load(draw)$ 
set_draw_defaults( draw_realpart=false,font_size=10)$
fpprintprec:5$ ratprint:false$  kill(all)$

# Nonlinear Pricing: 
    Capturing Consumer Surplus

The workbook deals with "nonlinear pricing" -- pricing when the firm
can either charge different prices for different units of a good or 
can charge a fee in addition to the per-unit price.

## Review and Introduction

Begin with a firm that faces a constant-elasticity demand for its product
and whose marginal cost curve is linear. Parameters are also provided.

p(x,A,a):=(x/A)^(1/a);  
mr(x,A,a):=(1+1/a)*(x/A)^(1/a);
mc(x,C,c):=C+c*x;   [A0,a0,C0,c0]:[5000, -1.5,5,2]$

find_root( ) yields the profit maximizing quantity, x0, and the 
quantity at which p = mc, xe0. The prices at these two quantities 
are determined.

x0:find_root(mr(x,A0,a0)-mc(x,C0,c0),x,.01,200)$  
p0:p(x0,A0,a0)$
xe0: find_root(p(x,A0,a0)-mc(x,C0,c0),x,.01,200)$  
pe0:p(xe0,A0,a0)$
matrix(["x0", "p0", "xe0", "pe0"], [x0, p0, xe0, pe0]);

The graph shows the equilibrium price and quantity. It also
shows the quantity at which p = mc. The triangular area
with the inverse demand curve and the marginal cost curve
as two sides, between x0 and xe0, is the Deadweight Loss.

wxdraw2d(yrange = [0, 1.5*p0], xlabel="x", ylabel=
"$ per unit",label_alignment=left,label(
  ["Consumer Surplus", 0, 1.1*p0]), label(
  ["Deadweight Loss", x0, pe0]),points_joined=true, 
 line_type=dots,points([[0, p0], [x0,p0],[x0,0]]),
 line_type=solid,line_width=2, key="Marginal Revenue",
 explicit(mr(x,A0,a0),x,0,1.25*xe0),color=black,
 key="Demand", explicit(p(x, A0, a0),x, 0, 1.25*xe0),
 color =red,key ="Marginal Cost",
 explicit(mc(x, C0, c0), x, 0, 1.25*xe0),
 title="Unexploited Values", dimensions = [640,480])$

### Pricing Mechanisms for Capturing Surplus

The text explores two important ways that a producer might use pricing 
schemes to appropriate some of the unexploited surpluses that the graph
reveals: market segmentation and nonlinear pricing.

### Conditions for Capturing Surplus

The text identifies and discusses three conditions that are required
for a firm to add to its profits by using either of 
these mechanisms.
1. The firm must be in an industry of one or more price-searchers.
2. Different units of the firm's product must have different values (measured
as willingness to pay) to buyers, and the seller must have information about
these differences.
3.The market must be segmented in some way, so that the buyer can exploit the
information about value differences.

### Market Segmentation

The term price discrimination refers to the basing of pricing on 
the information described above. The text discusses three major 
categories of price discrimination.