Question

1. Preferences and Utility

a. Bob enjoys cookies (x) according to the utility function U(x)=20x- 2 tx , where t is a

parameter that reflects how hungry he is. Cookies are costless in Bob’s world and so there is

no income constraint. Using the envelope theorem, calculate how Bob’s maximum utility

from eating cookies varies with t.

b. Are the following utility functions quasi-concave? Show why

i. 𝑈(𝑋, 𝑌) = ln(𝑋) + ln(𝑌)

ii. 𝑈(𝑋, 𝑌) = min(𝑋, 𝑌) (Hint: You can use a diagram or sample values here)

2. Utility Maximization

A consumer faces income constraints and has CES preferences of the following form:

U(x, y) x y    

a. Find the consumer’s demand for x as a function of prices and income.

b. Are these preferences homothetic? Explain why or why not.

c. Calculate the consumer’s income elasticity of demand.

Answer 1

1:(a)

$U(x)=20x- tx^{2}$

Using the envelope theory,

$\frac{dU}{dx}=20−2xt=0$

Or

$x=\frac{10}{t}$

(b)

Both utility functions are quasi concave

(I)

$U = ln x + ln y (1) MUx =\frac{U}{x} = \frac{1 }{x} MUy =\frac{U }{y }= \frac{1 }{ y} MRS = \frac{MUx }{ MUy }= (\frac{1 }{ x}) / (\frac{1 }{ y}) =\frac{ y }{ x}$

.

(ii)

$x′≥min(x′,y′)≥a$

$x′≥min(x′,y′)≥a$ and $x′′≥min(x′′,y′′)≥a$

$x″≥min(x″,y″)≥a,$ so$λx′+(1−λ)x′′≥a$

$λx′+(1−λ)x″≥a.$ Likewise, $λy′+(1−λ)y′′≥aλy′+(1−λ)y″≥a.$

Therefore, it follows that 

$min(λx′+(1−λ)x′′,(λy′(1−λ)y′))≥a$

$min(λx′+(1−λ)x″,(λy′+(1−λ)y″))≥a$

and consequently, 

$λ(x′,y′)+(1−λ)(x″,y″)$  is in P

2:(a)

If the consumer is consuming only two goods "x" and "y" and the price of good "x" is "P_x" and the price of good "y" is "P_y" and if the income of the consumer is represented as "M" then the equation of the budget constraint of the consumer can be expressed as follows

$P_xx +P_yy =M ..."1"$

The utility function of the consumer is

$U(x,y) = xy$

The lagrangian expression, in this case, is shown below

$ℒ = xy + λ(M-P_xx-P_yy) ..."2"$ differentiate the equation "2" with respect to "X" and "Y" and setting them equal to zero

$\frac{∂ℒ}{∂x}=y−λPx =0...."3"$

$∂ℒ∂y=x-λPy =0...."4"$

divide equation "3" by equation "4"

$\frac{Y}{X}=\frac{λPx}{λPyY}$

$Y =x(\frac{Px}{Py}) ...."5"$

substitute the equation "5" inside the budget constraint of the consumer in order to get the demand function of good "x"

budget constraint of the consumer in order to get the demand function of good "x"

$P_xx +P_yy =M$

$P_xx +P_yx(\frac{Px}{Py})=M$

$x =\frac{M}{2Px}....."6"$

The equation $x =\frac{M}{2Px}$ shows the demand function of good "x" as the function of income and prices.

(b)

Preferences are homothetic. It satisfies properties of homothetic function.

Homethetic function is a real valued function f(x₁.......x_n) in homogeneous degrees of k for all t0

$U(xy)=xy$

Suppose we increase x+y by λ we get

$U(λy,λx)=(λn)(λy)$

$=λ2U(xy)$

(c)

$Em=\frac{1}{2p_x}\frac{m}{\frac{m}{2p_x}}=1$