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ECO 204 Chapter 12: Practice Problems & Solutions for The Long Run Cost Minimization Problem in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (a) Does the company have increasing, constant or decreasing returns with respect to labor and materials?
Answer
( ) Suppose all inputs are scaled by a factor
. We need to check if the output with times the initial variable inputs
and same level of fixed inputs is greater than, equal to, or less than times the output with the initial inputs.
The output with times the initial variable inputs and same level of fixed inputs is:
( )
( ( ) ( ) )
( ) Clearly:
corresponds to increasing returns in labor and materials since:
times more labor and materials
corresponds to constant returns in labor and materials since:
times more labor and materials
corresponds to decreasing returns in labor and materials since:
times more labor and materials
For example, suppose so that if we double labor and materials levels, the new output is:
( ) ( ) Observe:
corresponds to increasing returns in labor and materials since:
and materials
corresponds to constant returns in labor and materials since: corresponds to decreasing returns in labor and materials since: 67
ECO 204 Chapter 12: Practice Problems & Solutions for The Long Run Cost Minimization Problem in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (b) Suppose
,
,
,
,
. Derive the optimal demand
for labor, materials and the short run cost function. Is the cost function consistent with the company’s returns to labor
and materials?
Answer
The CMP Lagrangean is:
[ I will do the calculations algebraically and numerically side by side:
Algebra Answer Numerical Answer
[ [ st The 1 FOC implies: st The 1 FOC implies: nd The 2 FOC implies:
nd The 2 FOC implies: Equating the two expressions for and rearranging: { } { }
Equating the two expressions for and rearranging: {
{ rd Substitute this in the 3 FOC: ( ) }
} rd Substitute this in the 3 FOC: ( )
[( ) 68
ECO 204 Chapter 12: Practice Problems & Solutions for The Long Run Cost Minimization Problem in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. [
[( ) From which: From which (labor depends on wages):
[
[( ) Now, the cost function is:
()
()
()
()
()
Here: And: The increases with output, concomitant with decreasing returns. In fact, since: This production function indeed has decreasing returns to labor and materials.
(c) Will a dollar increase in impact cost more or less or the same as a dollar increase in ? Answer
This calls for the envelope theorem:
[ If wages increases by a dollar, the impact on optimal cost (subject to constraint) is simply:
69
ECO 204 Chapter 12: Practice Problems & Solutions for The Long Run Cost Minimization Problem in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ()
() If the price of materials increases by a dollar, the impact on optimal cost (subject to constraint) is simply:
()
() In this problem (by coincidence) . Thus the company is impacted equally by a dollar increase in wages versus a dollar increase in price of materials. 70
ECO 204 Chapter 12: Practice Problems & Solutions for The Long Run Cost Minimization Problem in ECO 204 (this version 20122013)...
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