# Finding an exchange rate and competitive equilibrium given an initial allocation and utility function [closed]

I am attempting to solve the following question.

"Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham (H) and cheese (C).

Smith is a choosy eater and will eat ham and cheese only in the fixed proportions of 2 slices of cheese to 1 slice of ham. His utility function is given by $U_S = H^{1/2}C^{1/2}$.

Jones is more flexible in his dietary tastes and has a utility function given by $U_J = H^{1/3}C^{2/3}$.

Total endowments are 100 slices of ham and 200 slices of cheese.

Suppose Smith initially had 40H and 80C. What would the equilibrium position be?"

This is where I get completely lost. I'm told that given these initial allocations, the exchange rate is $$P_H/P_C = 4/3$$ and then the competitive equilibrium is allocation is $H_S = 50, \space C_S = 200/3,\space H_J = 50, \space C_J = 400/3$.

I solved for the contract curve and obtained $$C_S = 200H_S/{(200-H_S)}$$ I'm honestly completely lost as to how they obtain that exchange rate and the competitive equilibrium. Can someone please explain to me how to find it?

Thanks in advance for the help!

Equilibrium is achieved when demand equals supply in every market. We solve these types of problems by finding the set of prices which results in market clearing in every market. By Walras' law we know that if demand equals supply in the ham market it the cheese market must also clear, so we can find the set of prices which results in demand equaling supply in the ham market (working with the cheese market would also be valid).

The first step is to find each agent's gross demand for ham. We can find this by solving the utility maximization problem but since these utility functions are Cobb-Douglas we know the general form is

$$\frac{\alpha}{(\alpha+\beta)}\frac{m}{p_h}$$

where $m=40p_h+80p_c$ for Smith and $m=60p_h+120p_c$ for Jones since income is derived from the value of the endowment.

Gross demand gives how much each agent demands given a set of prices. Supply is equal to our total endowment which gives the market clearing condition $$\frac{40p_h+80p_c}{2p_h}+\frac{60p_h+120p_c}{3p_h}=100.$$

We can always normalise one of the prices to one so let $p_c=1$. Solving the resulting expression gives $p_h=4/3$. To find the allocation just substitute these prices into the gross demands for each agent.

• How did you know that you are able to normalize and where did you get that "general form" from? – King Tut Apr 26 '16 at 13:29
• Since we are interested in relative prices you can always set one of the prices equal to one and all other prices are then measured relative to this price. It may help to note that if we multiply all the prices by a positive constant then this set of prices also results in equilibrium. The general form comes from choosing $x$ and $y$ to maximize the Cobb-Douglas utility function $U(x,y)=x^\alpha y^\beta$ subject to the budget constraint $p_x x+p_y y=m$. – hk39 Apr 26 '16 at 14:07
• sorry still unclear about the general form. How do you move from the budget constraint and utility function to $\frac{\alpha}{(\alpha+\beta)}\frac{m}{p_h}$ ? – King Tut Apr 26 '16 at 16:55
• Assuming you are familiar with the Lagrange method, to solve the maximization problem set up the Lagrangian. $L(x,y,\lambda)=x^\alpha y^\beta -\lambda(p_x x+p_y y - m)$. Partially differentiate with respect to $x$, $y$ and $\lambda$ respectively these expressions equal to zero for the first order conditions. You should then be able to solve this system of simultaneous equations for $x$ and $y$ to get the general form for the demand functions. – hk39 Apr 26 '16 at 18:06