1. Suppose you can watch movies in the theater (t) and streaming at home (s), and earn utility according to the utility function:

u(t,s)=4ts

Where your marginal utilities are: MUt=4sMUs=4t

a. Put t on the horizontal axis and s on the vertical axis. Write an equation for MRSt,s

MRSt,s=MUtMUs=4s4t=st

b. Would bundles of (2,2) and (1,4) be on the same indifference curve?

Simply plug in each bundle to see how much utility it generates. If they have the same utility, they are on the same indifference curve.

u(t,s)=4tsu(2,2)=4(2)(2)u(2,2)=16

u(t,s)=4tsu(1,4)=4(1)(4)u(1,4)=16

Since both bundles generate utility of 16, they are both on the same curve.

c. Sketch this indifference curve.

2. You can get utility from consuming Soda (s) and Hot dogs (h), according to the utility function:

u(s,h)=√sh

The marginal utilities are:

MUs=0.5s−0.5h0.5MUh=0.5s0.5h−0.5

You have an income of $12, the price of Soda is $2, and the price of a Hot dog is $3. Put Soda on the horizontal axis and Hot dogs on the vertical axis.

a. What is your utility-maximizing bundle of Soda and Hot dogs?

Use the definition of the optimum, where the slope of the indifference curve (left) is equal to the slope of the brudget constraint (right):

MRSs,h=psphDefinition of the optimumMUsMUh=psphDefinition of MRS on left0.5s−0.5h0.50.5s0.5h−0.5=(2)(3)Plugging in what we know0.50.5s(−0.5−0.5)h(0.5−[−0.5])=23Using exponent rules for divisions−1h1=23Simplifying and cancellinghs=23Using exponent rules for negative exponentsh=23sMultiplying both sides bys

So we know that we will be buying 23 sodas for every 1 hot dog. This is the optimal ratio of consumption between the two goods.

To find the exact quantities of s and h, plug what we just found into the budget constraint:

pss+phh=mThe budget constraint equation2s+3h=12Plugging in what we are given2s+3(23s)=12Plugging in what we found relating b to a2s+2s=12Multiplying4s=12Addings=3Dividing by 4

Now that we know the quantity of sodas, we can use our knowledge of the ratio of sodas to hot dogs to find the quantity of hot dogs.

h=23sh=23(3)h=2

b. How much utility does this provide?

Plug answers from part A into the utility funtion:

u(s,h)=√shu(s,h)=√(3)(2)u(s,h)=√6