$13.$ Suppose that Hannah's utility function is $\backslash ($ U$\_\{$H$\}=\backslash )\backslash (3$ T$+4$ C $\backslash )$ and that Jose's utility function is $\backslash ($ U$\_\{$J$\}=\backslash )\backslash (4$ T$+3$ C $\backslash ),$ where $\backslash ($ T $\backslash )$ is pounds of tea per year and $\backslash ($ C $\backslash )$ is pounds of coffee per year. Suppose there are fixed amounts of $28$ pounds of coffee per year and $21$ pounds of tea per year. Suppose also that the initial allocation is $15$ pounds of coffee to Hannah $($leaving $13$ pounds to Jose$)$ and $10$ pounds of tea to Hannah $($leaving $11$ pounds of tea to Jose$).$

a$.$ What do the utility functions say about the marginal rates of substitution of coffee for tea?

b$.$ Draw the Edgeworth Box showing indifference curves and the initial allocation.

c$.$ Draw the contract curve on the Edgeworth Box. Explain why it looks different from the contract curves depicted in the text.

d$.$ Is the initial allocation of coffee and tea Pareto efficient?