Suppose the production function of an economy is given by:
Y = A̅K3/4 L1/4
A. Prove the production function is a homogeneous and follows a constant returns to scale.
Interpret the nature of production function.
B. Create a table and find out what are the five equations and five unknowns based on table-
that is reported in your text (pp.).
C. Solve these equations to get the solution to the model. Put your solution in the form of a Table
and explain the hiring rule of labor and capital based on graph of factor market equilibrium.
D. Estimate for the output per person in the long-run. Show it graphically. Explain why it follows
diminishing returns to scale.
E. Estimate TFP in the long-run, and explain TFP growth is crucial to make differences among
countries in terms of per capita GDP.
F. Point out the limitation of this model to explain long-run economic growth.
Given,
Demand curve of good Y:
"QY=10\u22125P_Y+P_X"
Where,
QY=Quantity of good Y
PY=Price of good Y
PX=Price of good X
PY=price of the good Y=K1
PX=price of the substitute good=K2
The following formulas will be used:
Price elasticity of demand "=\\frac{\u2202Q_Y}{\u2202P_Y}\u00d7\\frac{P_Y}{Q_Y}"
Cross−price elasticity of demand "=\\frac{\u2202Q_Y}{\u2202P_x}\u00d7\\frac{P_x}{Q_Y}"
Calculation of price elasticity of demand:
"\\frac{\u2202Q_Y}{\u2202P_Y}=\u22125"
Let us substitute the values of PY and PX in the demand function for quantity
of good Y
"Q_Y=10\u22125\u00d71+2\\\\Q_Y=10\u22125+2\\\\Q_Y=7"
Price elasticity of demand"=-5\\times{1}{7}"
Price elasticity of demand=−0.71
Calculation of cross-price elasticity of demand:
"\\frac{\u2202Q_Y}{\u2202P_X}=1\\\\P_X=K2\\\\Q_Y=7"
Cross−price elasticity of demand"=1\\times \\frac{2}{7}"
Cross−price elasticity of demand=0.29
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