Question

A consumer buys two goods, good X and a composite good Y. The utility function is given as ?(?,?) = ? + ?√? .   
1) Derive the demand function for good X.(5 marks)

2) Is good X a normal or an inferior good? Why? ( 5 marks)

3) Suppose that initially ?? = $1 and then it falls and becomes ?? = $0.5. Also suppose that Income=$10. Calculate the substitution effect, income effect, and the price effect and show you answer graphically (10 marks）

please show your work, and try to make it clear as possible, thanks

Answer 1

Utility function, U2, y) = x+2VY Price of good r is pr Price of good y is Py income of consumer is m

Part (1)

Consumer's utility ismarimized at point where MU: _PI MUу Ру Marginal utility from good x, MU: = Marginal utility from good y, MUY = au MU, PI MU, Py 2* This is demand function of good y Substituting this in budget constraint P.2.C +Pyy = m Prc + Py" po * Pr=m P2C+T=m + PrPyx + pz = mpy x (Px;Py, m) = mpy – pŽ Prpy This is demand function of good z

Part (2)

(pr.py, m) = mpy – p3 PxPy < _ mpy_p? PxPyP:Py 01 - Py 1 əm Popy PI Since price of good is always positive :: 130 - PI

Demand for good x rises when income of consumer rises and falls when income falls

Part (3)

We are asked to find substitution effect and income effect when price of good x falls from $1 to $0.5 and income of consumer is $10

Since we are not given price of good y, we are assuming it to be $1 for calculation purpose and subsequently for graphical presentation

With Pr = 1, Py = 1, m = 10 PX 12 mpy – p? 10*1 – 12 10 - 1 1*1 = T : initial =9 :: initial bundle contains I = 9 and y = 1

When price of good x decreases, relative price of good y increases. Therefore, consumer will substitute good y with good x. This is the substitution effect which induces consumer to consume more of good x and less of good y. Bundle consumed after substitution effect is called the decomposition bundle which gives the same utility as the initial bundle that is it lies on the original indifference curve but tangent to budget line with changed prices

Utiltiy of initial bundle U(9, 1) = 1 +2y=9+2V1= 9+2*1 = 9+2 = 11 Now with changed prices, utility maximization condition is MU: _PI MU, Py 0.5 ✓y= y = 0.5% = 0.25 With y = 0.25 and 2 + 2y = 11 2 + 2y = 11 2 + 2* V0.25 = 11 x + 2 * 0.5 = 11 = 11-1= 10

Decomposition bundle contains x = 10 and y= 0.25

On the other hand, due to fall in price of good x there is an increase in purchasing power of consumer.This is called as income effect which induces consumer to consume more of both the goods, But here the income effect induces consumer to consume more of good x only

After price change the utility marimization condition is 0.52 y = 72 = 0.25 Substituting this in budget constraint PrX +Pyy = m 0.5.+1* 0.25 = 10 0.5.r = 10 – 0.25 0.5x = -9.75 I=- 19.5 0.5

Final bundle contains r= 19.5 and y = 0.25

Substitution effect is difference between decomposition bundle and initial bundle which induces consumer to consume 1 unit more of good x and 0.75 unit less of good y

Income effect is difference between decomposition bundle and final bundle which induces consumer to consume same units of good y but 9.5 units more of good x

Price effect is sum of substitution effect and income effect which overall induces consumer to consume 10.5 units more of good x and 0.75 units less of good y

As price of good x falls from $1 to $0.5 Substitution effect = 10 – 9=1 Income effect = 19.5 – 10 = 9.5 Price effect = 19.5 - 9 = 10.5

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