John has the following utility function: $U(x,y)=x^{0\mathrm{.}5}{y}^{0\mathrm{.}5}$

a$.$ Determine the equation of the price consumption curve.

b$.$ If price of $y=$$$10$ and John has a weekly income of $$200,$ what is quantity demand of $x$ if the price of $x=$$$20?$

c$.$ Determine demand function of $x$ for John, assuming the price of $y$ and her income are constant. $($Hint: all points on the demand function MUST satisfy an equilibrium condition and x is a function of its own price$)$

Tickets to a rock concert sell for $$10.$ But at that price, the demand is substantially greater than the available number of tickets. Is the value or marginal benefit of an additional ticket greater than, less than, or equal to $$10?$ How might you determine that value?

Suppose the income elasticity of demand for food is $0\mathrm{.}5$ and the price elasticity of demand is $-1\mathrm{.}0.$ Suppose also that Felicia spends $$10,000$ a year on food, the price of food is $$2,$ and that her income is $$25,000.$

a$.$ If a sales tax on food caused the price of food to increase to $$2\mathrm{.}50,$ what would happen to her consumption of food? $($Hint: Because a larger price change is involved, you should assume that the price elasticity measures an arc elasticity, rather than a point elasticity.$)$

b$.$ Suppose that Felicia gets a tax rebate of $$2500$ to ease the effect of the sales tax. What would her consumption of food be now?

c$.$ Is she better or worse off when given a rebate equal to the sales tax payments? Draw a graph and explain.

Suppose that you are the consultant to an agricultural cooperative that is deciding whether members should cut their production of cotton in half next year. The cooperative wants your advice as to whether this action will increase members' revenues. Knowing that cotton $($C$)$ and watermelons $($W$)$ both compete for agricultural land in the South, you estimate the demand for cotton to be $C=3\mathrm{.}5-1\mathrm{.}0Pc+0\mathrm{.}25Pw+0\mathrm{.}5$I, where Pc is the price of cotton, Pw the price of watermelon, and I income. Should you support or oppose the plan? Is there any additional information that would help you to provide a definitive answer?

Consider a lottery with three possible outcomes:

$$125$ will be received with probability $0\mathrm{.}2$

$$100$ will be received with probability $0\mathrm{.}3$

$$50$ will be received with probability $0\mathrm{.}5$

a$.$ What is the expected value of the lottery?

b$.$ What is the variance of the outcomes?

c$.$ What would a risk$-$neutral person pay to play the lottery?