Exercises: Equilibrium in an Exchange Economy

Practice Problems

Practice Exercises

Exercise 1

Adam and Bob are the only two individuals in an island. Adam has been able to collect 10 coconuts and 5 fish, while Bob has 5 coconuts and 10 fish. Both individuals have the same utility function over coconut and fish consumption bundles represented by the following Constant Elasticity of Substitution (CES) utility function: \[ u(X_c, X_f) = \left( X_c^{\frac{\sigma - 1}{\sigma}} + X_f^{\frac{\sigma - 1}{\sigma}} \right)^{\frac{\sigma}{\sigma - 1}}, \qquad \sigma=2, \] where \(X_c\) is the number of coconuts and \(X_f\) the number of fish.

  1. Compute the Marshallian demand functions for coconuts and fish for both Adam and Bob as a function of prices.
  2. Compute the equilibrium prices of coconuts and fish in this economy.
  3. Compute the equilibrium consumption bundles for both Adam and Bob.
  4. For an arbitrary level \(\bar{u}\) of utility, compute the Hicksian demand functions for both Adam and Bob as a function of prices and utility.
  5. Compute the expenditure function as a function of prices and utility.
  6. Show that Marshallian and Hicksian demands coincide when utility is set to the Marshallian utility level.

Exercise 2

Only one hospital in a country provides medical services. Cities have preferences over medical services \(X\) and cash \(C\): \[ U(X,C) = X^{\frac{1}{2}} C^{\frac{1}{2}}. \] The government has allocated \(1\) million services to city 1 and \(2\) million to city 2. Cities have no cash. A policy lets cities sell medical services to outside buyers and receive 80% of the market price per service. Normalize the price of cash to \(1\) and let the (producer) price of services be \(p_X\). Compute each city’s optimal \((X,C)\) after the policy.

Exercise 3

In the market for tobacco (\(X_t\)) and alcohol (\(X_a\)), Carol and Dave have utilities \[ U_C(X_t, X_a) = X_t^{\frac{1}{3}} X_a^{\frac{2}{3}},\qquad U_D(X_t, X_a) = X_t^{\frac{2}{3}} X_a^{\frac{1}{3}}. \] Endowments: Carol \((4,1)\), Dave \((1,4)\).

  1. Compute Marshallian demands.
  2. Find equilibrium prices.
  3. Compute equilibrium consumption bundles.
  4. Suppose a 20% consumption tax on tobacco (buyers pay \((1+\tau)p_t\) with \(\tau=0.2\); sellers receive \(p_t\)).
      1. Characterize an equilibrium with the tax and lump-sum transfers \((T_C,T_D)\) such that markets clear.
      1. Give one revenue-neutral example with \(T_C+T_D\) equal to tax revenue and compute the resulting prices and allocations.
      1. If the government rebated revenue equally, what are the transfers?

Exercise 4

Apples (\(X_A\)) and pears (\(X_P\)) are traded by Eve and Frank. Block pricing applies to purchases: first \(\bar{A}\) apples at price \(\bar p_a\), additional apples at \(\bar p_a/2\); analogously, first \(\bar P\) pears at price \(\bar p_p\), additional pears at \(\bar p_p/2\). Eve’s endowment \((\omega_{EA},\omega_{EP})\), Frank’s \((\omega_{FA},\omega_{FP})\). Both have Cobb–Douglas utility \[ U(X_A,X_P)=X_A^\alpha X_P^{1-\alpha},\qquad \alpha\in(0,1). \]

  1. Write each consumer’s piecewise-linear budget and the corresponding Marshallian demand case by case.
  2. State market-clearing conditions and characterize equilibrium prices.
  3. Characterize equilibrium allocations.
  4. Comparative statics: how do equilibrium prices/allocations change as \(\bar A\) and \(\bar P\) increase?

Exercise 5

Two-good \((1,2)\), two-consumer \((A,B)\) exchange economy with initial allocations \[ \omega_A=(4,2),\qquad \omega_B=(2,4), \] total endowments \((6,6)\). Both have \(U(X_1,X_2)=X_1^\alpha X_2^{1-\alpha}\).

  1. Draw an Edgeworth box (\(6\times 6\)) and locate the initial endowment.
  2. Draw budget lines through the endowment for each consumer.
  3. Draw indifference curves for \(A\) and discuss their shape.
  4. What is the slope of a budget line?
  5. Compute equilibrium relative prices and show the equilibrium allocation.

Solutions

Notes on terminology for making Marshallian vs. Hicksian demands coincide

  • Homothetic preferences: if we scale a consumption bundle by any positive factor, the ranking of bundles does not change along that ray (indifference curves are “scaled copies”). Why it matters: income expansion paths are straight lines through the origin, and Marshallian demands are proportional to income. For homothetic preferences, Hicksian and Marshallian coincide at the Marshallian utility level.

Solution to Exercise 1

Step 0) Simplify the utility with an increasing transformation

With \(\sigma=2\), \[ u(X_c,X_f)=(X_c^{1/2}+X_f^{1/2})^2. \] Apply the strictly increasing transformation \(g(u)=\sqrt{u}\): \[ \tilde u(X_c,X_f)=\sqrt{u(X_c,X_f)}=X_c^{1/2}+X_f^{1/2}. \] This preserves preferences (same optima) and makes derivatives simple. It is linear in the transformed consumptions \(Z_c=\sqrt{X_c}\) and \(Z_f=\sqrt{X_f}\), which makes the MRS and the FOCs very easy.

1) Marshallian demands (for any consumer with wealth \(W(p)\) and prices \((p_c,p_f)\))

With \(\tilde u=\sqrt{X_c}+\sqrt{X_f}\), the FOC at an interior optimum sets MRS equal to the price ratio: \[ \text{MRS}_{c,f} =\frac{\partial \tilde u/\partial X_c}{\partial \tilde u/\partial X_f} =\frac{\tfrac{1}{2}X_c^{-1/2}}{\tfrac{1}{2}X_f^{-1/2}} =\left(\frac{X_f}{X_c}\right)^{1/2} =\frac{p_c}{p_f}. \] Hence \(X_f=X_c\left(\frac{p_c}{p_f}\right)^2\). With the budget \(p_c X_c+p_f X_f=W(p)\), solve: \[ X_c^M=\frac{W(p)\,p_f}{p_c(p_c+p_f)}, \qquad X_f^M=\frac{W(p)\,p_c}{p_f(p_c+p_f)}. \] Wealth from endowments: \[ W_A(p)=10p_c+5p_f,\qquad W_B(p)=5p_c+10p_f. \] Plug these into the formulas above for Adam and Bob.

2) Equilibrium prices

Normalize \(p_c=1\) and let \(p_f=p\). Aggregate endowments: \((15,15)\). Market-clearing for coconuts: \[ \begin{aligned} X_{c,A}^M+X_{c,B}^M=\frac{(10+5p)p}{1+p}+\frac{(5+10p)p}{1+p} =\frac{(15+15p)p}{1+p}&=15 \\ \Rightarrow\quad (15+15p)p&=15(1+p) \\ \Rightarrow\quad 15p^2&=15 \\ \Rightarrow\quad p^2&=1 \\ \Rightarrow\quad p&=1. \end{aligned} \] Thus \(p_c=p_f=1\). Why don’t we normalize with the simplex \(p_c+p_f=1\)? Look at the expression and choose the normalization that makes the algebra easiest.

3) Equilibrium allocations

With \(p_c=p_f=1\), \(W_A(1)=W_B(1)=15\) and \[ X_{c,i}^M=X_{f,i}^M=\frac{W_i(p)}{2}=\frac{15}{2}=7.5,\quad i\in\{A,B\}. \]

4) Hicksian (compensated) demands

Cost-minimization \(\min_{h_c,h_f}\{p_c h_c+p_f h_f \text{. s.t.}\ (h_c^{1/2}+h_f^{1/2})^2\ge \bar u\}\) is equivalent to \(h_c^{1/2}+h_f^{1/2}\ge \bar u^{1/2}\). The solution yields

\[ \begin{aligned} \mathcal{L} &=p_c h_c+p_f h_f+\mu\big(\bar u^{1/2}-(h_c^{1/2}+h_f^{1/2})\big) \\ \frac{\partial \mathcal{L}}{\partial h_c} &=p_c - \mu \frac{1}{2} h_c^{-1/2}=0 \quad\Rightarrow\quad \mu=2 p_c h_c^{1/2}, \\ \frac{\partial \mathcal{L}}{\partial h_f} &=p_f - \mu \frac{1}{2} h_f^{-1/2}=0 \quad\Rightarrow\quad \mu=2 p_f h_f^{1/2} \\ \Rightarrow\quad 2 p_c h_c^{1/2} &= 2 p_f h_f^{1/2} \quad\Rightarrow\quad h_f = h_c \left(\frac{p_c}{p_f}\right)^2. \end{aligned} \] Replace into the utility constraint (Local Non-Satiation ensures equality): \[ \begin{aligned} h_c^{1/2}+h_f^{1/2} &= \bar u^{1/2} \\ h_c^{1/2}+\left(\frac{p_c}{p_f}\right) h_c^{1/2} &= \bar u^{1/2} \\ h_c^{1/2}\left(1+\frac{p_c}{p_f}\right) &= \bar u^{1/2} \\ h_c^{1/2} &= \frac{\bar u^{1/2}}{1+\frac{p_c}{p_f}} \\ h_c &= \frac{\bar u}{\left(1+\frac{p_c}{p_f}\right)^2}. \end{aligned} \]

and similarly for \(h_f\): \[ h_f = \frac{\bar u}{\left(1+\frac{p_f}{p_c}\right)^2}. \]

5) Expenditure function

The expenditure function is \[ \begin{aligned} e(p_c,p_f,\bar u) &= p_c h_c + p_f h_f \\ &= \frac{p_c \bar u}{\left(1+\frac{p_c}{p_f}\right)^2} + \frac{p_f \bar u}{\left(1+\frac{p_f}{p_c}\right)^2} \\ &= \bar u \cdot \left( \frac{p_c}{\left(1+\frac{p_c}{p_f}\right)^2} + \frac{p_f}{\left(1+\frac{p_f}{p_c}\right)^2} \right). \end{aligned} \]

or simplified: \[ \begin{aligned} e(p_c,p_f,\bar u) &= \bar u \cdot \left( \frac{p_c}{\left(1+\frac{p_c}{p_f}\right)^2} + \frac{p_f}{\left(1+\frac{p_f}{p_c}\right)^2} \right) \\ &= \bar u \cdot \left( \frac{p_c p_f^2 + p_f p_c^2}{(p_c+p_f)^2} \right). \end{aligned} \]

6) Why Marshallian and Hicksian coincide at the Marshallian utility

In the optimal bundle, budget is spent entirely: \(W(p)=e(p_c,p_f,\bar u)\). Set \(\bar u\) to the Marshallian utility level \(\bar u=u(X_c,X_f)\):

\[ \begin{aligned} X_c&=\frac{W(p)\,p_f}{p_c(p_c+p_f)} \\ &=\frac{e(p_c,p_f,\bar u)\,p_f}{p_c(p_c+p_f)} \\ &=\frac{\bar u \cdot \left( \frac{p_c p_f^2 + p_f p_c^2}{(p_c+p_f)^2} \right) p_f}{p_c(p_c+p_f)} \\ &= \frac{\bar u (p_c p_f^3 + p_c^2 p_f^2)}{p_c (p_c+p_f)^3} \\ &= \frac{\bar u \; p_c \; p_f^2 \,(p_f + p_c)}{p_c (p_c+p_f)^3} \\ &=\frac{\bar u \cdot p_f^2}{(p_c+p_f)^2} \\ \end{aligned} \]

The utility is given by \(\bar u = (X_c^{1/2} + X_f^{1/2})^2\). And only in terms of \(X_c\) we have:

\[ \bar u = (X_c^{1/2} + (X_c\left(\frac{p_c}{p_f}\right)^2)^{1/2})^2 \]

which can be simplified as

\[ \begin{aligned} \bar u &= (X_c^{1/2} + (X_c\left(\frac{p_c}{p_f}\right)^2)^{1/2})^2 \\ &= (X_c^{1/2} + X_c^{1/2}\left(\frac{p_c}{p_f}\right))^2 \\ &= (X_c^{1/2}(1+\frac{p_c}{p_f}))^2 \\ &= X_c \cdot \left(1+\frac{p_c}{p_f}\right)^2. \end{aligned} \]

Plugin back \[ \begin{aligned} X_c &= \frac{\bar u \cdot p_f^2}{(p_c+p_f)^2} \\ &= \frac{X_c \cdot \left(1+\frac{p_c}{p_f}\right)^2 \cdot p_f^2}{(p_c+p_f)^2} \\ &= X_c \times \frac{\frac{p_f^2 (p_f+p_c)^2}{p_f^2}}{(p_c+p_f)^2} \\ &= X_c. \end{aligned} \]

and same for \(X_f\).

Solution to Exercise 2

A city with initial services \(X_0\) sells \(s\in[0,X_0]\) units at net price \(0.8\,p_X\) (it keeps 80% of the revenue): \[ X=X_0-s,\qquad C=0.8\,p_X\,s. \] Maximize \(U=\sqrt{XC}=\sqrt{(X_0-s)(0.8\,p_X\,s)}\).

FOC: \[ \begin{aligned} \frac{\partial U}{\partial s} &=-\frac{1}{2}\sqrt{\frac{0.8\,p_X}{X_0-s}}+\frac{1}{2}\sqrt{\frac{0.8\,p_X (X_0-s)}{s^2}}=0 \\ \Rightarrow\quad -\frac{1}{\sqrt{X_0-s}}+\frac{\sqrt{X_0-s}}{s}&=0 \\ \Rightarrow\quad s&=X_0-s \\ \Rightarrow\quad s&=\frac{X_0}{2}. \end{aligned} \] Thus each city sells half its initial services, based on their initial endowments. Final consumptions and cash:

  • City 1: \(X_1=\frac{1}{2}\), \(C_1=0.8\,p_X\cdot \frac{1}{2}=0.4\,p_X\).
  • City 2: \(X_2=1\), \(C_2=0.8\,p_X\cdot 1=0.8\,p_X\).

Solution to Exercise 3

Let prices be \((p_t,p_a)\) and normalize the numeraire \(p_a=1\). Denote the tobacco price by \(p_t=p\). Endowments are Carol \((4,1)\) and Dave \((1,4)\).

We use the slides’ wealth notation: for consumer \(i\in\{C,D\}\), \[ W_i(p)=p\cdot \omega_{it}+1\cdot \omega_{ia}. \] Thus, without taxes, \[ W_C(p)=4p+1,\qquad W_D(p)=p+4. \]

Notation: we call the no-transfer case the competitive equilibrium (prices only). With transfers present, we simply refer to an equilibrium with transfers.

1) Marshallian demands (step by step)

Utilities are Cobb–Douglas: \[ U_C(X_t,X_a)=X_t^{\tfrac13}X_a^{\tfrac23}\quad(\alpha_C=\tfrac13),\qquad U_D(X_t,X_a)=X_t^{\tfrac23}X_a^{\tfrac13}\quad(\alpha_D=\tfrac23). \] For Cobb–Douglas \(U=X_t^{\alpha}X_a^{1-\alpha}\) with prices \((p,1)\) and wealth \(W\), solve \[ \max_{X_t,X_a\ge 0} X_t^{\alpha}X_a^{1-\alpha}\quad \text{s.t.}\quad pX_t+1\cdot X_a=W. \] Take logs to linearize and form the Lagrangian \[ \mathcal L=\alpha\ln X_t+(1-\alpha)\ln X_a+\lambda\big(W-pX_t-X_a\big). \] FOCs: \[ \frac{\partial \mathcal L}{\partial X_t}=\frac{\alpha}{X_t}-\lambda p=0\;\Rightarrow\; X_t=\frac{\alpha}{\lambda p},\qquad \frac{\partial \mathcal L}{\partial X_a}=\frac{1-\alpha}{X_a}-\lambda=0\;\Rightarrow\; X_a=\frac{1-\alpha}{\lambda}. \] Budget \(pX_t+X_a=W\) gives, after substitution, \[ p\Big(\tfrac{\alpha}{\lambda p}\Big)+\tfrac{1-\alpha}{\lambda}=\frac{\alpha}{\lambda}+\frac{1-\alpha}{\lambda}=\frac{1}{\lambda}=W\;\Rightarrow\;\lambda=\frac{1}{W}. \] Hence Marshallian demands are \[ X_t^{M}=\alpha\,\frac{W}{p},\qquad X_a^{M}=(1-\alpha)\,W. \] Applying to each consumer using \(W_C(p)\) and \(W_D(p)\): \[ \begin{aligned} \text{Carol }(\alpha_C=\tfrac13):&\quad X_{t,C}(p)=\frac{1}{3}\,\frac{W_C(p)}{p}=\frac{1}{3}\,\frac{4p+1}{p},\quad X_{a,C}(p)=\frac{2}{3}\,W_C(p)=\frac{2}{3}(4p+1);\\ \text{Dave }(\alpha_D=\tfrac23):&\quad X_{t,D}(p)=\frac{2}{3}\,\frac{W_D(p)}{p}=\frac{2}{3}\,\frac{p+4}{p},\quad X_{a,D}(p)=\frac{1}{3}\,W_D(p)=\frac{1}{3}(p+4). \end{aligned} \]

2) Equilibrium prices (no tax) — step by step

Aggregate endowments are \((\Omega_t,\Omega_a)=(5,5)\). Market clearing in tobacco requires \[ X_{t,C}(p)+X_{t,D}(p)=\Omega_t=5. \] Substitute the demands and clear denominators carefully: \[ \begin{aligned} \frac{1}{3}\frac{4p+1}{p}+\frac{2}{3}\frac{p+4}{p}&=5 \\ \frac{4p+1+2(p+4)}{3p}&=5 \\ \frac{6p+9}{3p}&=5 \\ 6p+9&=15p \\ 9&=9p \\ p&=1. \end{aligned} \] Thus \((p_t,p_a)=(1,1)\). As a consistency check, alcohol also clears at \(p=1\): \[ X_{a,C}(1)+X_{a,D}(1)=\tfrac{2}{3}(5)+\tfrac{1}{3}(5)=\tfrac{10}{3}+\tfrac{5}{3}=5. \]

3) Equilibrium allocations (no tax)

At \(p=1\), wealths are \(W_C(1)=4\cdot 1+1=5\) and \(W_D(1)=1+4=5\). Then \[ \text{Carol: }\big(X_{t,C},X_{a,C}\big)=\Big(\tfrac{1}{3}\tfrac{5}{1},\;\tfrac{2}{3}\cdot 5\Big)=\Big(\tfrac{5}{3},\;\tfrac{10}{3}\Big),\quad \text{Dave: }\big(X_{t,D},X_{a,D}\big)=\Big(\tfrac{2}{3}\tfrac{5}{1},\;\tfrac{1}{3}\cdot 5\Big)=\Big(\tfrac{10}{3},\;\tfrac{5}{3}\Big). \] They sum to \((5,5)\), as required.

4) 20% consumption tax on tobacco — full characterization and examples

Let \(\tau=0.2\). Buyers face the consumption price \(p^{c}=(1+\tau)p=1.2\,p\), while sellers receive the producer price \(p\). Wealth from endowments is still \[ W_C(p)=4p+1,\qquad W_D(p)=p+4. \] Let lump-sum transfers be \((T_C,T_D)\). The budget for agent \(i\) is \[ p^{c}X_{t,i}+1\cdot X_{a,i}=W_i(p)+T_i. \] Repeating the Cobb–Douglas derivation with prices \((p^{c},1)\) and wealth \(W_i(p)+T_i\) gives \[ X_{t,i}=\alpha_i\,\frac{W_i(p)+T_i}{p^{c}},\qquad X_{a,i}=(1-\alpha_i)\,[W_i(p)+T_i]. \] Therefore \[ \begin{aligned} X_{t,C}&=\frac{1}{3}\,\frac{W_C(p)+T_C}{1.2\,p},&\quad X_{a,C}&=\frac{2}{3}\,[W_C(p)+T_C],\\ X_{t,D}&=\frac{2}{3}\,\frac{W_D(p)+T_D}{1.2\,p},&\quad X_{a,D}&=\frac{1}{3}\,[W_D(p)+T_D]. \end{aligned} \]

  • Market clearing: \[ X_{t,C}+X_{t,D}=5,\qquad X_{a,C}+X_{a,D}=5. \]

Transfers and equilibrium characterization (independent of funding): With lump-sum transfers \((T_C,T_D)\), the demand equations above and the market-clearing conditions determine the price \(p\) and the allocation for any given \((T_C,T_D)\). By the Second Welfare Theorem (with convex preferences and local non-satiation), different transfer vectors implement different competitive equilibria; the source of the transfers is irrelevant for this characterization.

(a) Algebraic equilibrium conditions (with transfers)

Tobacco clearing, multiplying both sides by \(1.2\,p\): \[ \frac{W_C+T_C}{3}+\frac{2(W_D+T_D)}{3}=5\cdot 1.2\,p=6p. \] Alcohol clearing: \[ \frac{2(W_C+T_C)}{3}+\frac{W_D+T_D}{3}=5. \] You can view these in two equivalent ways: - Given \((T_C,T_D)\), they determine the equilibrium price \(p\) and allocation. - Or, given a target price \(p>0\), they determine the transfers that implement an equilibrium at that \(p\).

Implementing transfers as functions of \(p\). Write the two clearing equations in levels and substitute \(W_C=4p+1\), \(W_D=p+4\): \[ \begin{aligned} &\text{tobacco:}&& (W_C+T_C)+2(W_D+T_D)=18p,\\ &\text{alcohol:}&& 2(W_C+T_C)+(W_D+T_D)=15. \end{aligned} \] That is, \[ \begin{aligned} T_C+2T_D&=18p-(4p+1)-2(p+4)=12p-9,\\ 2T_C+T_D&=15-2(4p+1)-(p+4)=9-9p. \end{aligned} \] Solving the linear system gives the transfers that implement \(p\): \[ T_C(p)=9-10p,\qquad T_D(p)=11p-9. \] Note these satisfy \(T_C(p)+T_D(p)=p\) as a consequence of market clearing; later, if you impose revenue neutrality, this identity coincides with the government budget condition.

(b) Revenue-neutral implementation and one unequal-rebate example

Revenue neutrality: the per-unit tax is \((1+\tau)p-p=\tau p\), market clearing gives taxed quantity \(\Omega_t=5\), hence revenue \[R=\tau p\cdot 5=0.2\,p\cdot 5=p,\] so the government budget is \[T_C+T_D=R=p.\]

One unequal-rebate example: \((T_C,T_D)=(0,\,p)\)

Use \(T_C=0\), \(T_D=p\), and \(W_C=4p+1\), \(W_D=p+4\). From tobacco clearing: \[ \begin{aligned} \frac{W_C}{3}+\frac{2(W_D+p)}{3}&=6p \\ W_C+2(W_D+p)&=18p \\ (4p+1)+2[(p+4)+p]&=18p \\ 4p+1+2(2p+4)&=18p \\ 8p+9&=18p \\ p&=\frac{9}{10}. \end{aligned} \] Then the allocations (write as exact fractions): \[ \begin{aligned} &p=\tfrac{9}{10},\quad p^{c}=\tfrac{27}{25},\quad W_C=\tfrac{23}{5},\quad W_D=\tfrac{49}{10},\quad T_C=0,\ T_D=\tfrac{9}{10}.\\[4pt] &X_{t,C}=\frac{1}{3}\,\frac{\tfrac{23}{5}}{\tfrac{27}{25}}=\frac{23}{15}\cdot\frac{25}{27}=\frac{115}{81},\quad X_{a,C}=\frac{2}{3}\cdot \tfrac{23}{5}=\tfrac{46}{15};\\ &X_{t,D}=\frac{2}{3}\,\frac{\tfrac{49}{10}+\tfrac{9}{10}}{\tfrac{27}{25}}=\frac{2}{3}\,\frac{\tfrac{29}{5}}{\tfrac{27}{25}}=\frac{58}{15}\cdot\frac{25}{27}=\frac{290}{81},\quad X_{a,D}=\frac{1}{3}\cdot \tfrac{29}{5}=\tfrac{29}{15}. \end{aligned} \] Checks: \(\tfrac{115}{81}+\tfrac{290}{81}=\tfrac{405}{81}=5\) and \(\tfrac{46}{15}+\tfrac{29}{15}=\tfrac{75}{15}=5\).

(c) Equal rebate transfers: \(T_C=T_D=\tfrac{p}{2}\)

Solve for \(p\) first. Tobacco clearing gives \[ \frac{W_C+\tfrac{p}{2}}{3}+\frac{2(W_D+\tfrac{p}{2})}{3}=6p \;\Rightarrow\; W_C+2W_D+\tfrac{3p}{2}=18p. \] Substitute \(W_C=4p+1\), \(W_D=p+4\) and solve step by step: \[ 4p+1+2(p+4)+\tfrac{3p}{2}=18p \;\Rightarrow\; 7.5p+9=18p \;\Rightarrow\; 9=10.5p \;\Rightarrow\; p=\frac{6}{7}. \] Now compute allocations (keep exact fractions): with \(p=\tfrac{6}{7}\) we have \(p^{c}=\tfrac{36}{35}\), \(W_C=\tfrac{31}{7}\), \(W_D=\tfrac{34}{7}\), and \(T_C=T_D=\tfrac{3}{7}\). Then \[ \begin{aligned} X_{t,C}&=\frac{1}{3}\,\frac{\tfrac{31}{7}+\tfrac{3}{7}}{\tfrac{36}{35}}=\frac{1}{3}\,\frac{\tfrac{34}{7}}{\tfrac{36}{35}}=\frac{34}{21}\cdot\frac{35}{36}=\frac{85}{54},&\quad X_{a,C}&=\frac{2}{3}\cdot \tfrac{34}{7}=\tfrac{68}{21},\\ X_{t,D}&=\frac{2}{3}\,\frac{\tfrac{34}{7}+\tfrac{3}{7}}{\tfrac{36}{35}}=\frac{2}{3}\,\frac{\tfrac{37}{7}}{\tfrac{36}{35}}=\frac{74}{21}\cdot\frac{35}{36}=\frac{185}{54},&\quad X_{a,D}&=\frac{1}{3}\cdot \tfrac{37}{7}=\tfrac{37}{21}. \end{aligned} \] Checks: \(\tfrac{85}{54}+\tfrac{185}{54}=\tfrac{270}{54}=5\) and \(\tfrac{68}{21}+\tfrac{37}{21}=\tfrac{105}{21}=5\). —

Solution to Exercise 4

Cost of purchasing and wealth from selling (block prices)

For any \(x\ge 0\), \[ E_A(x)= \begin{cases} \bar p_a\,x, & 0\le x\le \bar A,\\[4pt] \bar p_a\,\bar A+\dfrac{\bar p_a}{2}\,(x-\bar A), & x>\bar A, \end{cases} \qquad E_P(x)= \begin{cases} \bar p_p\,x, & 0\le x\le \bar P,\\[4pt] \bar p_p\,\bar P+\dfrac{\bar p_p}{2}\,(x-\bar P), & x>\bar P. \end{cases} \]

Wealth of \(i\in\{E,F\}\) when endowments are valued at the same schedule (notation consistent with \(W\)): \[ W_i \;=\; E_A(\omega_{iA})+E_P(\omega_{iP}),\qquad \Omega_A=\omega_{EA}+\omega_{FA},\ \ \Omega_P=\omega_{EP}+\omega_{FP}. \]

Individual Marshallian demand (Cobb–Douglas, derived from spending)

Cobb–Douglas \(U=X_A^\alpha X_P^{1-\alpha}\) implies the agent targets spendings \[ s_{A,i}=\alpha W_i,\qquad s_{P,i}=(1-\alpha)W_i. \] Convert spending into quantities by inverting \(E_A,E_P\).

  • Apples.
    If \(s_{A,i}\le \bar p_a\bar A\), all spending is at unit price \(\bar p_a\): \[ X_{A,i}=\frac{s_{A,i}}{\bar p_a}. \] If \(s_{A,i}>\bar p_a\bar A\), first buy \(\bar A\) units with \(\bar p_a\bar A\), then the remaining spending is \(s_{A,i}-\bar p_a\bar A\) and each extra unit costs \(\bar p_a/2\), so each currency unit buys \(2/\bar p_a\) units: \[ X_{A,i}=\bar A+\frac{s_{A,i}-\bar p_a\bar A}{\bar p_a/2} \;=\;\bar A+\frac{2}{\bar p_a}\big(s_{A,i}-\bar p_a\bar A\big). \]

  • Pears (analogous).
    If \(s_{P,i}\le \bar p_p\bar P\), \[ X_{P,i}=\frac{s_{P,i}}{\bar p_p}. \] If \(s_{P,i}>\bar p_p\bar P\), \[ X_{P,i}=\bar P+\frac{2}{\bar p_p}\big(s_{P,i}-\bar p_p\bar P\big). \]

Substitute \(s_{A,i}=\alpha W_i\) and \(s_{P,i}=(1-\alpha)W_i\) in each branch.

Market clearing

\[ X_{A,E}+X_{A,F}=\Omega_A,\qquad X_{P,E}+X_{P,F}=\Omega_P. \]

Equilibrium by aggregate regime (prices up to a numeraire)

Below, we present closed forms for the relative price and the implied allocations when the aggregate economy falls in one of four quantity regimes. In each case, compute \(W_i\) from endowments, plug into the branch formulas above, and the allocations will sum to \((\Omega_A,\Omega_P)\).

Case 1: Neither good uses the second block

\[ \Omega_A\le \bar A,\qquad \Omega_P\le \bar P. \] All agents are in the first-block branch for both goods: \[ X_{A,i}=\frac{\alpha W_i}{\bar p_a},\qquad X_{P,i}=\frac{(1-\alpha) W_i}{\bar p_p}. \] Clearing implies \[ \frac{\alpha(W_E+W_F)}{\bar p_a}=\Omega_A,\qquad \frac{(1-\alpha)(W_E+W_F)}{\bar p_p}=\Omega_P \ \Rightarrow\ \boxed{\ \frac{\bar p_a}{\bar p_p}=\frac{\alpha}{1-\alpha}\cdot\frac{\Omega_P}{\Omega_A}\ }. \]

Case 2: Only apples use the second block

\[ \Omega_A>\bar A,\qquad \Omega_P\le \bar P. \] Apples beyond-first-block, pears first-block. Using the spending-to-quantity conversion: \[ X_{A,i}=\bar A+\frac{2}{\bar p_a}\big(\alpha W_i-\bar p_a\bar A\big) =\frac{2\alpha}{\bar p_a}W_i-\bar A,\qquad X_{P,i}=\frac{(1-\alpha) W_i}{\bar p_p}. \] Summing and clearing: \[ \frac{2\alpha}{\bar p_a}(W_E+W_F)-2\bar A=\Omega_A,\qquad \frac{1-\alpha}{\bar p_p}(W_E+W_F)=\Omega_P, \] hence \[ \boxed{\ \frac{\bar p_a}{\bar p_p} =\frac{2\alpha}{1-\alpha}\cdot \frac{\Omega_P}{\Omega_A+2\bar A}\ }. \]

Case 3: Only pears use the second block

\[ \Omega_A\le \bar A,\qquad \Omega_P>\bar P. \] Apples first-block, pears beyond-first-block: \[ X_{A,i}=\frac{\alpha W_i}{\bar p_a},\qquad X_{P,i}=\bar P+\frac{2}{\bar p_p}\big((1-\alpha) W_i-\bar p_p\bar P\big) =\frac{2(1-\alpha)}{\bar p_p}W_i-\bar P. \] Summing and clearing: \[ \frac{\alpha}{\bar p_a}(W_E+W_F)=\Omega_A,\qquad \frac{2(1-\alpha)}{\bar p_p}(W_E+W_F)-2\bar P=\Omega_P, \] so \[ \boxed{\ \frac{\bar p_a}{\bar p_p} =\frac{\alpha}{2(1-\alpha)}\cdot \frac{\Omega_P+2\bar P}{\Omega_A}\ }. \]

Case 4: Both goods use the second block (deep interior)

\[ \Omega_A>\bar A,\qquad \Omega_P>\bar P. \] Both beyond-first-block: \[ X_{A,i}=\frac{2\alpha}{\bar p_a}W_i-\bar A,\qquad X_{P,i}=\frac{2(1-\alpha)}{\bar p_p}W_i-\bar P. \] Summing and clearing: \[ \frac{2\alpha}{\bar p_a}(W_E+W_F)-2\bar A=\Omega_A,\qquad \frac{2(1-\alpha)}{\bar p_p}(W_E+W_F)-2\bar P=\Omega_P, \] giving \[ \boxed{\ \frac{\bar p_a}{\bar p_p} =\frac{\alpha}{1-\alpha}\cdot \frac{\Omega_P+2\bar P}{\Omega_A+2\bar A}\ }. \]

Equilibrium allocations (all regimes)

  1. Compute \(W_i=E_A(\omega_{iA})+E_P(\omega_{iP})\) using the block-price cost functions above.
  2. Use the appropriate branch for each good:

  • First-block branch: \[ X_{A,i}=\frac{\alpha W_i}{\bar p_a},\qquad X_{P,i}=\frac{(1-\alpha) W_i}{\bar p_p}. \]

  • Beyond-first-block branch: \[ X_{A,i}=\bar A+\frac{2}{\bar p_a}\big(\alpha W_i-\bar p_a\bar A\big),\qquad X_{P,i}=\bar P+\frac{2}{\bar p_p}\big((1-\alpha) W_i-\bar p_p\bar P\big). \]

These allocations satisfy \(X_{A,E}+X_{A,F}=\Omega_A\) and \(X_{P,E}+X_{P,F}=\Omega_P\) by construction.

Comparative statics in \(\bar A,\bar P\)

When a good uses the second block in equilibrium, the marginal price relevant for most purchased units is \(\bar p/2\), and the relative price adjusts so that, holding endowments fixed, \[ \frac{\partial}{\partial \bar A}\left(\frac{\bar p_a}{\bar p_p}\right)>0 \quad\text{if apples use the second block},\qquad \frac{\partial}{\partial \bar P}\left(\frac{\bar p_a}{\bar p_p}\right)<0 \quad\text{if pears use the second block}. \]

Solution to Exercise 5

1–4) Box, budgets, indifference curves, slope

  • Box \(6\times 6\), initial endowment at \((4,2)\) for \(A\) (hence \((2,4)\) for \(B\)).
  • Budget lines through the endowment with slope \(-\frac{p_1}{p_2}\).
  • Indifference curves for \(A\) from \(X_1^\alpha X_2^{1-\alpha}=\bar u\): \[ X_2=\bar u^{\frac{1}{1-\alpha}}\,X_1^{-\frac{\alpha}{1-\alpha}}, \] which are convex to the origin (diminishing MRS).

5) Equilibrium price ratio and allocation

Let \(p_2=1\), \(p_1=p\). Wealth: \(W_A(p)=4p+2\), \(W_B(p)=2p+4\). Demand for good 1: \(X_{1,A}=\alpha W_A(p)/p\), \(X_{1,B}=\alpha W_B(p)/p\). Clear good 1 (total supply \(6\)): \[ \alpha\Big(\frac{4p+2}{p}+\frac{2p+4}{p}\Big)=6 \ \Rightarrow\ \alpha\Big(6+\frac{6}{p}\Big)=6 \ \Rightarrow\ p=\frac{\alpha}{1-\alpha}. \] Then \[ W_A(p)=\frac{2}{1-\alpha},\qquad W_B(p)=\frac{4}{1-\alpha}, \] and allocations \[ X_{1,A}=\frac{\alpha W_A(p)}{p}=2(1+\alpha),\quad X_{2,A}=(1-\alpha) W_A(p)=2(1+\alpha);\\ X_{1,B}=\frac{\alpha W_B(p)}{p}=2(2-\alpha),\quad X_{2,B}=(1-\alpha) W_B(p)=2(2-\alpha). \] These satisfy market clearing \((2(1+\alpha)+2(2-\alpha)=6)\) for each good. In the symmetric case \(\alpha=\tfrac12\) we have \(p=1\), \(W_A=W_B=6\), and both consumers choose \((3,3)\).

Budget lines pass through the endowment. For A, with \(p_2=1\), the budget is \[ p\,X_{1,A}+X_{2,A}=W_A(p)=p\cdot 4+1\cdot 2, \] so when \(p=1\) it is \(X_{1,A}+X_{2,A}=6\) (slope \(-1\) through \((4,2)\)), as shown in the diagram.

Edgeworth box diagram (illustrative)

Assumption for this diagram: \(\alpha=\tfrac{1}{2}\) (so \(p_1/p_2=1\)). Then \(W_A=W_B=6\) and A’s budget is \(X_1+X_2=6\) through the endowment \((4,2)\).

A: X₁ (0 → 6) A: X₂ (0 → 6) B: X₁ (6 → 0) B: X₂ (6 → 0) 0 6 0 6 Contract curve Endowment A:(4,2) = B:(2,4) Assumes α = 1/2 (p₁/p₂ = 1) Budget (through endowment) ICₐ (tangent) ICₐ (through endowment)