Chapter 6 Henderson and Quandt Microeconomics, Questions 6.1-6.3_ Solutions
6-1 Two hundred consumers derive utility from the consumption of two goods. Each has the
utility function U = 10 +5 +. Each has a fixed income of 100 dollars. Assume that the
price of Q₂ is 4 dollars per unit. Express the aggregate demand for Q₁ as a function of p₁. Is the
aggregate demand curve downward sloping?
To express the aggregate demand for Q1 as a function of its price, we need to find the optimal consumption bundle for each consumer at different prices of Q1. We can do this by using the consumer’s utility maximization problem:
Maximize U = 10q1 + 5q2 + q1q2
Subject to the budget constraint: p1q1 + p2q2 = m
Where p1 is the price of Q1, p2 is the price of Q2, m is the income, and q1 and q2 are the quantities of Q1 and Q2 consumed, respectively.
Solving the utility maximization problem using Lagrange multipliers, we get the following demand functions:
Q1 = (10 – p2) / (2 + p1)
Q2 = (10 – p1) / (2 + p2)
Substituting p2 = 4 and m = 100, we get:
Q1 = (10 – 4) / (2 + p1) = 6 / (2 + p1)
The aggregate demand for Q1 is the sum of individual demand functions for all consumers. Since there are 200 consumers with identical preferences and budget constraints, we can simply multiply the individual demand function by 200:
Q1 = 1200 / (2 + p1)
The aggregate demand curve for Q1 is downward sloping, as the quantity demanded of Q1 decreases as its price increases. This is due to the negative slope of the individual demand function for Q1, which is a result of the substitution effect being larger than the income effect for a normal good.
6-2 Construct a short-run supply function for an entrepreneur whose short-run cost function is
To construct the short-run supply function for the entrepreneur, we need to find the quantity of output that the entrepreneur is willing to supply at different prices. We can do this by finding the marginal cost function and comparing it to the price.
The marginal cost function is the derivative of the cost function with respect to output, which is:
MC(q) = 0.12q^2 – 1.6q
To find the short-run supply function, we need to find the output level that corresponds to the minimum of the average variable cost (AVC) curve. The AVC curve is given by:
AVC(q) = (0.04q^3 – 0.8q^2 + 10g + 5) / q = 0.04q^2 – 0.8q + 10g/q + 5/q
To find the minimum of the AVC curve, we take the derivative with respect to q and set it equal to zero:
dAVC(q)/dq = 0.08q – 0.8 – 10g/q^2 – 5/q^2 = 0
Solving for q, we get:
Q = (2 + 2.5g)^(1/3)
The short-run supply function is then given by:
Q = f(p) = (2 + 2.5g)^(1/3) if p ≥ AVC(q) = 0.04q^2 – 0.8q + 10g/q + 5/q
The entrepreneur will produce the quantity q if the market price is at least equal to the AVC at that quantity. If the price is lower than the AVC, the entrepreneur will shut down and produce nothing. Therefore, the short-run supply function is upward-sloping, as the entrepreneur will only produce more output as the market price increases.
6-3 A good Q is produced using only one input X. The market for Q is supplied by 100 identical
competitive firms each of which has the production function q = where 0<β<1. Each firm
behaves as if the price of X were constant. However, the industry as a whole faces an upward
sloping supply curve for X:r = b(100x) where b>0. Derive the industry's long-run supply curve.
In the long run, firms can adjust their inputs to optimize their production and profits. Therefore, in the long run, each firm will choose the optimal level of input x to produce q at the minimum possible cost. This minimum cost is given by the long-run marginal cost (LRMC) curve, which is the cost of producing one more unit of output in the long run.
Since each firm has the same production function, their LRMC curves are identical. The LRMC curve for each firm is given by:
LRMC = w/(βx^(1-β))
Where w is the wage rate paid for input X.
In the long run, firms will enter or exit the industry depending on whether they are making profits or losses. If the industry is making positive profits, new firms will enter the industry, increasing the supply of Q and driving down the price. If the industry is making losses, some firms will exit the industry, reducing the supply of Q and driving up the price.
The industry’s long-run supply curve is the horizontal sum of the LRMC curves of all firms that are producing in the long run. The industry will produce at the point where the LRMC of the last firm equals the market price of Q.
To find the industry’s long-run supply curve, we first find the equilibrium input level x* for each firm by equating the LRMC to the price of input X:
w/(βx*^(1-β)) = b(100x*)
Solving for x*, we get:
X* = [(w/b)/100^(1/β)]^(1/(1-β))
Substituting x* into the production function, we get the equilibrium output level q* for each firm:
Q* = x*^β * 100
The industry’s total output Q* is the sum of the output of all firms:
Q* = 100q* = 100[(w/b)/100^(1/β)]^(β/(1-β))
The industry’s long-run supply curve is then the inverse function of Q*(w), which gives the equilibrium market price of Q for each level of industry output:
w/b = [Q*(w)/100]^(1-β)/100^β
Taking the inverse of this equation, we get the industry’s long-run supply curve:
Q*(w) = [w/b * 100^(1/β)]^(1/(1-β))
The industry’s long-run supply curve is upward sloping, as the quantity supplied of Q increases as the price increases. The slope of the long-run supply curve depends on the elasticity of the LRMC curve and the elasticity of the supply of input X.