pls answer asap 4.2 A producer of chocolates, Choco, produces chocolates (x) (in
ID: 1165617 • Letter: P
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pls answer asap
4.2 A producer of chocolates, Choco, produces chocolates (x) (in kg) according to the production function x(K, Z) K Zi by using a mass of cacao (k) and sugar (Z). The full competition). price of one unit of Z is pz and the price of one unit of K is pk(Assumption: a. Calculate the optimal amounts of K and Z that Choco uses as a function of the amount of chocolates sold (x) (assume cost minimization). b. What are Choco's costs Co) as a function of the amount of chocolates sold? c. How does Choco's supply function of chocolates péx) look like?Explanation / Answer
a).
So, here the production function of “chocolate” is given by “X=K^1/4*Z^1/4”. So, the marginal product of these input are given by.
=> “MPk = (1/4)*K^(-3/4)*Z^(1/4)” and “MPz = (1/4)*Z^(-3/4)*K^(1/4)”. Now, the price of “K” and “Z” are “Pk=1/8” and “Pz=1/2”. So, at the equilibrium “MRTS” must be equal to “Pk/Pz”.
=> MRTS = Pk/Pz, => [(1/4)*K^(-3/4)*Z^(1/4)]/[ (1/4)*Z^(-3/4)*K^(1/4)] = (1/8)/(1/2).
=> [K^(-3/4)*Z^(1/4)] / [Z^(-3/4)*K^(1/4)] = (2/8).
=> Z / K = ¼, => Z = (1/4)*K.
Now, the production function is given by, X = K^1/4*Z^1/4.
=> X = K^1/4*[(1/4)*K]^1/4, => X = K^1/2*(1/4)^1/4, => X^2 = K/2, => K = 2*X^2.
Now, Z = K/4, => Z = 2*X^2/4 = X^2/2, => Z = X^2/2.
So, here the optimum “K” and “Z” in terms of “X” are given by “2*X^2” and “X^2/2” respectively.
b).
So, here the cost function is given by, “C = Pk*K + Pz*Z = (1/8)*K + (1/2)*Z.
=> C = (1/8)*[2*X^2] + (1/2)*[X^2/2] = X^2/4 + X^2/4, => C = X^2/2.
c).
Now, here the supply function is given by, “P=MC”, => P = 2*X/2 = X, => “P = X”. So, here the supply function is given by “P=X“.
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