Please answer ASAP Suppose a consumer, Sheldon Cooper, has the following utility

Question

Please answer ASAP

Suppose a consumer, Sheldon Cooper, has the following utility function for comic books (C) , and tangerine chicken (T): Use the utility function to define the indifference curve that passes through all of the bundles of C and T that provide the consumer with exactly 10 units of utility and solve the indifference curve for T as a function of C. (Hint: set the utility function equal to 10) Find the marginal rate of substitution as a function of C (for just the indifference curve in part (a)) by differentiating the indifference curve with respect to C; is the indifference curve downward sloping? Does the indifference curve have a diminishing marginal rate of substitution? Prove using calculus or with a numerical example, and remember that we're concerned with how the absolute value of the MRS changes. Find the marginal rate of substitution as a function of both C and T by using the definition involving marginal utilities. (This is the more general representation of MRS as it applies to all possible bundles and so to the complete set of ICs.) Suppose Jeremy spends all of his income on crab cakes, C and lobsters L, and his utility function is given by: If Jeremy's income is Y = 144, the price of crab cakes is pc = $4 and the price of lobster ispL = $2: Assuming lobster is plotted on the y-axis, find Jeremy's marginal rate of substitution and the marginal rate of transformation he faces. Assuming an interior solution exists, solve for the Jeremy's optimal bundle of crab cakes and lobster using just the marginal rates of substitution and transformation and Jeremy's budget constraint. Derive expressions for the marginal utility per dollar Jeremy receives from consuming crab cakes and the marginal utility per dollar Jeremy receives from consuming lobster and show that the two are equal at the optimal bundle. Finally, set up Jeremy's utility maximization problem and use the method of Lagrange multipliers to solve for the optimal quantities of crab cakes and lobster. Do you get the same answers as before? Solve for the optimal value of the Lagrange multiplier. Does it look familiar?

Explanation / Answer

(a) for indifference curve: U(C,T) = 2(CT)^.5 = 10

=> C*T = 5^2 = 25 => T = 25/C

thus T(c) = 25/c

(b) T'(c) = d(25/c)/dc = -25/c^2

clearly the slope is negative and hence downward sloping

(c) T''(c) = 50/c^3 > 0 , so the indifference curve doesnt have a diminishing MRS

(d) MRS(C,T) = MU(C)/MU(T) , where MU(C) is the marginal ultility wrt C

MU(C) = d(U(C,T))/dC = (T/C)^0.5

MU(T) = (C/T)^0.5

thus MRS(C,T) = T/C

Hope this helps

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