ufun2.r

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utility function

    

    

      

    

  

  

    

      

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  1. IN: 
      

    • consumption $ c \in \mathbb{R} $, 
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    • params = list(cutoff=$c^*$,CRRA=1.4,diag.plot=FALSE)
      • 

    

    2. 

  2. OUT: 

$$ u(c), u'(c), u''(c) $$



      

    • for consumption $ c \leq c^* $, return taylor series approx around $c^*$, denoted $\tilde{u}(c)$

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    3. 


    

    

      

    

  

  

    

      

        #
      

      
Definition:



$$ \begin{eqnarray*}
u(c) & = & \begin{cases}
\frac{c^{1-\sigma}-1}{1-\sigma} & \text{if }\sigma\neq1,c>c^{*}\\
\ln c & \text{if }\sigma=1,c>c^{*}\\
\tilde{u}(c) & \text{if }c\leq c^{*}
\end{cases}
\end{eqnarray*} $$


    

    

      
utility <- function(x,params){
	
	if (params$CRRA != 1){

    

  

  

    

      

        #
      

      
if $ CRRA \neq 1 $


    

    

      
		floor.x                    <- x             # copy actual consumption (can be negative) into floored consumption
		floor.x[x < params$cutoff] <- params$cutoff # replace consumption below cutoff with cutoff

    

  

  

    

      

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$$ diff.cons = c - c^* $$


    

    

      
		diff.cons                  <- x - floor.x   # difference between actual and floored consumption

    

  

  

    

      

        #
      

      

$$ floor.x^{1-CRRA} $$


    

    

      
		tmp.x     <- floor.x^(1-params$CRRA)

    

  

  

    

      

        #
      

      
gradient of utility: $ \frac{du}{dx} $


    

    

      
		du.dx     <- tmp.x/floor.x

    

  

  

    

      

        #
      

      
hessian of utility: $ \frac{d^2 u}{dx^2} $


    

    

      
		ddu.dxx   <- -params$CRRA*du.dx/floor.x

    

  

  

    

      

        #
      

      
third derivative of u: $ \frac{d^3 u}{dx^3} $


    

    

      
		dddu.dxxx <- -(1+params$CRRA)*ddu.dxx/floor.x

    

  

  

    

      

        #
      

      
CRRA utility at floor.x: $ u = \frac{floor.x^{1- CRRA} - 1}{1-CRRA} $


    

    

      
		util      <- (tmp.x - 1)/(1-params$CRRA)      
	} else {

    

  

  

    

      

        #
      

      
else if CRRA==1, log utility

    

    

      
		floor.x                    <- x
		floor.x[x < params$cutoff] <- params$cutoff # replace consumption below cutoff with cutoff
		diff.cons                  <- x - floor.x   # difference between actual and floored consumption

		util      <- log(floor.x) # u(cons)
		du.dx     <- 1/floor.x    # u'(cons)
		ddu.dxx   <- -1/floor.x^2 # u"(cons)
		dddu.dxxx <- 2/floor.x^3  # u'''(cons)
	}

    

  

  

    

      

        #
      

      
cubic approximation:


if $ c\leq c^*$, apply the following transformation. notice that $diff.cons = c-c^* = 0$ if $c>c^*$.

    

    

      

    

  

  

    

      

        #
      

      

$$ \tilde{u}(c)  =  u\left(c^{*}\right)+u'\left(c^{*}\right)\left[c-c^{*}\right]+\frac{1}{2}u''\left(c^{*}\right)\left[c-c^{*}\right]^{2}+\frac{1}{6}u'''\left(c^{*}\right)\left[c-c^{*}\right]^{3} $$


    

    

      
	return.util <- util + du.dx*diff.cons + 0.5*ddu.dxx*diff.cons^2 + 1/6*dddu.dxxx*diff.cons^3

    

  

  

    

      

        #
      

      

$$ \frac{d\tilde{u}(c)}{dc}  =  u'\left(c^{*}\right)+u''\left(c^{*}\right)\left[c-c^{*}\right]+\frac{1}{2}u'''\left(c^{*}\right)\left[c-c^{*}\right]^{2} $$


    

    

      
	return.grad <- du.dx + ddu.dxx*diff.cons + 0.5*dddu.dxxx*diff.cons^2

    

  

  

    

      

        #
      

      

$$ \frac{d^{2}\tilde{u}(c)}{dc^{2}}  =  u''\left(c^{*}\right)+u'''\left(c^{*}\right)\left[c-c^{*}\right] $$


    

    

      
	return.hess <- ddu.dxx + dddu.dxxx*diff.cons

    

  

  

    

      

        #
      

      
make plot if diag.plot was set to TRUE

    

    

      
	if (params$diag.plot){
		par(mfcol=c(3,2))
		plot(x=x,return.util,type="l",main="approx. utility",xlab="consumption",ylab="utility")
		abline(v = params$cutoff,col="red")
		grid()
		plot(x=x,return.grad,type="l",main="approx. gradient",xlab="consumption",ylab="gradient")
		abline(v = params$cutoff,col="red")
		grid()
		plot(x=x,return.hess,type="l",main="approx. hessian",xlab="consumption",ylab="hessian")
		abline(v = params$cutoff,col="red")
		grid()
		plot(x=x,util,type="l",main="capped utility",xlab="consumption",ylab="capped utility")
		grid()
		plot(x=x,du.dx,type="l",main="capped gradient",xlab="consumption",ylab="capped gradient")
		grid()
		plot(x=x,ddu.dxx,type="l",main="capped hessian",xlab="consumption",ylab="capped hessian")
		grid()
		par(mfcol=c(1,1))
	}

    

  

  

    

      

        #
      

      
return list of values

    

    

      
	return(list(utility=return.util,gradient=return.grad,hessian=return.hess))
}