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9.2. PRICING TO MARKET 289optimality, transform the household’s problem into an unconstraineddynamic choice problem by rewriting the budget constraint (9.110) interms of consumption and substituting this result into the utility function(9.107). Do the same for the foreign agent. The resulting Þrst-orderconditions can be re-arranged to yield, 9δ t P t+1 C t+1 = βP t C t , (9.112)µ St+1δ t Pt+1C ∗t+1∗· ¸M 1t γCt=P t 1 − δ tM ∗ tP ∗t=⎡⎣γC ∗ tS t1 − δ tS t+1S tW t= βP ∗t C ∗ t , (9.113)², (9.114)⎤⎦1², (9.115)h t (z) = 1 , (9.116)ρ P t C tW ∗th ∗ t (z) =1 ρ Pt ∗ Ct∗ . (9.117)Domestic household demand for domestic z−goods and for foreign z ∗ -goods are" # −θpt (z)c t (z) = C t . (9.118)P t⇐(183)9 Differentiating the utility function with respect to B t gives∂U t= −δ t β+ =0,∂B t P t C t P t+1 C t+1which is re-arranged as (9.112). Differentiating the utility function with respect toM t gives∂U t∂M t=−1P t C t+βP t+1 C t+1+ γ P tµMtP t −²=0.Re-arranging this equation and using (9.112) to substitute out P t+1 C t+1 = βP t C t /δ tresults in (9.114). The Þrst-order condition for hours is∂U t∂h t=W tP t C t− ρh t =0,from which (9.117) follows directly. Derivations of the Euler-equations for the foreigncountry follow analogously.