simplify:  U(X,Y) = ( ½ m) (1/PX )1/2  (1/PY )1/2   = ( ½ m) (PX PY)-1/2

this is the indirect utility function (U as a function of prices and income) STEP 4:  invert the function to isolate M:    M = U /[ ½  (PX PY)-1/2 ] = 2U (PX PY)1/2;

Notice that M is increasing in prices. That is, to maintain a given U, I need more income to offset any rise in prices.

STEP 5  set U = 102.0621 and set Px = 6 and Py = 3.  Hence we are finding the income that attains the original target utility but with new prices.

SOLUTION: So M = 2(102.0621) (6x3)1/2  = 866.0254  done

d) (2 point) Given the “new” income in (c) with Px = $6 and Py = $3, find the new optimum. Confirm that it yields the same utility as in (a).

X= ½ m /Px

Y = ½ M/Py

Hence U = (83.31/2)(1661/2) = 102.0676

so falls as M falls

falls as M falls

close enough to the original value

e) (3 point) What are the Hicks Substitution and Income Effects of the fall in the price of y? eg find ∆X and ∆Y.

See figure below>

The HICKS SUBSTITUTION EFFECT is the change in consumption when we change relative prices but hold utility constant.

For X: 72.17 – 83.33 = -11.17

For Y: 144.34 – 125 = + 19.34

Notice that Y rises and X falls since we are moving along the original indifference curve.  Eg these are Hicks Substitutes.

The HICKS INCOME EFFECT is the change in X and Y when we hold prices at the new values but let the actual income be 1000 instead of the 866.

For X:   83.33 – 72.17 = +11.17    notice that this exactly offsets the Hicks substitution effect

For Y: 166.66 - 144.34 = + 22.32

Note that this implies both goods are Normal Goods.

Let A = original consumption   (83, 125) with U = 102

Let B = new consumption          (83, 166) with U = 117

Let C = (72, 144)  with U = 102: this is the income-adjusted consumption that holds to original utility