Question

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.

Answer 1

Given,

Demand curve of good Y:

$QY=10−5P_Y+P_X$

Where,

Q_Y=Quantity of good Y

P_Y=Price of good Y

P_X=Price of good X

P_Y=price of the good Y=K1

P_X=price of the substitute good=K2

The following formulas will be used:

Price elasticity of demand $=\frac{∂Q_Y}{∂P_Y}×\frac{P_Y}{Q_Y}$

Cross−price elasticity of demand $=\frac{∂Q_Y}{∂P_x}×\frac{P_x}{Q_Y}$

Calculation of price elasticity of demand:

$\frac{∂Q_Y}{∂P_Y}=−5$

Let us substitute the values of P_Y and P_X in the demand function for quantity

 of good Y

$Q_Y=10−5×1+2\\Q_Y=10−5+2\\Q_Y=7$

Price elasticity of demand$=-5\times{1}{7}$

Price elasticity of demand=−0.71

Calculation of cross-price elasticity of demand:

$\frac{∂Q_Y}{∂P_X}=1\\P_X=K2\\Q_Y=7$

Cross−price elasticity of demand$=1\times \frac{2}{7}$

Cross−price elasticity of demand=0.29