Midterm1_2022_Fall-v3-Solutions

2. In their paper Determinants of Technical Efficiency Among Coconut Smallholder Production in Johor, Malaysia: A Cobb Douglas Stochastic Frontier Production Approach 1 , Professors Z. Omar, F. A. Fatah analyze coconut output, Y , using the regression equation ln ( Y ) = c 0 + β 1 ln ( X 1 ) + β 2 ln ( X 2 ) + β 3 ln ( X 3 ) where ln ( x ) is the natural logarithm of x and where “ Y is coconut output, X 1 is the planting area in acres, X 2 is the quantity of fertilizer in kilograms, and X 3 is the quantity of fungicide in litres.” (a) (3 pts) What is the production function for coconuts? Briefly explain the basis for your equation. Solution: The regression analysis of a production function is based on the logarithm of the production function. Given this we can interpret the regression equation above as the logarithm of the production for coconuts. From this we can write the production function as Y = exp ( c 0 + β 1 ln ( X 1 ) + β 2 ln ( X 2 ) + β 3 ln ( X 3 )) (1) so Y = e c 0 X β 1 1 X β 2 2 X β 3 3 (2) (b) (3 pts) Derive an expression for the marginal product of fungicide. Solution: The marginal product is the partial derivative of production with respect to the associated variable, in this case X 3 . So MP-fungicide = ∂Y ∂X 3 (3) = ∂ ∂X 3 e c 0 X β 1 1 X β 2 2 X β 3 3 (4) = e c 0 X β 1 1 X β 2 2 ∂X β 3 3 ∂X 3 (5) so MP-fungicide = β 3 e c 0 X β 1 1 X β 2 2 X β 3 - 1 3 (6) or MP-fungicide = β 3 Y X - 1 3 (7) 1 IOP Conf. Series: Earth and Environmental Science 757 (2021) 012013 Page 3

6. The labor force growth rates for Yemen and Egypt are 2.2%/year and 1.68%/years, respectively. Consider the shock and response of Yemen’s economy if their labor force growth rate suddenly became that of Egypt using the Solow-Swan model. (a) (3 pts) At the moment of the shock, did the actual level of per-capita income shift, did the steady-state level of per-capita income shift, did both shift, or did neither shift? Briefly support your answer with the appropriate equation(s). Solution: The income per-capita, or Y L , curves are given by Y L = κ ( t ) α E ( t ) (23) where κ ( t ) = K ( t ) E ( t ) L ( t ) (24) for the actual curve and κ ( t ) = κ * = s g E + g L + δ 1 1 - α (25) for the steady-state associated balanced-growth curve. Since g L is the only variable that changed, it follows that the steady-state level of per-capita income shifted. (b) (5 pts) Draw a carefully annotated graph of Y L for the actual and balanced growth paths. Since the graph is qualitative, the numerical values for income per-capita are not important, but your graph should clearly show the difference between these two paths in the time before, during, and after the shock. Solution: Your graph should look something like this: Page 6