Problem 1: Write each production function given below in terms of output per person y a Y/L and...

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Problem 1: Write each production function given below in terms of output per person y a Y/L and capital per person k = K/L. Plot these per person versions in a graph with y on the vertical axis and k on the horizontal axis. (You can assume A is a constant positive number). a) Y AK IOL2/3 and Y AK3/40/4 (plot them on the same graph). b)Y.K. c)Y=K+ AL d) = K — AL
Problem 2: Consider the production model that we studied in class (Chapter 4) and assume that the production function is now given by Y = AK3/ XXXXXXXXXXEverything else in the model remains unchanged. a) Reproduce the analogue of Table 4.1 for this new economy. That is, state carefully what are your endogenous variables, parameters, and what equations you have in order to solve for equilibrium. b) Fully describe the solution of the model (in other words provide formulas that relate each endogenous variable of the model with the known parameters). c) Let A = 1, it = 100, L = 1000. What are the equilibrium values of the wage, the rental rate of capital, and total output? d) How does the equilibrium wage that you reported in (c) change if L = 1500 and everything else remains the same? Why? e) Again consider general values for A, K, L. What is the equilibrium value of output per person?
Problem 3: The (steady state) equilibrium value of output per person in the Solow growth model (Chapter 5) is given by omin
The equilibrium value of output per person in the production model (Chapter 4) is given by
y* = Aka/3 (2)

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David · Accepted Answer

Answer to problem1: 
a) Take Y =  ̅K1/3L2/3 
Divide both side by L, we get 
Y/L =  ̅(K/L)1/3   or y =  ̅(k)1/3  …………… (1)  
Now take Y =  ̅K3/4L1/4 
Again divide both side by L, we get 
Y/L =  ̅(K/L)3/4 =  y =  ̅(k)3/4 ………………… (2) 
Graphs of (1) and (2) are as follows;
(b) Y = K 
Divide both side by L, we get 
y = k
(b) Y = K+ ̅L 
Divide both side by L, we get 
y = k+ ̅
(d)  
 Y = K- ̅L 
Divide both side by L, we get 
y = k- ̅
Answer to Problem2: 
a) Y =  ̅K3/4L1/4 
Divide both side by L, we get 
Y/L =  ̅(K/L)3/4 =  y =  ̅(k)3/4 
There is only one endogenous variable; total output (Y) 
Parameters are labor, capital,  ̅, wage, rental rate of capital 
We have to maximize Y =  ̅K3/4L1/4 with respect to constraints; L≤ ̅ and K≤ ̅ 
b)  
Marginal product of labor = (1/4)  ̅ (K/L)3/4 …………… (3) 
Marginal product of Capital = (3/4)  ̅ (L/K)1/4 …………… (4) 
At optimum Marginal (product of labor/ Marginal product of Capital) =  real wage (w)/rental cost 
of capital (r) 
So at optimum we would have following condition; 
(1/4)  ̅ (K/L)3/4/(3/4)  ̅ (L/K)1/4 = w/r or 
(1/3)(K/L) = w/r or 
K = 3L(w/r) 
Substituting this in the production function Y =  ̅K3/4L1/4, we get 
Y =  ̅[3L(w/r)]3/4 L1/4 
c) 
At the equilibrium, wage = marginal product of labor 
So from (3), wage (w) = (1/4)  ̅ (K/L)3/4 
At  ̅ =1,  ̅ = 1000 and  ̅ = 100 
Hence equilibrium value of wage (w) = (1/4)*(100/1000)^(3/4) = 0.04446 
Similarly, rental cost of capital (r) = marginal product of capital 
So from (4), rental cost of capital (r) = (3/4)  ̅ (L/K)1/4 
Hence equilibrium value of rental cost of capital (r) = (3/4)*(1000/100)^(1/4) = 1.33371 
Total output (Y) = 1*((100)^(3/4))*((1000)^(1/*4)) = 177.

Problem 1: Write each production function given below in terms of output per person y a Y/L and capital per person k = K/L. Plot these per person versions in a graph with y on the vertical axis and k...

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