Consider an exchange economy with two consumers Charlotte and Dylan, and two goods quinoa (Q) and raspberries (R) Charlotte has an initial endowment of 99 6 units of quinoa and 76 raspberries Dylan has 124 4 units of quinoa and 78 1 raspberries Charlotte's utility function is given by U C Q C 1 2 R C 1 2 , where Q C and R C are her consumption of Q and R, respectively Dylan's utility function is given by U D Q D 1 3 R D 2 3 , where Q D and R D are his consumption of Q and R, respectively Suppose that the market price of quinoa is p Q 2 and the market price of raspberries is p R 1 Keeping the price of raspberries normalized at p R 1 , find the market clearing price of quinoa, p Q (Hint write the value of the initial endowment for each consumer as a function of p Q Then, using the utility maximizing condition along with the budget constraint, find the optimal choice of Q for each consumer as a function of p Q Finally, use the market clearing condition, which states that the total consumption of Q must be equal to the total endowment of Q in the economy, to solve for p Q Keep your numbers as precise as possible at all stages of the calculation, and only round your final solution to the second decimal point )

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Consider an exchange economy with two consumers: Charlotte and Dylan, and two goods: quinoa (Q) and raspberries (R). Charlotte has an initial endowment of 99.6

Consider an exchange economy with two consumers: Charlotte and Dylan, and two goods: quinoa (Q) and raspberries (R). Charlotte has an initial endowment of 99.6 units of quinoa and 76 raspberries. Dylan has 124.4 units of quinoa and 78.1 raspberries.

Charlotte's utility function is given byU_C=Q_C^1/2R_C^1/2, whereQ_CandR_Care her consumption of Q and R, respectively. Dylan's utility function is given byU_D=Q_D^1/3R_D^2/3, whereQ_DandR_Dare his consumption of Q and R, respectively.

Suppose that the market price of quinoa isp_Q=2and the market price of raspberries isp_R=1.

Keeping the price of raspberries normalized atp_R=1, find themarket clearingprice of quinoa,p_Q. (Hint: write the value of the initial endowment for each consumer as a function ofp_Q. Then, using the utility maximizing condition along with the budget constraint, find the optimal choice ofQfor each consumer as a function ofp_Q. Finally, use the market clearing condition, which states that the total consumption ofQmust be equal to the total endowment ofQin the economy, to solve forp_Q. Keep your numbers as precise as possible at all stages of the calculation, and only round your final solution to the second decimal point.)