All previous examples assumed a constant returns to scale technologies.
 
 
Example 10: Decreasing returns to scale
 
 
Sector-specific inputs can represent decreasing returns to scale. Rents accrue to a fictitious factor called "capital".
 
Note:  If we wanted to think of the specific factor earnings as the rents of firms with decreasing returns, we could introduce separate agents with endowments equal 
 to the specific factors returns to the specific factors would then be the profits of the firms.
 
 Production Sectors Consumers
 Markets | X Y W | CONS
 ------------------------------------------------------
 PX | 100 -100 |
 PY | 100 -100 |
 PW |  200 | -200
 PL | -50 -50 | 100
 PKX | -50 | 50
 PKY | -50 | 50
 ------------------------------------------------------

For constant returns to scale technologies the cost function, in equilibrium, defines the price of output: p = c(r, w) 
where w is the exogenous wage rate and r is the residual return to the sector's fixed factor.
 
 
The price elasticity of supply at the benchmark point is:
 
(eq. I) ES_X = dX/ d(p/w) * (p/w)/X
   
If we use the calibrated CES cost function and equate cost to the price of output, we get an expression for the price elasticity of supply as follows:
 
(eq. II) ES_X = ESUBL * (1 - SHAREKX) / SHAREKX
 
To check the price elasticity of supply (ES_X):
1) we change the price of good X with a production subsidy (SX), 
2) then we compute the producer price of good X (PXP) as the consumer price PX.L plus the subsidy: PXP = (1 + SX) * PX.L. 
3) We divide the change in supply of X (X.L - 1) by the change in the ratio of producer price of X to price of labor; and multiply by the benchmark ratio (1) divided by the benchmark level of supply of good X (1) to produce a finite difference approximation of the elasticity: 
 
(eq. III) ES_X = (X.L - 1)/ ((PXP/PL.L) - 1)