Question

Dilek is also hiking the Appalachian trail but was not lucky and didn't bump into any of the other hikers but she is very resourceful. She ran out of both water and energy bars. Still she realized she can get water by walking to a nearby river and can get food by picking berries Both production activities require only time to be carried out which is where she gets lucky since time is all she has, more specifically she has 16 hours of working time. The technology that she has to produce water and food is described as follows: y= 0.25 Ly Also note that she has the exact same function for her decision making process. If you successfully answer the first question of this problem you should get a PPF that yields to preferences as Manu hence you can use his utility MRT 16(16 a2)/2 1) Find the algebraic expression for the PPF and graph it 2) Find the efficient allocation of water and food for Dilek 3) Find the equilibrium of Dilek's economy. For this if you should get the following Cost Function for Dilek's Firm: C(r, y)wa216wy And this means that the marginal costs are: MC 2wr and MC, 32wy the definitions correctly you use

Answer 1

1).

Consider the given problem here the production function of “water” and “food” are given below.

=> “X=Lx^0.5” and “Y=0.25*Ly^0.5”. Now, she has “16 hours of working time”.

=> the above production function can be written as follows.

=> “Lx=X^2” and “Ly=(4Y)^2 = 16*Y^2”, => the equation of PPF is given below.

=> Lx+Ly=16, => X^2 + 16*Y^2 = 16, be the equation of PPF.

Y Food T1 T2 X Water

So, the above fig shows the PPF of the economy.

2).

Now, the PPF is given by, => X^2+16*Y^2=16, => 2X*dX + 16*2Y*dY = 0, => X*dX + 16*Y*dY = 0.

=> dY/dX = (-X/16Y), => MRT = X/16Y. Now, the utility function is given by, => U=X^0.5*Y.

=> MUx=0.5*X^(-0.5)*Y and MUy=X^0.5, => MRS = MUx/MUy = [0.5*X^(-0.5)*Y]/X^0.5 = Y/2X.

=> MRS = Y/2X. now, at the equilibrium “MRS” is equal to “MRT”.

=> MRS = MRT, => Y/2X = X/16Y, => Y/X = X/8Y, => 8Y^2 = X^2. The PPF equation is given by.

=> X^2 + 16Y^2 = 16, => 8Y^2 + 16Y^2 = 16, => 24*Y^2 = 16, => Y^2 = 16/24 = 2/3, => Y^2 = 2/3.

=> Y = 0.82 units = Food.

Now, X^2 = 8Y^2, => X^2 = 8(2/3) = 16/3, => X=2.31 units = Water.

So, the efficient allocation of water and food are given by “Y=0.82 units” and “X=2.31 units” respectively.

3).

Consider the given fig shows the equilibrium of this economy.So, here “E” be the equilibrium of this economy where, the production of “X” and “Y” are “2.31” and “0.82” respectively. Now, “Lx=X^2=16/3” and “Ly=16*Y^2=16*(2/3) = 32/3. So, here “16/3 hours goes to the production of X” and “32/3 hours goes to the production of Y”.

Y Food T1 E 0.82 2.31 T2 X Water