Consumer Equilibrium

• Now we have all the tools we need to analyze consumer choice Consumer Choice • Suppose a consumer faces the constraints – p1x1 + p2x2 ≤ m and x1 ≥ 0, x2 ≥ 0. • The consumer’s problem is to choose and to ECON 370: Microeconomic Theory maximize utility (i.e. to attain highest possible indifference curve) s.t. the above constraints, Summer 2004 – Rice University • That is, the consumer is choosing the most Stanley Gilbert preferred point in the budget set.

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Optimal Choice Rational Constrained Choice

• Provided that this is an “interior” equilibrium, x2 – That is, that it involves strictly positive amounts of all goods • And assuming all our assumptions (A1 – A4) hold: • Then we have equality of Optimal Choice – Slope of budget constraint (-p1 / p2) – Slope of indifference curve (MRS)

– Ratio of marginal utilities (MU1/MU2) * * • And our budget is exhausted: p1x1 + p2x2 = m Preferences • Interpretation

– Willingness to trade (-slope of IC) = p1 / p2 – Marginal utility per dollar equalized across goods Budget x1

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1 Characteristics of Optimal Choice What about savings?

• We said that generally all money is • Interior solution x2 spent • Slope of indifference curve = Slope of • That is p x * + p x * = m budget constraint 1 1 2 2 • How do we deal with savings? •(that is) MRS = -p / p 1 2 – By assuming our consumption choices * * • budget is exhausted: p1x1 + p2x2 = m include spending today v. spending in the future. Saving for Retirement

Spending Today x1

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Finding the Optimum Example

• We will demonstrate two methods for finding the • Assume for this example that optimal choice – Preferences are Cobb-Douglass α 1 – α –MRS = p1 / p2 – U = x y

– Maximizing the Utility function – Budget constraint is pxx + pyy ≤ m • (At least) one other that I will not cover:

MU MU 1 = 2 p1 p2

– Mathematically, it is indistinguishable from the first

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2 MRS Solution MRS Solution (cont)

In order to calculate MRS, we first calculate Marginal Utilities We will use x* and y* to represent our solution α ∂U ∂U α α y* p −1 1−α MU = = ()1−α x y−α x (from previous page) MU x = = αx y y * = ∂x ∂y 1−α x py Solving for y*: Then we calculate MRS as: α * 1−α px * xα −1y1−α y = x dy ∂u ∂x MU x α α y MRS = = = = −α = α py dx ∂u ∂y MU y ()1−α x y 1−α x

* * Since MRS equals the slope of the budget line, we get: Since all money is spent, we also have: pxx + pyy = m

α y px * = * 1−α p * * 1−α * p x So: p x + p x x = p x + p x = x = m 1−α x py x y x x α py α α

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MRS Solution (cont 2) Maximization Solution

Finally: • Our problem is: α 1 – α * –max{u(x, y)} = max{x y } px x * αm If= m Then x = – Subject to the requirement that pxx + pyy ≤ m. α px • Remembering that all money will be spent, we have: * 1−α px * Substituting this into y = x (from before) – pxx + pyy= m α py • From here, the simplest way to proceed is to substitute the (1−α )m modified constraint into problem to be solved: Gives y* = – y = (m – pxx) / py py • Substituting into the original problem gives: * αm * (1−α )m α 1−α α 1−α α −1 So, our solution is: x = y = max{x y }= max{x (m − px x) py } px py

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3 Maximization (cont) Maximization (cont 2) α Remembering our calculus, maximizing a function requires: Since all this mess equals zero, we can eliminate common terms: 2 df (x) d f (x) α −1 α −1 1−α α −α = 0 and ≤ 0 αpy [ x (m − px x) − px (1−α )x (m − px x) ]= 0 dx dx2 ⎛ m − px x ⎞ α α 1−α α −1 ⇒ ⎜ ⎟ − ()1−α px = 0 So max{x ()m − px x py } ⎝ x ⎠ α d α 1− ⇒ x ()m − p x pα −1 = 0 ⇒ α()m − px x − (1−α )px x = αm −αpx x − px x +αpx x = αm − px x = 0 dx x y

α −1 α −1 1−α α −α αm ⇒ py [ x ()()m − px x − px 1−α x (m − px x) ]= 0 ⇒ x = Exactly as before px

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Maximization (cont 3) Cobb-Douglas Observation α Verifying the Second Order Condition: • Note that for Cobb-Douglas utility function d – Demands are linear in income pα −1[ xα −1()()m − p x 1−α − p 1−α xα (m − p x)−α ] dx y x x x – Expenditure shares are constant α – Expenditure shares sum to one (reorganizing it a bit)

α 1−α −α α −1⎡ d ⎛ mα ⎞ d ⎛ m ⎞ ⎤ = py ⎢ ⎜ − px ⎟ − px ()1−α ⎜ − px ⎟ ⎥ ⎣ dx ⎝ x ⎠ dx ⎝ x ⎠ ⎦ * αm * (1−α )m ⎡ m m −α α m m −α −1⎤ x = y = α −1 ⎛ ⎞ ⎛ ⎞ < 0 p p = py ⎢− ()1− 2 ⎜ − px ⎟ − px ()1−α 2 ⎜ − px ⎟ ⎥ x y ⎣ x ⎝ x ⎠ x ⎝ x ⎠ ⎦

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4 “Corner” Solutions Corner Example

• An “interior” solution was earlier described as: – involving strictly positive amounts of all goods x2 – And all our assumptions (A1 – A4) hold Optimal Choice • If a solution does not meet these requirements, it is called a “Corner” Solution • Other methods have to be used to solve it

Preferences

Budget x1

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