Question

Imran has $158 to spend on goods x and y. His utility function is given by U(x,y)=min{1x,5y}. The unit price of good y is $2. The unit price of good x is initially $1, but then changes to $7.

What is the income effect of the price change on the consumption of good x?

Enter a numerical value below. You may round to the second decimal if necessary. Include a negative sign if the answer involves a decrease in quantity.

Answer 1

Imran has M = $158

U(x,y) = min{1x,5y}

P_y = $ 2 , P_x = $ 1 , P_x^/ = $ 7

We know that,

If we maximize U(x,y) = min{ax,by}

Subject to: P_x x + P_y y =M

Then, x^* = bM/(aP_y + bP_x) and y^* = aM/(aP_y + bP_x)

Now for P_y = $ 2 , P_x = $ 1 ;

x^* = (5*158)/(1*2+ 5*1) = 112.86(approx)

y^* = (1*158)/7 = 22.57(approx)

Now for P_y = $ 2 , P_x^/= $ 7 ;

x^*/ = (5*158)/(1*2+ 5*7) = 21.35(approx)

y^*/ = (1*158)/37 = 4.27(approx)

Now, we are calculating the fictitious level of income, M^/ so that original bundle is affordable at new price (7, 2).

P_x^/ x^* + P_y y^* = M^/

^or, M^/ = (7*112.86) + (2*22.57) = 835.16(approx)

Now, we are calculating the intermediate level of consumption {x₁ , y₁} at fictitious level of income M^/ and new price vector (7, 2).

So, x₁ = b M^//(aP_y + bP_x^/) = (5*835.16)/(1*2 + 5*7) = 112.86(approx)

y₁ = a M^//(aP_y + bP_x^/) = (1*835.16)/(1*2 + 5*7) = 22.57(approx)

So, Substitution Effect = x₁ - x^* = 112.86 – 112.86 = 0 [ We know that for a perfect complement function, SE = 0]

Total Price Effect = x^*/ - x^* = 21.35 – 112.86 = (- 91.51)

As, Total price Effect = Substitution Effect + Income Effect

So, IE = TPE – SE

Income Effect = (- 91.51) – 0 = (- 91.51)

The income effect of the price change on the consumption of of good x is (- 91.51).