Now, it is January $2023.$ Dee will buy $50,000$ pounds of lean hog in March $2023.$ To hedge the hog price, she buys a March $2023$ maturity lean hog contract at the future price of $$1\mathrm{.}20$ per pound. Suppose that when the contract matures in March $2023,$ the market price of hog turns out to be $$1$ per pound. This future contract calls for the purchase of $50,000$ pounds.

a$)$ Calculate Dee's payoff from purchasing $50,000$ pounds of hogs in the spot market in March $2023($note: a payment for purchase is a negative payoff$)$;

b$)$ Calculate Dee's long payoff from the future market in March $2023$;

c$)$ Calculate Dee's total payoff from both the spot and future markets in March $2023$;

d$)$ If the market price of hog goes up to $$1\mathrm{.}50$ per pound in March $2023,$ what's Dee's total payoff? Is it the same as the total payoff in part c$)?$ Why?

e$)$ Draw the future contract payoff graph for Dee