# Consider two consumers C1 and C2 that have preferences over food (F) and engineered goods (E). The consumers both have preferences U(F, E) = log F + α log E and have initial endowments F e 1 , F e 2 , E e 1 and E e 2 . If it helps you in your computations define I1 = F e 1 + pEe 1 and I2 = F e 2 + pEe 2 . (a) Given prices pF = 1 and pE = p, write down the budget constraint of consumer 1. (b) Solve consumer 1’s problem and find the his optimal consumption F ∗ 1 , E ∗ 1 . Express it as a function of p, F e 1 and E e 1 . (c) Find the consumption F ∗ 2 , E ∗ 2 of consumer 2 as a function of p, F e 2 , E e 2 (d) Find the price p ∗ that equilibrates the market for good F. Express p ∗ as a function of α, F e 1 + F e 2 and E e 1 + E e 2 . (e) Check that p ∗ equilibrates the market for good E. We now add a third consumer C3 that has initial endowments F e 3 , E e 3 and utility U(F, E) = log F + α log E. (f) Given prices pF = 1 and pE = p find the individual demand of consumer 3 as a function of p, F e 3 and E e 3 . 3 (g) Given price p what is the individual demand of consumers 1 and 2 ? (h) Write down the equation that characterizes equilibrium on the market for F.

Consider two consumers C1 and C2 that have preferences over food (F) and engineered goods (E). The consumers both have preferences U(F, E) = log F + α log E and have initial endowments F e 1 , F e 2 , E e 1 and E e 2 . If it helps you in your computations define I1 = F e 1 + pEe 1 and I2 = F e 2 + pEe 2 .

(a) Given prices pF = 1 and pE = p, write down the budget constraint of consumer 1.

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(b) Solve consumer 1’s problem and find the his optimal consumption F ∗ 1 , E ∗ 1 . Express it as a function of p, F e 1 and E e 1 .

(c) Find the consumption F ∗ 2 , E ∗ 2 of consumer 2 as a function of p, F e 2 , E e 2

(d) Find the price p ∗ that equilibrates the market for good F. Express p ∗ as a function of α, F e 1 + F e 2 and E e 1 + E e 2 .

(e) Check that p ∗ equilibrates the market for good E. We now add a third consumer C3 that has initial endowments F e 3 , E e 3 and utility U(F, E) = log F + α log E.

(f) Given prices pF = 1 and pE = p find the individual demand of consumer 3 as a function of p, F e 3 and E e 3 . 3

(g) Given price p what is the individual demand of consumers 1 and 2 ?

(h) Write down the equation that characterizes equilibrium on the market for F.

**Consider two consumers C1 and C2 that have preferences over food (F) and engineered goods (E). The consumers both have preferences U(F, E) = log F + α log E and have initial endowments F e **1 ,** F e **2 , E e** 1 and E e **2 .** If it helps you in your computations define I1 = F e 1 + pEe 1 and I2 = F e 2 + pEe 2 .**

**(a) Given prices pF = 1 and pE = p, write down the budget constraint of consumer 1.**

**(b) Solve consumer 1’s problem and find **the his** optimal consumption F ∗ **1 ,** E ∗ **1 .** Express it as a function of p, F e 1 and E e **1 .

**(c) Find the consumption F ∗ **2 ,** E ∗ 2 of consumer 2 as a function of p, F e **2 , E e** 2**

**(d) Find the price p ∗ that equilibrates the market for good F. Express p ∗ as a function of α, F e 1 + F e 2 and E e 1 + E e **2 .

**(e) Check that p ∗ equilibrates the market for good E. We now add a third consumer C3 that has initial endowments F e **3 , E e** 3 and utility U(F, E) = log F + α log E.**

**(f) Given prices pF = 1 and pE = p find the individual demand of consumer 3 as a function of p, F e 3 and E e **3 .** 3**

**(g) Given price p what is the individual demand of consumers 1 and **2 ?

**(h) Write down the equation that characterizes equilibrium on the market for F**.

Consider two consumers C1 and C2 that have preferences over food (F) and engineered goods (E). The consumers both have preferences U(F, E) = log F + α log E and have initial endowments F e 1 , F e 2 , E e 1 and E e 2 . If it helps you in your computations define I1 = F e 1 + pEe 1 and I2 = F e 2 + pEe 2 .

(a) Given prices pF = 1 and pE = p, write down the budget constraint of consumer 1.

(b) Solve consumer 1’s problem and find the his optimal consumption F ∗ 1 , E ∗ 1 . Express it as a function of p, F e 1 and E e 1 .

(c) Find the consumption F ∗ 2 , E ∗ 2 of consumer 2 as a function of p, F e 2 , E e 2

(d) Find the price p ∗ that equilibrates the market for good F. Express p ∗ as a function of α, F e 1 + F e 2 and E e 1 + E e 2 .

(e) Check that p ∗ equilibrates the market for good E. We now add a third consumer C3 that has initial endowments F e 3 , E e 3 and utility U(F, E) = log F + α log E.

(f) Given prices pF = 1 and pE = p find the individual demand of consumer 3 as a function of p, F e 3 and E e 3 . 3

(g) Given price p what is the individual demand of consumers 1 and 2 ?

(h) Write down the equation that characterizes equilibrium on the market for F.

*Consider two consumers C1 and C2 that have preferences over food (F) and engineered goods (E). The consumers both have preferences U(F, E) = log F + α log E and have initial endowments F e *1 ,* F e *2 , E e* 1 and E e *2 .* If it helps you in your computations define I1 = F e 1 + pEe 1 and I2 = F e 2 + pEe 2 .*

*(a) Given prices pF = 1 and pE = p, write down the budget constraint of consumer 1.*

*(b) Solve consumer 1’s problem and find *the his* optimal consumption F ∗ *1 ,* E ∗ *1 .* Express it as a function of p, F e 1 and E e *1 .

*(c) Find the consumption F ∗ *2 ,* E ∗ 2 of consumer 2 as a function of p, F e *2 , E e* 2*

*(d) Find the price p ∗ that equilibrates the market for good F. Express p ∗ as a function of α, F e 1 + F e 2 and E e 1 + E e *2 .

*(e) Check that p ∗ equilibrates the market for good E. We now add a third consumer C3 that has initial endowments F e *3 , E e* 3 and utility U(F, E) = log F + α log E.*

*(f) Given prices pF = 1 and pE = p find the individual demand of consumer 3 as a function of p, F e 3 and E e *3 .* 3*

*(g) Given price p what is the individual demand of consumers 1 and *2 ?

*(h) Write down the equation that characterizes equilibrium on the market for F.*

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